additive one dimensional cellular automata are chaotic according to devaney s definition of chaos abstract we study the chaotic behavior of a particular class of dynamical systems cellular automata we specialize the definition of chaos given by devaney for general dynamical systems to the case of cellular automata a dynamical system x f is chaotic according to devaney s definition of chaos if its transition map f is sensitive to the initial conditions topologically transitive and has dense periodic orbits on x our main result is the proof that all the additive one dimensional cellular automata defined on a finite alphabet of prime cardinality are chaotic in the sense of devaney

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

the attractor basin portrait of a cellular automaton local space time structures such as domains and the intervening dislocations dominate a wide class of cellular automaton ca behavior for such spatially extended dynamics regular domains vicinities and attractors are introduced as organizing principles to identify the discretized analogs of attractors basins and separatrices structures used in classifying dissipative continuous state dynamical systems we describe the attractor basin portrait of nonlinear elementary ca rule 18 whose global dynamics is largely determined by a single regular attracting domain the latter s basin is analyzed in terms of subbasin and portal structures associated with particle annihilation the conclusion is that the computational complexity of such ca is more apparent than real transducer machines are constructed that automatically identify domain and dislocation structures in space time count the number of dislocations in a spatial pattern and implement an isomorphism between rule 18 and rule 90 we use a transducer to trace dislocation trajectories and confirm that in rule 18 isolated dislocation trajectories as well as a dislocation gas agree extremely well with the classical model of annihilating diffusive particles the ca efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space time

the kink of cellular automaton rule 18 performs a random walk we give an exact characterization of the movement of a single kink in the elementary cellular automaton rule 18 it is a random walk with independent increments as well as independent delay times its statistical parameters are computed to confirm the earlier simulation results by grassberger

mouvement brownien cones et processus stables letw wt t 0 denote a two dimensional brownian motion starting at 0 and for 0 4 here we construct a measure a local time supported onh for 4 2 the brownian motionw time changed by the inverse of this local time is shown to be a two dimensional stable process with index 2 2 this results extends spitzer s construction of the cauchy process which is recovered by taking 2 a formula which describes the behaviour ofw before a timet h is established and applied to the proof of a conjecture of burdzy we also obtain a two dimensional version of the famous theorem of levy concerning the maximum process of linear brownian motion precisely for 0 2 letst denote the vertex of the smallest wedge of the typez c which contains the path ofw up to timet the processst wt is shown to be a reflected brownian motion in the wedgec with oblique reflection on the sides finally we investigate various extensions of the previous results to brownian motion inrd d 3 letc be the cone associated with an open subset of the spheresd 1 and leth be defined ash above sufficient conditions are given forh to contain nonzero times in terms of the first eigenvalue of the dirichlet laplacian on

energy fields of partially coherent sources random acoustic fields and their energetic quantities acoustic active and reactive intensities potential and kinetic energy densities are described in terms of the mutual coherences between sources conditions to correctly construct the coherence matrix of sources in a multivariate random process are given it is shown that the description of a sound field using the coherence matrix of source is equivalent to the superposition of a number of independent coherent fields which do not correspond to the original localized sources a method based on processing the principal components of the coherence matrix of sources is given to reduce the number of necessary fields the coherence function between acoustic pressure and particle velocity and the curl of active intensity are proposed as two indicators for estimating the degree of coherence and the polarization of acoustic fields these indicators are analyzed theoretically and experimentally the description of the structure of partially coherent fields is generalized by the definition of the field matrix whose rank is an indicator of the local complexity of an acoustic field

phase and amplitude gradient method for the estimation of acoustic vector quantities an alternative pressure sensor based method for estimating the acoustic intensity the phase and amplitude gradient estimation page method is presented this method uses the same hardware as the standard finite difference method but does not suffer from the frequency dependent bias inherent to the finite difference method a detailed derivation of the page method and the finite difference method is presented both methods are then compared using simple acoustic fields the ability to unwrap the phase component of the page method is discussed which leads to accurate intensity estimates above previous frequency limits the uncertainties associated with both methods of estimation are presented it is shown that the page method provides more accurate intensity estimates over a larger frequency bandwidth

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

on the convex hull of planar brownian snake the planar brownian snake is a continuous strong markov process taking values in the space of continuous functions in r 2 that are stopped at some time for a fixed time the snake is distributed like a planar brownian motion with a random lifetime this paper characterizes the convex hull of the trace of the snake paths that exit the half plane at the origin it is shown that the convex hull at 0 is roughly a factor of x smoother than the convex hull of a piece of planar brownian motion at its minimum y value

comparison of multimicrophone probe design and processing methods in measuring acoustic intensity three multimicrophone probe arrangements used to measure acoustic intensity are the four microphone regular tetrahedral the four microphone orthogonal and the six microphone designs finite sum and finite difference processing methods can be used with such probes to estimate pressure and particle velocity respectively a numerical analysis is performed to investigate the bias inherent in each combination of probe design and processing method probes consisting of matched point sensor microphones both embedded and not embedded on the surface of a rigid sphere are considered results are given for plane wave fields in terms of root mean square average bias and maximum bias as a function of angle of incidence an experimental verification of the analysis model is described of the combinations considered and under the stated conditions the orthogonal probe using the origin microphone for the pressure estimate is shown to have the lowest amount of intensity magnitude bias lowest intensity direction bias comes from the six microphone probe using an average of the 15 intensity components calculated using all microphone pairs also discussed are how multimicrophone probes can advantageously use correction factors calculated from a numerical analysis and how the results of such an analysis depend on the chosen definition of the dimensionless frequency

moments of inertia and the shapes of brownian paths we compute the joint probability law of the principal moments of inertia of brownian paths open or closed using constrained path integrals and random matrix theory the case of two dimensional paths is discussed in detail in particular we show that the ratio of the average values of the largest and smallest moments is equal to 4 99 open paths and 3 07 closed paths we also present results of numerical simulations which include investigation of the relationships between the moments of inertia and the arithmetic area enclosed by a path

aperiodicity in one dimensional cellular automata abstract cellular automata are a class of mathematical systems characterized by discreteness in space time and state values determinism and local interaction a certain class of one dimensional binary site valued nearest neighbor automata is shown to generate infinitely many aperiodic temporal sequences from arbitrary finite initial conditions on an infinite lattice the class of automaton rules that generate aperiodic temporal sequences are characterized by a particular form of injectivity in their interaction rules included are the nontrivial linear automaton rules that is rules for which the superposition principle holds certain nonlinear automata with injectivity properties similar to those of linear automata and a wider subset of nonlinear automata whose interaction rules satisfy a weaker form of injectivity together with certain symmetry conditions the evolution of this last subset of automata can be viewed as consisting of multiple domains within which behavior exactly mimics that of a linear automaton a linearization technique is used to establish that these domains coalesce in finite time to produce at most two domains and thereby to assert the aperiodicity of their temporal sequences

predicting nonlinear cellular automata quickly by decomposing them into linear ones we show that a wide variety of nonlinear cellular automata cas can be decomposed into a quasidirect product of linear ones these cas can be predicted by parallel circuits of depth o log2 t using gates with binary inputs or o log t depth if sum mod p gates with an unbounded number of inputs are allowed thus these cas can be predicted by idealized parallel computers much faster than by explicit simulation even though they are nonlinear this class includes any ca whose rule when written as an algebra is a solvable group we also show that cas based on nilpotent groups can be predicted in depth o log t or o 1 by circuits with binary or sum mod p gates respectively we use these techniques to give an efficient algorithm for a ca rule which like elementary ca rule 18 has diffusing defects that annihilate in pairs this can be used to predict the motion of defects in rule 18 in o log2 t parallel time

global analysis of self similarity features of cellular automata selected examples abstract the analysis of self similarity properties of the evolution patterns of linear cellular automata is investigated using the theory of hierarchical iterated function systems and substitution systems we present several selected examples

points cones du mouvement brownien plan le cas critique letb bt t 0 be a planar brownian motion and let 0 for anyt 0 the pointz bt is called a one sided cone point with angle if there exist e 0 and a wedgew z with vertexz and angle such thatbs w z for everys t t e burdzy and shimura have shown independently that one sided cone points with angle exist when 2 but not when 2 the present paper deals with the critical case 2 we show that cone points with angle 2 do not exist

the convex hull for a random acceleration process in two dimensions we compute exactly the mean perimeter l t and the mean area a t of the convex hull of a random acceleration process of duration t in two dimensions we use an exact mapping that relates via cauchy s formulae the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process x t y t to the computation of the extreme value statistics of the associated one dimensional component process x t the latter can be computed exactly for the one dimensional random acceleration process even though the process is non markovian physically our results are relevant in describing the average shape of a semi flexible ideal polymer chain in two dimensions

le spectre de certaines mosaiques poissoniennes du plan et l enveloppe convexe du pont brownien the spectral empirical function of a homogeneous isotropic poisson mosaic process is a functional of the perimeter of the convex hull of planar brownian bridge some geometrical identities follow

mean perimeter of the convex hull of a random walk in a semi infinite medium we study various properties of the convex hull of a planar brownian motion defined as the minimum convex polygon enclosing the trajectory in the presence of an infinite reflecting wall recently in a rapid communication phys rev e textbf 91 050104 r 2015 we announced that the mean perimeter of the convex hull at time t rescaled by sqrt dt is a non monotonous function of the initial distance to the wall in the present article we first give all the details of the derivation of this mean rescaled perimeter in particular its value when starting from the wall and near the wall we then determine the physical mechanism underlying this surprising non monotonicity of the mean rescaled perimeter by analyzing the impact of the wall on two complementary parts of the convex hull finally we provide a further quantification of the convex hull by determining the mean length of the portion of the reflecting wall visited by the brownian motion as a function of the initial distance to the wall

percolation and disorder resistance in cellular automata we rigorously prove a form of disorder resistance for a class of one dimensional cellular automaton rules including some that arise as boundary dynamics of two dimensional solidification rules specifically when started from a random initial seed on an interval of length l with probability tending to one as l the evolution is a replicator that is a region of space time of density one is filled with a spatially and temporally periodic pattern punctuated by a finite set of other finite patterns repeated at a fractal set of locations on the other hand the same rules exhibit provably more complex evolution from some seeds while from other seeds their behavior is apparently chaotic a principal tool is a new variant of percolation theory in the context of additive cellular automata from random initial states

universality and time scale invariance for the shape of planar levy processes for a broad class of planar markov processes viz l evy processes satisfying certain conditions valid e g in the case of brownian motion and l evy flights we establish an exact universal formula describing the shape of the convex hull of sample paths we show indeed that the average number of edges joining paths points separated by a time lapse ensuremath delta ensuremath tau ensuremath in left ensuremath delta ensuremath tau 1 ensuremath delta ensuremath tau 2 right is equal to 2 phantom rule 0 16em 0ex text ln left ensuremath delta ensuremath tau 2 ensuremath delta ensuremath tau 1 right regardless of the specific distribution of the process s increments and regardless of its total duration t the formula also exhibits invariance when the time scale is multiplied by any constant apart from its theoretical importance our result provides insights regarding the shape of two dimensional objects e g polymer chains modeled by the sample paths of stochastic processes generally more complex than brownian motion in particular for a total time or parameter duration t the average number of edges on the convex hull cut off to discard edges joining points separated by a time lapse shorter than some ensuremath delta ensuremath tau lt will be given by 2 phantom rule 0 16em 0ex text ln frac t ensuremath delta ensuremath tau thus it will only grow logarithmically rather than at some higher pace

the one dimensional exactly 1 cellular automaton replication periodicity and chaos from finite seeds in the exactly 1 cellular automaton also known as rule 22 every site of the one dimensional lattice is either in state 0 or in state 1 and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three site neighborhood at the current time we analyze this rule started from finite seeds i e those initial configurations that have only finitely many 1 s three qualitatively different types of evolution are observed replication periodicity and chaos we focus on rigorous results assisted by algorithmic searches for the first two behaviors in particular we explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds some empirical observations about chaotic seeds are also presented

study of the shape of random walks ii inertia moment ratios and the two dimensional asphericity a study of some features of the quantities used to describe the shape of random walks focusing on the probability distribution of the asphericity of discrete random walks in two dimensions is presented a connection is established between the asphericity distribution and the probability distribution corresponding to the ratio of the principal inertia moments the probability distributions of arbitrary inertia moment ratios are analysed for varying spatial dimension and an analytic expression for them is found which presents excellent agreement with monte carlo data in all the cases considered this function is then used to obtain the corresponding two dimensional asphericity distribution which also proved to be a very good approximation to the true distribution obtained from an independent monte carlo simulation

local asymptotic laws for the brownian convex hull we prove a general theorem for the precise rate at which the convex hull of brownian motion gets created the latter result relates large deviation theory to p levy s geometric proof of strassen s law of the iterated logarithm this also answers a question of s evans moreover we give a partial solution to a question of j hammersley and p levy regarding the slowness of the growth of the hull process several examples some classical and some new are given to illustrate the theorems finally we present applications to the convex hull of random walks ind dimensions

nonintersection exponents for brownian paths ii estimates and applications to a random fractal let x and y be independent two dimensional brownian motions x 0 0 0 y 0 e 0 and let p e p x 0 1 y 0 1 o q e y 0 1 does not contain a closed loop around 0 asymptotic estimates when e 0 of p e q e and some related probabilities are given let f be the boundary of the unbounded connected component of r 2 z 0 1 where z t x t tx 1 for t 0 1 then f is a closed jordan arc and the hausdorff dimension of f is less or equal to 3 2 1 4 2

percolation dimension of fractals abstract percolation dimension is introduced in this note it characterizes certain fractals and its definition is based on the hausdorff dimension it is shown that percolation dimension and boundary dimension are in a sense independent from the hausdorff dimension and therefore privide an additional tool for classification of fractals

on brownian slow points it is shown that for a wiener process x t both the quantities mathop inf limits t mathop overline lim limits h text to text 0 x t text text h x t sqrt h text and mathop text sup limits t mathop underline lim limits h text to text 0 text x t text text h x t sqrt h are almost surely equal to 1

invertible behavior in elementary cellular automata with memory elementary cellular automata ecas have been studied for their ability to generate complex global behavior despite their simplicity one variation of ecas is obtained by adding memory to each cell in a neighborhood this process generates a provisional configuration in which the application of an evolution rule establishes the dynamics of the system this version is known as an eca with memory ecam most previous work on ecams analyzed the complex behavior taking chaotic ecas however the present paper investigates reversible ecams as obtained from reversible and permutative ecas these ecas have at least one ancestor for every configuration thus the correct permutation of states may specify the memory function to obtain reversible ecams for permutative ecas which are often irreversible we demonstrate that the use of a quiescent state and the correct manipulation of de bruijn blocks produce reversible ecams

brownian excursions and minimal thinness i on donne des criteres de type integral et un critere de type kolmogorov pour les excursions browniennes comme corollaire

computing fractal dimensions for additive cellular automata abstract efficient methods are given for finding the statistics of symbol blocks occuring in the graphs of additive cellular automata after large numbers of iterations in particular the methods yield fractal dimensions entropies and a recurrence relation for the number of nonzero symbols the methods are based on the construction of a nonnegative integer matrix a closely related to the automation the largest eigenvalue of a tells the fractal dimension while the corresponding eigenvector gives the relative frequencies of blocks

no triple point of planar brownian motion is accessible burdzy s research supported in part by nsf grant dms 9322689 and ams centennial research fellowship werner s research supported by an eec hcm leibniz fellowship

robust periodic solutions and evolution from seeds in one dimensional edge cellular automata we study one dimensional cellular automata ca with values 0 and 1 we assume that such ca are started from semi infinite configurations those that have 0 s to the left of some site and we focus on the identification of robust periodic solutions rps which when observed from the left edge of the light cone advance into any environment with positive velocity we then utilize rps and related concepts to analyze ca dynamics from seeds i e initial configurations with finitely many 1 s

a systematic method to obtain 3d finite difference formulations for acoustic intensity and other energy quantities abstract a systematic method is described to obtain formulations based on the finite difference approximation for computation of the energy quantities of three dimensional 3d sound fields from measurements it uses up to the first order terms of the taylor series expansion of the sound pressure about the microphone positions of a particular probe configuration the use of a symbolic computation allows approximate expressions to be obtained for all energy quantities as a function of the cross spectral densities of the microphone signals the application of this approach is illustrated by the three components of the active acoustic intensity vector and the energy densities for different types of 3d probes consisting of 4 6 microphones the inherent effect associated with the finite sum and finite difference approximations is evaluated for each of these probes by considering 1d sound fields in order to make a comparison of their performance

convex hull of n planar brownian paths an exact formula for the average number of edges we establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent brownian paths in the plane this requires the introduction of a counting criterion which amounts to cutting off edges that are in a specific sense small the main argument consists in a mapping between planar brownian convex hulls and configurations of constrained independent linear brownian motions this new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of brownian convex hulls in the plane

flat points of two dimensional brownian motion supposez is a two dimensional brownian motion it is shown that a s there existt0 and 0 such thatz t0 is an extremal point of the convex hull of z t t0 t t0 and also an extremal point of the convex hull of z t t0 t t0 and moreover the tangent lines to the convex hulls atz t0 form a non zero angle

replication in one dimensional cellular automata abstract in a cellular automaton ca replication is the ability to indefinitely generate copies of a finite collection of patterns starting from finite seeds a transparent feature of additive ca replication mechanisms are less clear in the absence of additivity this paper investigates such dynamics through several examples for the 1 or 2 rule and its generalizations replication is inevitable and we investigate self organization properties in the perturbed exactly 1 rule we study frequency of replicators and the new phenomenon is called quasireplication the last ca is the extended 1 or 3 rule which allows for replication on different backgrounds we employ a mixture of rigorous and empirical techniques

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

frequency domain expressions for the estimation of time averaged acoustic energy density this paper builds on earlier work by the same authors to derive expressions for the time averaged acoustic energy density in the frequency domain using the auto and cross spectral densities of multiple microphone elements expressions for the most common three dimensional geometric arrangements are derived simplified expressions for use with two channel spectrum analysers are also presented

acoustic isogeometric boundary element analysis an isogeometric boundary element method based on t splines is used to simulate acoustic phenomena we restrict our developments to low frequency problems to establish the fundamental properties of the proposed approach using t splines the computer aided design cad and boundary element analysis are integrated without recourse to geometry clean up or mesh generation a regularized burton miller formulation is used resulting in integrals which are at most weakly singular we employ a collocation based approach to generate the linear system of equations the method is verified against closed form solutions and direct comparisons are made with conventional lagrangian discretizations it is demonstrated that the superior accuracy of the isogeometric approach emanates from the exact geometric description encapsulated in the t spline the method is then applied to a real world application to illustrate the potential for integrated engineering design and analysis

experimental validation of acoustic intensity bandwidth extension by phase unwrapping the phase and amplitude gradient estimator page method for active acoustic intensity uses pairwise microphone transfer functions to obtain the phase gradient which improves the calculation bandwidth over the traditional weighted quadspectral method additionally for broadband sources the page theory indicates that the transfer function phase can be unwrapped to further extend the usable frequency range to beyond the spatial nyquist frequency here two experiments demonstrate intensity bandwidth extension by more than an order of magnitude using phase unwrapping first plane wave tube results show accurate one dimensional intensity calculations with the microphones separated by five wavelengths 30 times the traditional limit second two dimensional measurements of a laboratory scale jet with a four microphone probe yield physically reasonable calculations at frequencies 15 times the traditional limit

geometric properties of 2 dimensional brownian paths let a be the set of all points of the plane visited by 2 dimensional brownian motion before time 1 with probability 1 all points of a are twist points except a set of harmonic measure zero twist points may be continuously approached in a only along a special spiral although negligible in the sense of harmonic measure various classes of cone points are dense in a with probability 1 cone points may be approached in a within suitable wedges

brownian excursions from hyperplanes and smooth surfaces on obtient une decomposition en produit gauche de la loi d excursion brownienne a n dimensions n 2 a partir d un hyperplan