fourier integral operators fractal sets and the regular value theorem we prove that if e r2d for d 2 is an ahlfors david regular product set of sufficiently large hausdorff dimension denoted by dimh e and is a sufficiently regular function then the upper minkowski dimension of the set w e l w tl 1 l m does not exceed dimh e m in line with the regular value theorem from the elementary differential geometry our arguments are based on the mapping properties of the underlying fourier integral operators and are intimately connected with the falconer distance conjecture in geometric measure theory we shall see that our results are in general sharp in the sense that if the hausdorff dimension is smaller than a certain threshold then the dimensional inequality fails in a quantifiable way the constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry

modelling cellular automata with partial differential equations a system of 10 coupled nonlinear partial differential equations is exhibited which simulates an arbitrary two dimensional nine neighbor square lattice cellular automata a number of theoretical implications of the result and the techniques used in its construction as well as possible practical consequences are discussed

analysis of self organized criticality in ehrenfest s dog flea model the self organized criticality in ehrenfest s historical dog flea model is analyzed by simulating the underlying stochastic process the fluctuations around the thermal equilibrium in the model are treated as avalanches we show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q gaussian the distribution which is obtained naturally in the context of nonextensive statistical mechanics if one avoids the finite size effects by increasing the system size we provide clear numerical evidence that the relation between the exponent ensuremath tau of avalanche size distribution obtained by maximum likelihood estimation and the q value of appropriate q gaussian obeys the analytical result recently introduced by caruso et al phys rev e 75 055101 r 2007 this allows us to determine the value of q parameter a priori from one of the well known exponents of such dynamical systems

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

universal computation and physical dynamics abstract a dynamical system is said to be computationally universal if it can be programmed through its initial condition to perform and digital computation such systems include traditional abstract computers such as turing machines and cellular automata as well as more physical models such as hard sphere gases and even a single particle in an appropriately shaped box because of the ability of any two such systems to simulate one another they are all equivalent in the range of computations they can perform and the global dynamics of any one of them provides a microcosm of all cause effect relations that can be expressed by deductive logic or numerical simulation this allows universal computers to be used to define in an objective manner that sort of complexity which increases when a self organizing system organizes itself

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

power spectrum of mass and activity fluctuations in a sandpile we consider a directed abelian sandpile on a strip of size 2 ifmmode times else texttimes fi n driven by adding a grain randomly at the left boundary after every t timesteps we establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary base representation of the position of a random walker on a ring of size 3 n we find that while the fluctuations of mass have a power spectrum that varies as 1 f for frequencies in the range 3 ensuremath 2n ensuremath ll f ensuremath ll 1 t the activity fluctuations in the same frequency range have a power spectrum that is linear in f

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

on the distance sets of ahlfors david regular sets abstract i prove that if k r 2 is a compact s ahlfors david regular set with s 1 then dim p d k 1 where d k x y x y k is the distance set of k and dim p stands for packing dimension the same proof strategy applies to other problems of similar nature for instance one can show that if k r 2 is a compact s ahlfors david regular set with s 1 then there exists a point x 0 k such that dim p k k x 0 1

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

cellular automata as a paradigm for ecological modeling we review cellular automata as a modeling formalism and discuss how it can be used for modeling spatial ecological processes the implications of this modeling paradigm for ecological observation are stressed finally we discuss some shortcomings of the cellular automaton formalism and mention some extensions and generalizations which may remedy these shortcomings

pinned distance sets k simplices wolff s exponent in finite fields and sum product estimates an analog of the falconer distance problem in vector spaces over finite fields asks for the threshold 0 such that delta e gtrsim q whenever e gtrsim q alpha where e subset mathbb f q d the d dimensional vector space over a finite field with q elements not necessarily prime here delta e x 1 y 1 2 dots x d y d 2 x y in e iosevich and rudnev trans am math soc 359 12 6127 6142 2007 established the threshold frac d 1 2 and in hart et al trans am math soc 363 3255 3275 2011 proved that this exponent is sharp in odd dimensions in two dimensions we improve the exponent to tfrac 4 3 consistent with the corresponding exponent in euclidean space obtained by wolff int math res not 10 547 567 1999 the pinned distance set delta y e x 1 y 1 2 dots x d y d 2 x in e for a pin y in e has been studied in the euclidean setting peres and schlag duke math j 102 193 251 2000 showed that if the hausdorff dimension of a set e is greater than tfrac d 1 2 then the lebesgue measure of y e is positive for almost every pin y in this paper we obtain the analogous result in the finite field setting in addition the same result is shown to be true for the pinned dot product set pi y e x cdot y x in e under the additional assumption that the set e has cartesian product structure we improve the pinned threshold for both distances and dot products to frac d 2 2d 1 the pinned dot product result for cartesian products implies the following sum product result let a subset mathbb f q and z in mathbb f q if a geq q frac d 2d 1 then there exists a subset e subset a times dots times a a d 1 with e gtrsim a d 1 such that for any a 1 dots a d 1 in e a 1a a 2a dots a d 1 a za frac q 2 where a j a a ja a in a j 1 dots d 1 a generalization of the falconer distance problem is to determine the minimal 0 such that e contains a congruent copy of a positive proportion of k simplices whenever e gtrsim q alpha here the authors improve on known results for k 3 using fourier analytic methods showing that may be taken to be frac d k 2

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

self replicating pulses and sierpinski gaskets in excitable media in our previous papers we have shown by computer simulations that a sierpinski gasket pattern appears in a bonhoeffer van der pol type reaction diffusion system in this paper we show another class of regular self similar structure which is found in four different excitable reaction diffusion systems this result strongly implies that the existence of the self similar spatiotemporal evolution is universal in excitable reaction diffusion media

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

random boolean network model exhibiting deterministic chaos this paper considers a simple boolean network with n nodes each node s state at time t being determined by a certain number of parent nodes which may vary from one node to another this is an extension of a model studied by andrecut and ali int j mod phys b 15 17 2001 who consider the same number of parents for all nodes we make use of the same boolean rule as andrecut and ali provide a generalization of the formula for the probability of finding a node in state 1 at a time t and use simulation methods to generate consecutive states of the network for both the real system and the model the results match well we study the dynamics of the model through sensitivity of the orbits to initial values bifurcation diagrams and fixed point analysis we show that the route to chaos is due to a cascade of period doubling bifurcations which turn into reversed period halving bifurcations for certain combinations of parameter values

1 f α spectra in elementary cellular automata and fractal signals we systematically compute the power spectra of the one dimensional elementary cellular automata introduced by wolfram on the one hand our analysis reveals that one automaton displays 1 f spectra though considered as trivial and on the other hand that various automata classified as chaotic or complex display no 1 f spectra we model the results generalizing the recently investigated sierpinski signal to a class of fractal signals that are tailored to produce 1 f ensuremath alpha spectra from the widespread occurrence of elementary cellular automata patterns in chemistry physics and computer sciences there are various candidates to show spectra similar to our results

from cellular automaton to difference equation a general transformation method which preserves time evolution patterns we propose a general method to construct a partial difference equation which preserves any time evolution patterns of a cellular automaton the method is based on inverse ultradiscretization with filter functions

sierpinski signal generates 1 f alpha spectra we investigate the row sum of the binary pattern generated by the sierpinski automaton interpreted as a time series we calculate the power spectrum of this sierpinski signal analytically and obtain a unique rugged fine structure with underlying power law decay with an exponent of approximately 1 15 despite the simplicity of the model it can serve as a model for 1 f ensuremath alpha spectra in a certain class of experimental and natural systems such as catalytic reactions and mollusc patterns

continuous time symmetric hopfield nets are computationally universal we establish a fundamental result in the theory of computation by continuous time dynamical systems by showing that systems corresponding to so called continuous time symmetric hopfield nets are capable of general computation as is well known such networks have very constrained lyapunov function controlled dynamics nevertheless we show that they are universal and efficient computational devices in the sense that any convergent synchronous fully parallel computation by a recurrent network of n discrete time binary neurons with in general asymmetric coupling weights can be simulated by a symmetric continuous time hopfield net containing only 18n 7 units employing the saturated linear activation function moreover if the asymmetric network has maximum integer weight size wmax and converges in discrete time t then the corresponding hopfield net can be designed to operate in continuous time t e for any e 0 such that wmax212n e21 e in terms of standard discrete computation models our result implies that any polynomially space bounded turing machine can be simulated by a family of polynomial size continuous time symmetric hopfield nets

smoothness of projections bernoulli convolutions and the dimension of exceptions erd s 1939 1940 studied the distribution of the random series p 0 n and showed that is singular for infinitely many 1 2 1 and absolutely continuous for a e in a small interval 1 1 solomyak 1995 proved a conjecture made by garsia 1962 that is absolutely continuous for a e 1 2 1 in order to sharpen this result we have developed a general method that can be used to estimate the hausdorff dimension of exceptional parameters in several contexts in particular we prove for any 0 1 2 the set of 0 1 such that is singular has hausdorff dimension strictly less than 1 for any borel set a rd with hausdorff dimension dim a d 1 2 there are points x a such that the pinned distance set x y y a has positive lebesgue measure moreover the set of x where this fails has hausdorff dimension at most d 1 dim a let k denote the middle cantor set for 1 2 and let k r be any compact set peres and solomyak 1998 showed that for a e 0 1 2 such that dim k dimk 1 the sum k k has positive length we show that the set of exceptional in this statement has hausdorff dimension at most 2 dim k dimk 0 for any borel set e rd with dim e 2 almost all orthogonal projections of e onto lines through the origin have nonempty interior and the exceptional set of lines where this fails has dimension at most d 1 dim e if is a borel probability measure on rd with correlation dimension greater than m 2 then for a prevalent set of c1 maps f rd rm in the sense described by hunt sauer and yorke 1992 the image of under f has a density with at least fractional derivatives in l2 rm

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

multifractal analyses of row sum signals of elementary cellular automata abstract we first apply the wt mfdfa mfdfa and wtmm multifractal methods to binomial multifractal time series of three different binomial parameters and find that the wtmm method indicates an enhanced difference between the fractal components than the known theoretical result next we make use of the same methods for the time series of the row sum signals of the two complementary eca pairs of rules 90 165 and 150 105 for ten initial conditions going from a single 1 in the central position up to a set of ten 1 s covering the ten central positions in the first row since the members of the pairs are actually similar from the statistical point of view we can check which method is the most stable numerically by recording the differences provided by the methods between the two members of the pairs for various important quantities of the scaling analyses such as the multifractal support the most frequent holder exponent and the hurst exponent and considering as the better one the method that provides the minimum differences according to this criterion our results show that the mfdfa performs better than wt mfdfa and wtmm in the case of the multifractal support while for the other two scaling parameters the wt mfdfa is the best the employed set of initial conditions does not generate any specific trend in the values of the multifractal parameters

time evolution of the rule 150 cellular automaton activity from a fibonacci iteration the rule 150 cellular automaton is a remarkable discrete dynamical system as it shows 1 f spectra if started from a single seed j nagler and j c claussen phys rev e 71 067103 2005 despite its simplicity a feasible solution for its time behavior is not obvious its self similarity does not follow a one step iteration like other elementary cellular automata here it is shown how its time behavior can be solved as a two step vectorial or string iteration which can be viewed as a generalization of fibonacci iteration generating the time series from a sequence of vectors of increasing length this allows us to compute the total activity time series more efficiently than by simulating the whole spatiotemporal process or even by using the closed expression the results are further extended to the generalization of rule 150 to the two dimensional case and to bethe lattices and the relation to corresponding integer sequences is discussed