the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

the dual tree of a recursive triangulation of the disk in the recursive lamination of the disk one tries to add chords one after another at random a chord is kept and inserted if it does not intersect any of the previously inserted ones curien and le gall ann probab vol 39 pp 2224 2270 2011 have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process m based on a new approach resembling ideas from the so called contraction method in function spaces we prove that when properly rescaled the planar dual of the discrete lamination converges almost surely in the gromov hausdorff sense to a limit real tree t which is encoded by m this confirms a conjecture of curien and le gall 1 introduction and main results in 18 curien and le gall introduce the model of random recursive triangulations of the disk the construction goes as follows atn 1 two points are sampled independently with uniform distribution on the circle they are connected by a chord a straight line which splits the disk into two fragments later on at each step two independent points are sampled uniformly at random on the circle and are connected by a chord if the latter does not intersect any of the previously inserted chords in other words the two points are connected by a chord if they both fall in the same fragment this gives rise to a sequence of laminations of the disk for us a lamination will be a collection of chords which may only intersect at their end points at timen the lamination ln consists of the union of the chords inserted up to timen as an increasing closed subset of the disk ln converges and it is proved in 18 that

polygonal and polyhedral contour reconstruction in computed tomography this paper is about three dimensional 3 d reconstruction of a binary image from its x ray tomographic data we study the special case of a compact uniform polyhedron totally included in a uniform background and directly perform the polyhedral surface estimation we formulate this problem as a nonlinear inverse problem using the bayesian framework vertice estimation is done without using a voxel approximation of the 3 d image it is based on the construction and optimization of a regularized criterion that accounts for surface smoothness we investigate original deterministic local algorithms based on the exact computation of the line projections their update and their derivatives with respect to the vertice coordinates results are first derived in the two dimensional 2 d case which consists of reconstructing a 2 d object of deformable polygonal contour from its tomographic data then we investigate the 3 d extension that requires technical adaptations simulation results illustrate the performance of polygonal and polyhedral reconstruction algorithms in terms of quality and computation time

bouncing ball with a finite restitution chattering locking and chaos we study the dynamic evolution of a bouncing ball on a vibrating platform such as the membrane of a loudspeaker as a function of its coefficient of restitution ensuremath alpha and demonstrate primarily that the presence of chaos in this system is by no means inevitable and its development is by no means obvious indeed we show that generic trajectories starting under experimental conditions terminate in a region of chattering where the memory of earlier dynamics is lost before repeating themselves in a periodic way as a consequence except possibly for the strictly elastic case ensuremath alpha 1 the evolution to chaos via a period doubling route will not be observed our arguments are corroborated by numerical studies concerning especially the divergence of the mean period of generic trajectories as the elastic limit is approached ensuremath alpha ensuremath rightarrow 1

learnability with respect to fixed distributions consider a first order typed language with semantics s for expressions and types adding subtyping means that a partial order on types is defined and that the typing rules are extended to the effect that expression e has type t whenever e has type s and s t we show how to adapt the semantics s in a simple set theoretic way obtaining a semantics s that satisfies in addition to some obvious requirements also the property that s s is included in s t whenever s t

the size of random fragmentation trees we consider the random fragmentation process introduced by kolmogorov where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution the dislocation law this is repeated recursively for all pieces more precisely we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold and we study the associated random tree dean and majumdar found a phase transition for this process the number of fragmentations is asymptotically normal for some dislocation laws but not for others depending on the position of roots of a certain characteristic equation this parallels the behavior of discrete analogues with various random trees that have been studied in computer science we give rigorous proofs of this phase transition and add further details the proof uses the contraction method we extend some previous results for recursive sequences of random variables to families of random variables with a continuous parameter we believe that this extension has independent interest

bayesian inversion of geoelectrical resistivity data summary enormous quantities of geoelectrical data are produced daily and often used for large scale reservoir modelling to interpret these data requires reliable and efficient inversion methods which adequately incorporate prior information and use realistically complex modelling structures we use models based on random coloured polygonal graphs as a powerful and flexible modelling framework for the layered composition of the earth and we contrast our approach with earlier methods based on smooth gaussian fields we demonstrate how the reconstruction algorithm may be efficiently implemented through the use of multigrid metropolis coupled markov chain monte carlo methods and illustrate the method on a set of field data

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

edge preserving tomographic reconstruction with nonlocal regularization tomographic image reconstruction using statistical methods can provide more accurate system modeling statistical models and physical constraints than the conventional filtered backprojection fbp method because of the ill posedness of the reconstruction problem a roughness penalty is often imposed on the solution to control noise to avoid smoothing of edges which are important image attributes various edge preserving regularization methods have been proposed most of these schemes rely on information from local neighborhoods to determine the presence of edges in this paper we propose a cost function that incorporates nonlocal boundary information into the regularization method we use an alternating minimization algorithm with deterministic annealing to minimize the proposed cost function jointly estimating region boundaries and object pixel values we apply variational techniques implemented using level sets methods to update the boundary estimates then using the most recent boundary estimate we minimize a space variant quadratic cost function to update the image estimate for the positron emission tomography transmission reconstruction application we compare the bias variance tradeoff of this method with that of a conventional penalized likelihood algorithm with local huber roughness penalty

non homogeneous polygonal markov fields in the plane graphical representations and geometry of higher order correlations we consider polygonal markov fields originally introduced by arak and surgailis probab theory relat fields 80 543 579 1989 our attention is focused on fields with nodes of order two which can be regarded as continuum ensembles of non intersecting contours in the plane sharing a number of features with the two dimensional ising model we introduce non homogeneous version of polygonal fields in anisotropic environment for these fields we provide a class of new graphical constructions and random dynamics these include a generalized dynamic representation generalized and defective disagreement loop dynamics as well as a generalized contour birth and death dynamics next we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal markov fields in their consistent regime

reconstruction from projections based on detection and estimation of objects parts i and ii performance analysis and robustness analysis the problem of reconstructing a multidimensional field from noisy limited projection measurements is approached using an object based stochastic field model objects within a cross section are characterized by a finite dimensional set of parameters which are estimated directly from limited noisy projection measurements using maximum likelihood estimation in part i the computational structure and performance of the ml estimation procedure are investigated for the problem of locating a single object in a deterministic background simulations are also presented in part ii the issue of robustness to modeling errors is addressed

object based 3 d reconstruction of arterial trees from magnetic resonance angiograms by exploiting a priori knowledge of arterial shape and smoothness subpixel accuracy reconstructions are achieved from only four noisy projection images the method incorporates a priori knowledge of the structure of branching arteries into a natural optimality criterion that encompasses the entire arterial tree an efficient optimization algorithm for object estimation is presented and its performance on simulated phantom and in vivo magnetic resonance angiograms is demonstrated it is shown that accurate reconstruction of bifurcations is achievable with parametric models

consistent polygonal fields we extend our results 3 on the construction of polygonal markov fields on the plane taking finite number of values and having a given symmetric markov process as the marginal process on each line to the case when is reversible

suppression of period doubling in the dynamics of a bouncing ball abstract we present a theoretical and experimental study of a bouncing ball system where the effect of a near resonant perturbation is to suppress the onset of the first period doubling bifurcation near the bifurcation point the full dynamical equations are reduced to a discrete time map governing the dynamics on a slowly oscillating center manifold the derivation emphasizes the geometry of the phase space dynamics and serves to clarify several points of a recently proposed theory regarding the effects of strong near resonant perturbations our experimental results agree with the simple theory for moderate perturbations for larger perturbations higher order effects must be included

randomness in the bouncing ball dynamics abstract the dynamics of a vibrated bouncing ball is studied numerically in the reduced impact representation where the velocity of the bouncing ball is sampled at each impact with the plate asynchronous sampling its random nature is thus fully revealed i the chattering mechanism through which the ball gets locked on the plate is accomplished within a limited interval of the plate oscillation phase and ii is well described in impact representation by a special structure of looped nested bands and iii chattering trajectories and strange attractors may coexist for appropriate ranges of the parameter values structure and substructure of the chattering bands are well explained in terms of a simple impact map rule these results are of potential application to the analysis of high temperature vibrated granular gases

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

irregular diffusion in the bouncing ball billiard abstract we call a system bouncing ball billiard if it consists of a particle that is subject to a constant vertical force and bounces inelastically on a one dimensional vibrating periodically corrugated floor here we choose circular scatterers that are very shallow hence this billiard is a deterministic diffusive version of the well known bouncing ball problem on a flat vibrating plate computer simulations show that the diffusion coefficient of this system is a highly irregular function of the vibration frequency exhibiting pronounced maxima whenever there are resonances between the vibration frequency and the average time of flight of a particle in addition there exist irregularities on finer scales that are due to higher order dynamical correlations pointing towards a fractal structure of this curve we analyze the diffusive dynamics by classifying the attracting sets and by working out a simple random walk approximation for diffusion which is systematically refined by using a green kubo formula

reconstructing polygons from moments with connections to array processing we establish a set of results showing that the vertices of any simply connected planar polygonal region can be reconstructed from a finite number of its complex moments these results find applications in a variety of apparently disparate areas such as computerized tomography and inverse potential theory where in the former it is of interest to estimate the shape of an object from a finite number of its projections whereas in the latter the objective is to extract the shape of a gravitating body from measurements of its exterior logarithmic potentials at a finite number of points we show that the problem of polygonal vertex reconstruction from moments can in fact be posed as an array processing problem and taking advantage of this relationship we derive and illustrate several new algorithms for the reconstruction of the vertices of simply connected polygons from moments

three dimensional reconstruction from projections with incomplete and noisy data by object estimation an estimation approach to three dimensional reconstruction from parallel ray projections with incomplete and very noisy data is described using a stochastic dynamic model for an object of interest in a probed domain of known background density the reconstruction problem is reformulated as a n onlinear state estimation problem an approximate minimum mean square error globally optimal algorithm for the solution of this problem is presented the algorithm which is recursive in a hybrid frequency space domain operates directly on the fourier transformed projection data eliminating altogether the attempt to invert the projection integral equation the simulation example considered in this paper demonstrates that good object estimates may be obtained with as few as five views in a limited sector of 90 and at a signal to noise ratio as low as 0 db

on a functional contraction method methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space our approach is an extension of the so called contraction method to the space c 0 1 of continuous functions endowed with uniform topology and the space d 0 1 of cadlag functions with the skorokhod topology the contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences it is based on stochastic fixed point equations where probability metrics can be used to obtain contraction properties and allow the application of banach s fixed point theorem we develop the use of the zolotarev metrics on the spaces c 0 1 and d 0 1 in this context applications are given in particular a short proof of donsker s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed

stein s method for diffusion approximations stein s method of obtaining distributional approximations is developed in the context of functional approximation by the wiener process and other gaussian processes an appropriate analogue of the one dimensional stein equation is derived and the necessary properties of its solutions are established the method is applied to the partial sums of stationary sequences and of dissociated arrays to a process version of the wald wolfowitz theorem and to the empirical distribution function

pac learning with generalized samples and an applicaiton to stochastic geometry an extension of the standard probably approximately correct pac learning model that allows the use of generalized samples is introduced a generalized sample is viewed as a pair consisting of a functional on the concept class together with the value obtained by the functional operating on the unknown concept it appears that this model can be applied to a number of problems in signal processing and geometric reconstruction to provide sample size bounds under a pac criterion a specific application of the generalized model to a problem of curve reconstruction is considered and some connections with a result from stochastic geometry are discussed

pac learning from general examples abstract in this paper we study a new view on the pac learning model in which the examples are more complicated than in the standard model there an example usually is an element of the learning domain and its label indicates whether it belongs to the target concept here the examples can be subsets and their labels indicate some relation to the target concept e g whether they intersect it or not we show how this setting can be easily transformed into the standard pac model however for an analysis it is much more natural to stick to the original formulation then the central notion is that of the relative dimension of a target class with respect to a sample class which replaces the vapnik chervonenkis dimension the investigation of structural aspects of the relative dimension is followed by its applications to learning environments it turns out that computing or bounding the relative dimension leads to interesting combinatorial problems even for simple target and sample classes sometimes the analysis is easier if one represents the concepts as unions or intersections of simpler ones we present sharp bounds on the relative and the vapnik chervonenkis dimension of the complicated class in terms of the simpler one

gibbs delaunay tessellations with geometric hardcore conditions in this paper we prove the existence of infinite gibbs delaunay tessellations on 2 the interaction depends on the local geometry of the tessellation we introduce a geometric hardcore condition on small and large cells consequently we can construct more regular infinite random delaunay tessellations

bouncing ball problem stability of the periodic modes exploring all its ramifications we give an overview of the simple yet fundamental bouncing ball problem which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity the dynamics is modeled on the basis of a discrete map of difference equations which numerically solved fully reveals a rich variety of nonlinear behaviors encompassing irregular nonperiodic orbits subharmonic and chaotic motions chattering mechanisms and also unbounded nonperiodic orbits for periodic motions the corresponding conditions for stability and bifurcation are determined from analytical considerations of a reduced map through numerical examples it is shown that a slight change in the initial conditions makes the ball motion switch from periodic to chaotic orbits bounded by a velocity strip v gamma 1 epsilon where gamma is the nondimensionalized shaking acceleration and epsilon the coefficient of restitution which quantifies the amount of energy lost in the ball table collision

noisy precursors of nonlinear instabilities this paper studies the effect of external noise on systems displaying nonlinear instabilities of periodic orbits each class of instability is found to have its own characteristic signature as displayed by the power spectrum results are derived for each of the codimension one instabilities familiar from bifurcation theory

global uniform symptotic attractive stability of the non autonomous bouncing ball system abstract the non autonomous bouncing ball system consists of a point mass in a constant gravitational field which bounces inelastically on a flat vibrating table a sufficient condition for the global uniform attractive stability of the equilibrium of the non autonomous bouncing ball system is proved in this paper by using a lyapunov like method which can be regarded as an extension of lyapunov s direct method to lyapunov functions which may also temporarily increase along solution curves the presented lyapunov like method is set up for non autonomous measure differential inclusions and constructs a decreasing step function above the oscillating lyapunov function furthermore it is proved that the attractivity of the equilibrium of the bouncing ball system is symptotic i e there exists a finite time for which the solution has converged exactly to the equilibrium for this attraction time an upper bound is given in this paper

the use of three and four dimensional surface harmonics for rigid and nonrigid shape recovery and representation the use of spherical harmonics for rigid and nonrigid shape representation is well known this paper extends the method to surface harmonics defined on domains other than the sphere and to four dimensional spherical harmonics these harmonics enable us to represent shapes which cannot be represented as a global function in spherical coordinates but can be in other coordinate systems prolate and oblate spheroidal harmonics and cylindrical harmonics are examples of surface harmonics which we find useful nonrigid shapes are represented as functions of space and time either by including the time dependence as a separate factor or by using four dimensional spherical harmonics this paper compares the errors of fitting various surface harmonics to an assortment of synthetic and real data samples both rigid and nonrigid in all cases we use a linear least squares approach to find the best fit to given range data it is found that for some shapes there is a variation among geometries in the number of harmonics functions needed to achieve a desired accuracy in particular it was found that four dimensional spherical harmonics provide an improved model of the motion of the left ventricle of the heart

on stochastic perturbations of iterations of circle maps abstract the purpose of this paper is to show the effect of stochastic perturbations when iterating a circle map we show that if we iterate a circle map with irrational rotation number and allow a small stochastic rotation between each iteration each rotation angle being independent of the others but not necessarily having the same distribution then in general the shape of the n th iterate will tend towards the shape of one point function as n tends to infinity that is in the long run the circle will be contracted into exactly one point a point which however moves around on the circle

spontaneous magnetization in the plane the arak process is a solvable stochastic process which generates coloured patterns in the plane patterns are made up of a variable number of random non intersecting polygons we show that the distribution of arak process states is the gibbs distribution of its states in thermodynamic equilibrium in the grand canonical ensemble the sequence of gibbs distributions forms a new model parameterised by temperature we prove that there is a phase transition in this model for some non zero temperature we illustrate this conclusion with simulation results we measure the critical exponents of this off lattice model and find they are consistent with those of the ising model in two dimensions

probability distribution functions for a single particle vibrating in one dimension experimental study and theoretical analysis abstract we consider the form of the rebound velocity 0 particle velocity and height h probability density functions pdfs for the one dimensional motion of a single particle on a sinusoidally oscillating base the motion is considered in the limit of high excitation vibration frequency collision rate experimentally we find that these pdfs are well approximated by p 0 0 0 exp 0 2 a gaussian p exp 2 and a boltzmann type function p h h exp 2 gh where is a constant and g is the acceleration due to gravity we develop an analytical model which accurately predicts the general form for the rebound velocity pdf the other two pdfs are then analytically shown to follow as a consequence scaling laws for the particle granular temperature with peak base velocity and particle base restitution coefficient determined from previous work can also be predicted from the pdf a fine scale spiky structure in the rebound velocity pdf is found using numerical simulations to be a consequence of resonance phenomena between the particle and vibrating base good agreement between scaling laws from the theory and simulation is found but insufficient data is obtainable to derive accuracy exponents experimentally

chaotic dynamics of a bouncing ball abstract a detailed study of a mapping on a two dimensional manifold is made the mapping describes a system subject to periodic forcing in particular an imperfectly elastic ball bouncing on a vibrating platform quasiperiodic motion on a one dimensional manifold is proven and observed numerically at low forcing while at higher forcing smale horseshoes are present we examine the evolution of the attracting set with changing parameter spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises a new type of chaos in which a trajectory alternates between two distinct chaotic regions is described and explained in terms of manifold collisions throughout we are concerned to examine the behaviour of lyapunov exponents typical behaviour of lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed at higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given

dobrushin kotecký shlosman theorem for polygonal markov fields in the plane we consider the so called length interacting arak surgailis polygonal markov fields with v shaped nodes a continuum and isometry invariant process in the plane sharing a number of properties with the two dimensional ising model for these polygonal fields we establish a low temperature phase separation theorem in the spirit of the dobrushin koteck shlosman theory with the corresponding wulff shape deteremined to be a disk due to the rotation invariant nature of the considered model as an important tool replacing the classical cluster expansion techniques and very well suited for our geometric setting we use a graphical construction built on contour birth and death process following the ideas of fernandez ferrari and garcia

a unified approach to statistical tomography using coordinate descent optimization over the past years there has been considerable interest in statistically optimal reconstruction of cross sectional images from tomographic data in particular a variety of such algorithms have been proposed for maximum a posteriori map reconstruction from emission tomographic data while map estimation requires the solution of an optimization problem most existing reconstruction algorithms take an indirect approach based on the expectation maximization em algorithm we propose a new approach to statistically optimal image reconstruction based on direct optimization of the map criterion the key to this direct optimization approach is greedy pixel wise computations known as iterative coordinate decent icd we propose a novel method for computing the icd updates which we call icd newton raphson we show that icd newton raphson requires approximately the same amount of computation per iteration as em based approaches but the new method converges much more rapidly in our experiments typically five to ten iterations other advantages of the icd newton raphson method are that it is easily applied to map estimation of transmission tomograms and typical convex constraints such as positivity are easily incorporated

total curvature and intersection tomography abstract we show how to reconstruct certain one dimensional generalized sets in the plane which we callk sets from their projections a projection onto a linegat the pointpcounts with multiplicities the number of points in the set which lie on the line which is perpendicular togand passes throughp we also prove a formula analogous to crofton s formula relating the total absolute curvature of such a set to the average total variation of its projections the formulation of the class ofk sets captures a notion of finite total absolute curvature that is independent of parametrizations we define the total absolute curvature of ak set to be the minimal total absolute curvature of a collection of curves which represent the set

sampling schemes for multidimensional signals with finite rate of innovation in this paper we consider the problem of sampling signals that are nonband limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation streams of diracs and piecewise polynomials are the examples of such signals and thus are known as signals with finite rate of innovation fri we know that the classical band limited sine sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band limited however the recent results on fri sampling suggest that it is possible to sample and perfectly reconstruct such nonband limited signals using a rich class of kernels in this paper we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials satisfy strang fix conditions in fact the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples using these moments and the annihilating filter method prony s method the innovative part of the signal and therefore the signal itself is perfectly reconstructed in particular we present local directional derivatives based and global complex moments based radon transform based sampling schemes for classes of fri signals such as sets of diracs bilevel and planar polygons quadrature domains e g circles ellipses and cardioids 2d polynomials with polygonal boundaries and n dimensional diracs and convex polytopes this work has been explored in a promising way in super resolution algorithms and distributed compression and might find its applications in photogrammetry computer graphics and machine vision

stochastic processes that generate polygonal and related random fields a reversible ergodic markov process taking values in the space of polygonally segmented images is constructed the stationary distribution of this process can be made to correspond to a gibbs type distribution for polygonal random fields as introduced by arak and surgailis 1989 and a few variants thereof such as those arising in bayesian analysis of random fields extensions to generalized polygonal random fields are presented where the segmentation boundaries are not necessarily straight line segments

turbulence in multistep methods for initial value problems solving a scalar initial value problem with a numerical k step scheme amounts to iterating a k dimensional discrete dynamical system in this paper we investigate regular and chaotic behaviour of some dynamical systems associated with adams bashforth formulas applied to a simple nonlinear test equation

dissipation and its consequences in the scaling exponents for a family of two dimensional mappings the effects and consequences of dissipation in the scaling exponents describing the behaviour of average properties over the chaotic dynamics for a family of two dimensional mappings are studied the mapping is parametrized by an exponent in one of the dynamical variables and by a parameter 0 1 which denotes the amount of the dissipation the lyapunov exponents are obtained for different values of and in the range 0 1 the behaviour of the approaching orbits to the chaotic attractors is described analytically to be of exponential type the deviation around the average action for chaotic orbits was described by a single set of scaling exponents obtained for different leading the model to fall into the same universality class as that of the dissipative bouncer model

some dynamical properties of a classical dissipative bouncing ball model with two nonlinearities abstract some dynamical properties for a bouncing ball model are studied we show that when dissipation is introduced the structure of the phase space is changed and attractors appear increasing the amount of dissipation the edges of the basins of attraction of an attracting fixed point touch the chaotic attractor consequently the chaotic attractor and its basin of attraction are destroyed given place to a transient described by a power law with exponent 2 the parameter space is also studied and we show that it presents a rich structure with infinite self similar structures of shrimp shape

recursive self similarity for random trees random triangulations and brownian excursion recursive self similarity for a random object is the property of being decomposable into independent rescaled copies of the original object certain random combinatorial objects trees and triangulations possess approximate versions of recursive self similarity and then their continuous limits possess exact recursive self similarity in particular since the limit continuum random tree can be identified with brownian excursion we get a nonobvious recursive self similarity property for brownian excursion

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

polygonal web representation for higher order correlation functions of consistent polygonal markov fields in the plane we consider polygonal markov fields originally introduced by arak in 4th ussr japan symposium on probability theory and mathematical statistics abstracts of communications 1982 arak and surgailis in probab theory relat fields 80 543 579 1989 our attention is focused on fields with nodes of order two which can be regarded as continuum ensembles of non intersecting contours in the plane sharing a number of salient features with the two dimensional ising model the purpose of this paper is to establish an explicit stochastic representation for the higher order correlation functions of polygonal markov fields in their consistency regime the representation is given in terms of the so called crop functionals defined by a mobius type formula of polygonal webs which arise in a graphical construction dual to that giving rise to polygonal fields the proof of our representation formula goes by constructing a martingale interpolation between the correlation functions of polygonal fields and crop functionals of polygonal webs

partial match queries in random k d trees we solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random k d trees our approach is new and of some generality in particular it is applicable to many problems involving differential equations or difference equations with polynomial coefficients

convergence of algorithms for reconstructing convex bodies and directional measures we investigate algorithms for reconstructing a convex body k in r n from noisy measurements of its support function or its brightness function in k directions u 1 u k the key idea of these algorithms is to construct a convex polytope p k whose support function or brightness function best approximates the given measurements in the directions u 1 u k in the least squares sense the measurement errors are assumed to be stochastically independent and gaussian it is shown that this procedure is strongly consistent meaning that almost surely p k tends to k in the hausdorff metric as k here some mild assumptions on the sequence u i of directions are needed using results from the theory of empirical processes estimates of rates of convergence are derived which are first obtained in the l 2 metric and then transferred to the hausdorff metric along the way a new estimate is obtained for the metric entropy of the class of origin symmetric zonoids contained in the unit ball similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in k directions u 1 u k here the dudley and prohorov metrics are used the methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body

inelastic collapse of a ball bouncing on a randomly vibrating platform a theoretical study is undertaken of the dynamics of a ball which is bouncing inelastically on a randomly vibrating platform of interest are the distributions of the number of flights n f and the total time ensuremath tau c until the ball has effectively collapsed i e coalesced with the platform in the strictly elastic case both distributions have power law tails characterized by exponents that are universal i e independent of the detail of the platform noise distribution however in the inelastic case both distributions have exponential tails p n f ensuremath sim mathrm exp ensuremath ensuremath theta 1 n f and p ensuremath tau c ensuremath sim mathrm exp ensuremath ensuremath theta 2 ensuremath tau c the decay exponents ensuremath theta 1 and ensuremath theta 2 depend continuously on the coefficient of restitution and are nonuniversal however as one approaches the elastic limit they vanish in a manner which turns out to be universal an explicit expression for ensuremath theta 1 is provided for a particular case of the platform noise distribution

rbf nets for approximating an object s boundary by image random sampling abstract our aim is to discuss the problem of approximating the boundary of an object in a binary image in contrast to earlier papers on similar topics we avoid finding contours and then their approximations instead we sample an image in a relatively small number of points 1 3 of pixels at random the collected data is used for imposing constraints on parameters of a radial basis functions rbf neural net it is proved that if the rbf net structure is sufficiently rich to approximate an object boundary then estimates of rbf net parameters tend to their true values as the number of samples approaches to infinity then a seemingly linear algorithm of estimating linear weights is proposed and its consistency is also proved its implementation which is based on an iterative use of the linear programming is briefly discussed and the results of its testing are shortly reported

a family of dissipative two dimensional mappings chaotic regular and steady state dynamics investigation abstract some dynamical properties for a family of two dimensional mappings controlled by three parameters are considered in this paper for null dissipation and depending on the other control parameters diffusion in phase space is observed a connection with the standard mapping is made to determine the range of control parameters leading to unlimited diffusion in the dissipative case for low nonlinearity we obtain the fixed points and an expression to find any periodic orbit with any period if it exists for large nonlinearity we investigate the lyapunov exponents and characterise the steady state of the system for a long time

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

reconstructing convex sets from support line measurements algorithms are proposed for reconstructing convex sets given noisy support line measurements it is observed that a set of measured support lines may not be consistent with any set in the plane a theory of consistent support lines which serves as a basis for reconstruction algorithms that take the form of constrained optimization algorithms is developed the formal statement of the problem and constraints reveals a rich geometry that makes it possible to include prior information about object position and boundary smoothness the algorithms which use explicit noise models and prior knowledge are based on maximum likelihood and maximum a posteriori estimation principles and are implemented using efficient linear and quadratic programming codes experimental results are presented this research sets the stage for a more general approach to the incorporation of prior information concerning the estimation of object shape

on the contraction method with degenerate limit equation a class of random recursive sequences yn with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form x stackrel mathcal l x for nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the unique solution of the limit equation in this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation in particular we establish some new tools and a general convergence scheme which transfers information on mean and variance into a central limit law with normal limit we also obtain a convergence rate result for the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables

on the estimation of a convex set from noisy data on its support function abstract in problems of medical imaging and robotic vision measurements of a convex set are sometimes obtained via the set s support function the standard way in which the convex set is recovered from such data is to suppose that it is polygonal model the errors as normal random variables and apply constrained maximum likelihood methods typically the number of sides assumed of the polygon is equal to or a little less than the number of data points however from a statistical viewpoint the number of sides should really be interpreted as a smoothing parameter and chosen to optimize some measure of performance additionally if the true set is not a polygon then a polygonal estimate can be aesthetically unsatisfactory in this article we suggest periodic smoothing methods for estimating the convex set

on the origin of chaos in gearbox models abstract discrete dissipative mappings are derived from the dynamics of gearboxes modelled mathematically in mechanical engineering in a special regime we are able to derive a mapping which belongs to the class of fermi accelarator systems these systems exhibit chaotic behaviours therefore the chaos observed in numerical calculations of gearbox systems is understood from a theoretical point of view

markov fields with polygonal realizations we define a new class of markov random fields with two dimensional parameter z t 2 which take a finite number of values separated by random polygonal boundaries in some special cases these random fields are consistent in t and allow and alternative description in terms of the equilibrium evolution of a system of one dimensional particles

arak clifford surgailis tessellations basic properties and variance of the total edge length non homogeneous random tessellations in the plane with t shaped vertices are considered which are defined as gibbsian modifications of a poisson line process with arbitrary locally finite intensity measure in the homogeneous set up they can be regarded as a specific case of the general arak surgailis polygonal fields in the plane and share some properties with the iteration stable stit tessellations in the non homogeneous and anisotropic environment an explicit expression for the partition function is provided and first and second order properties of the random length measure of the tessellations restricted to a convex sampling window are derived in the more special isotropic regime a closed formula for the pair correlation function and the variance of the total edge length is obtained

motion of a ball dropped onto a one dimensional self affine surface we present a molecular dynamics study of the subsequent dissipation free motion of a ball dropped onto a self affine profile characterized by the hurst exponent h the distribution of sizes of the energetically accessible intervals to which the ball is confined as a function of the height h at which the ball is dropped has been determined furthermore the correlations between the distribution of slopes of the profiles and the distribution of directions of the velocity vector at impact were studied we found that the distribution of angles the velocity vector makes with the normal of the profile is independent of both h and the roughness amplitude the distribution of horizontal lengths of the parabolic trajectories of the ball between impacts were further studied and we found it to be a power law with exponent h 2 for small lengths and gaussian like for large lengths finally we discuss the scaling of the parameters of the particle trajectory with respect to rescaling of the self affine surface on which the ball bounces