on a new class of planar fractals the pascal sierpinski gaskets a new class of planar fractals called the pascal sierpinski gaskets is described of which the well known sierpinski gasket is a special case some of these gaskets are true mandelbrot fractals possessing non integral dimensions as well as self similarity the remaining ones are not self similar but appear to have nonintegral dimensions

fractal structures derivable from the generalisations of the pascal triangle generalisations of order k or 2 of the pascal triangle are used to construct generalised pascal sierpinski gaskets of orders k l or 2 it is shown that all such gaskets are self affine fractals but when k 2 and l is prime then the gasket is rigorously self similar and possesses a similarity dimension the evolutionary morphology of the gaskets of orders k l prime bears a resemblance to the growth of pyrolitic graphite films and other material structures

sample complexity for learning recurrent perceptron mappings recurrent perceptron classifiers generalize the usual perceptron model they correspond to linear transformations of input vectors obtained by means of autoregressive moving average schemes or infinite impulse response filters and take into account those correlations and dependences among input coordinates which arise from linear digital filtering this paper provides tight bounds on the sample complexity associated to the fitting of such models to experimental data the results are expressed in the context of the theory of probably approximately correct pac learning

further results on controllability of recurrent neural networks this paper studies controllability properties of recurrent neural networks the new contributions are 1 an extension of a previous result to a slightly different model 2 a formulation and proof of a necessary and sufficient condition and 3 an analysis of a low dimensional case for which the hypotheses made in previous work do not apply

fractal sequences derived from the self similar extensions of the sierpinski gasket the well known sierpinski gasket has been generalised by the authors into the generalised pascal sierpinski gaskets gpsg of orders k l where both k and l are or 2 it has been shown here that families of self similar sequences can be derived from these extensions of the sierpinski gasket when k 2 and l is a prime number

relatively free semigroups of intermediate growth abstract the problem of calculating the growth of a finitely generated f g semigroup satisfying the given system of identities is considered examples of relatively free semigroups having intermediate growth new growth criteria and constructions are given the main ideas employed in the proofs are based on some new points of view on growth our method is also useful for constructing various examples of f g nilpotent and nil semigroups having intermediate growth and arbitrarily small gelfand kirillov superdimension

universal power law tails for singularity dominated strong fluctuations integrals of the type i t identical to integral integral dxdy delta t h x y are considered where h is a function from an ensemble labelled by parameters i can represent circulation times of fluid particles in the plane orbital periods or semiclassical densities of states for one dimensional hamiltonian systems spectral densities for two dimensional crystals or the strength of a wave pulse produced by propagation of a deformed step discontinuity as h varies over the ensemble i develops strong fluctuations associated with legendre singularities for unrestricted functions h the probability distribution p i describing the fluctuations of i decays according to a universal i 9 law obtained by a scaling argument involving arnold s classification of catastrophes if h is only quadratic in y p varies as i 10 if the integral defining i is one dimensional p varies as i 3 if it is three dimensional i decays no faster than i 48 the probability distribution of i identical to mod delta i delta t mod decays as i 2

the smallest mealy automaton of intermediate growth abstract in this paper we study the automaton i 2 the smallest mealy automaton of intermediate growth first considered by the last two authors i i reznykov v i sushchansky the two state mealy automata over the two symbol alphabet of the intermediate growth mat zametki 72 2002 102 117 we describe the automatic transformation monoid defined by i 2 give a formula for the generating series for the ball volume growth function of i 2 and give sharp asymptotics for the growth function of i 2 namely i 2 n 2 5 2 3 3 4 2 n 1 4 exp n 6 with the ratios of left to right hand side tending to 1 as n

remarks on the singularity indices of arnol d and berry we give a coordinate free algebraic interpretation for singularity indices of quasihomogeneous functions and their unfoldings as first exploited systematically by berry in statistical optics we also make some observations on geometric interpretation on formulae for indices and on the difficulties of rigorous treatment for non quasihomogeneous functions

on the 3 state mealy automata over an m symbol alphabet of growth order nlogn 2logm abstract we consider sequence j m m 2 of 3 state mealy automata over an m symbol alphabet such that the growth of j m is intermediate of order n log n 2 log m for each automaton j m we describe the transformation monoid s j m defined by it provide generating series for the growth functions and consider some properties of s j m and j m

singularities and the distribution of density in the burgers adhesion model abstract we are interested in the tail behavior of the pdf of mass density within the one and d dimensional burgers adhesion model used e g to model the formation of large scale structures in the universe after baryon photon decoupling we show that large densities are localized near kurtoparabolic singularities residing on space time manifolds of codimension two d 2 or higher d 3 for smooth initial conditions such singularities are obtained from the convex hull of the lagrangian potential the initial velocity potential minus a parabolic term the singularities contribute universal power law tails to the density pdf when the initial conditions are random in one dimension the singularities are preshocks nascent shocks whereas in two and three dimensions they persist in time and correspond to boundaries of shocks in all cases the corresponding density pdf has the exponent 7 2 originally proposed by e et al phys rev lett 78 1997 1904 for the pdf of velocity gradients in one dimensional forced burgers turbulence we also briefly consider models permitting particle crossings and thus multi stream solutions such as the zel dovich approximation and the jeans vlasov poisson equation with single stream initial data they have singularities of codimension one yielding power law tails with exponent 3

lur e systems with multilayer perceptron and recurrent neural networks absolute stability and dissipativity sufficient conditions for absolute stability and dissipativity of continuous time recurrent neural networks with two hidden layers are presented in the autonomous case this is related to a lur e system with multilayer perceptron nonlinearity such models are obtained after parametrizing general nonlinear models and controllers by a multilayer perceptron with one hidden layer and representing the control scheme in standard plant form the conditions are expressed as matrix inequalities and can be employed for nonlinear h sub spl infin control and imposing closed loop stability in dynamic backpropagation

the ring of k regular sequences ii in this paper we continue our study of k regular sequences begun in 1992 we prove some new results give many new examples from the literature and state some open problems

on the spatial fourier transforms of the pascal sierpinski gaskets it has been shown here that the fourier transform which can be obtained experimentally of an evolving fractal can be manipulated to yield the fractal dimension although only the pascal sierpinski gaskets have been considered here because of their generality it is expected that this technique can be utilized to identify other planar fractals as well

complete controllability of continuous time recurrent neural networks this paper studies controllability for the class of control systems commonly called continuous time recurrent neural networks it is shown that under a generic condition on the input matrix the system is controllable for every possible state matrix the result holds when the activation function is the hyperbolic tangent

a learning result for continuous time recurrent neural networks abstract the following learning problem is considered for continuous time recurrent neural networks having sigmoidal activation functions given a black box representing an unknown system measurements of output derivatives are collected for a set of randomly generated inputs and a network is used to approximate the observed behavior it is shown that the number of inputs needed for reliable generalization the sample complexity of the learning problem is upper bounded by an expression that grows polynomially with the dimension of the network and logarithmically with the number of output derivatives being matched

self similarity in diffraction by a self similar fractal screen using the sierpinski gasket as an example it has been shown here that the paraxial fraunhoffer zone diffracted field of a self similar fractal screen also exhibits self similarity this also establishes that fractal structures can be used with great profit in problems involving array syntheses recently it has been shown that the sierpinski gasket is itself a member of a much wider class of gaskets and the potential for the use of fractal structures in electromagnetic em problems may be vast indeed

generalized sierpinski fractal multiband antenna a new set of fractal multiband antennas called mod p sierpinski gaskets is presented mod p sierpinski fractal antennas derive from the pascal triangle and present a log periodic behavior which is a consequence of their self similarity properties mod p sierpinski fractal antennas constitute a generalization of the classical sierpinski antenna