a maximum entropy framework for nonexponential distributions probability distributions having power law tails are observed in a broad range of social economic and biological systems we describe here a potentially useful common framework we derive distribution functions for situations in which a joiner particle k pays some form of price to enter a community of size where costs are subject to economies of scale maximizing the boltzmann gibbs shannon entropy subject to this energy like constraint predicts a distribution having a power law tail it reduces to the boltzmann distribution in the absence of economies of scale we show that the predicted function gives excellent fits to 13 different distribution functions ranging from friendship links in social networks to protein protein interactions to the severity of terrorist attacks this approach may give useful insights into when to expect power law distributions in the natural and social sciences

on the existence of invariant absolutely continuous measures let be a measure space with normalized measure f a nonsingular transformation we prove there exists anf invariant normalized measure which is absolutely continuous with respect to if and only if there exist 0 and 0 1 such that e implies f k e for allk 0

finite approximations of frobenius perron operators a solution of ulam s conjecture to multi dimensional transformations abstract we prove that ulam s piecewise constant approximation algorithm is convergent for computing an absolutely continuous invariant measure associated with a piecewise c2 expanding transformation or a jablonski transformation s 0 1 n rn 0 1 n this solves an extension of ulam s conjecture to multi dimensions and generalizes the convergence result given by t y li for one dimensional transformations

the spectral analysis of frobenius perron operators abstract it is known that the frobenius perron operator p s l 1 0 1 l 1 0 1 associated with a transformation s from 0 1 to itself with inf s 1 is quasi compact as an operator on the banach space bv 0 1 of functions of bounded variation in l 1 0 1 and thus p s bv 0 1 bv 0 1 possesses only the finite peripheral spectrum and in particular 1 is an isolated eigenvalue of p s in this paper we show that under mild conditions on s the spectrum of p s l 1 x l 1 x is either the closed unit disk c 1 or a cyclic subset of c 1

a manifold independent approach to understanding transport in stochastic dynamical systems abstract we develop a new collection of tools aimed at studying stochastically perturbed dynamical systems specifically in the setting of bi stability that is a two attractor system it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero noise case barriers in the phase space pseudo barriers thus creating a pre heteroclinic tangency chaos like behavior the stochastic dynamical system has a corresponding frobenius perron operator with a stochastic kernel which describes how densities of initial conditions move under the noisy map thus in studying the action of the frobenius perron operator we learn about the transport of the map we have employed a galerkin ulam like method to project the frobenius perron operator onto a discrete basis set of characteristic functions to highlight this action localized in specified regions of the phase space graph theoretic methods allow us to re order the resulting finite dimensional markov operator approximation so as to highlight the regions of the original phase space which are particularly active pseudo barriers of the stochastic dynamics our toolbox allows us to find 1 regions of high activity of transport 2 flux across pseudo barriers and also 3 expected time of escape from pseudo basins some of these quantities are also possible via the manifold dependent stochastic melnikov method but melnikov only applies to a very special class of models for which the unperturbed homoclinic orbit is available our methods are unique in that they can essentially be considered as a black box of tools which can be applied to a wide range of stochastic dynamical systems in the absence of a priori knowledge of manifold structures we use here a model of childhood diseases to showcase our methods our tools will allow us to make specific observations of 1 loss of reducibility between basins with increasing noise 2 identification in the phase space of active regions of stochastic transport 3 stochastic flux which essentially completes the heteroclinic tangle

a piecewise quadratic maximum entropy method for the statistical study of chaos abstract let the frobenius perron operator p s l 1 0 1 l 1 0 1 related to a nonsingular transformation s 0 1 0 1 have an invariant density f we propose a piecewise quadratic maximum entropy method for the numerical approximation of f the role of the partition of unity property of the quadratic functions for the numerical recovery of f has been depicted the proposed algorithm overcomes the ill conditioning shortage of the traditional maximum entropy method which only employs polynomials so that any number of moments can be used to increase the accuracy of the computed invariant density the convergence rate of the method is shown to be of order 3 numerical results are presented to justify the theoretical analysis of the method

piecewise linear least squares approximations of frobenius perron operators abstract we propose a piecewise linear numerical method based on least squares approximations for computing stationary density functions of frobenius perron operators associated with piecewise c2 and stretching mappings of the unit interval we prove the weak convergence of the method for a class of frobenius perron operators and the numerical results show that it is also norm convergent and has a better convergence rate than the piecewise linear markov approximation method

a direct approach to conformational dynamics based on hybrid monte carlo recently a novel concept for the computation ofessentialfeatures of the dynamics of hamiltonian systems such as molecular dynamics has been proposed the realization of this concept had been based on subdivision techniques applied to the frobenius perron operator for the dynamical system the present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework this approach allows us to define the phrase conformation in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations in a first step the frequency of conformational changes is characterized in statistical terms leading to the definition of some markov operatortthat describes the corresponding transition probabilities within the canonical ensemble in a second step a discretization oftvia specific hybrid monte carlo techniques is shown to lead to astochasticmatrixp with these theoretical preparaions an identification algorithm for conformations to be presented in a later paper is applicable it is demonstrated that the discretization oftcan be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked numerical results for the n pentane molecule and the triribonucleotide adenylyl 3 5 cytidylyl 3 5 cytidin are given and interpreted

approximate solution of fredholm integral equations by the maximum entropy method an approximate means of solving fredholm integral equations by the maximum entropy method is developed the fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum entropy methods has been successfully implemented in a previous paper by mead and papanicolaou l r mead and n papanicolaou j math phys 25 2404 1984 several explicit examples are given of approximate maximum entropy solutions of fredholm integral equations of the first and second kinds and of the wiener hopf type both the weaknesses and strengths of the method are discussed

convergence analysis of projection methods for frobenius perron operators based on matrix norm techniques abstract using matrix norm techniques we give a unified convergence analysis of a general projection method for computing stationary density functions of frobenius perron operators associated with piecewise c 2 and stretching transformations of the unit interval

a finite element method for the frobenius perron operator equation we present a finite element framework to numerically solve the absolutely continuous invariant measure associated with a non singular transformation both convergence analysis and error estimates of the method are given for a class of mappings in r n

a maximum entropy method for solving frobenius perron operator equations let s 0 1 0 1 be a nonsingular transformation that preserves an absolutely continuous invariant measure with density f in this paper we develop a general algorithm for approximating f using the maximum entropy approach

on the convergence of moment problems we study the problem of estimating a nonnegative density given a finite number of moments such problems arise in numerous practical applications as the number of moments increases the estimates will always converge weak as measures but need not converge weakly in l this is related to the existence of functions on a compact metric space which are not essentially riemann integrable in some suitable sense we characterize the type of weak convergence we can expect in terms of riemann integrability and in some cases give error bounds when the estimates are chosen to minimize an objective function with weakly compact level sets such as the bolzmann shannon entropy they will converge weakly in l when an lp norm 1 p is used as the objective the estimates actually converge in norm these results provide theoretical support to the growing popularity of such methods in practice

maximum entropy approach with higher moments for solving fokker planck equation abstract the maximum entropy method with higher number of moments is used to solve the fokker planck equation an adopted newton method is used to iterate the maximum entropy set of equations the method is used to calculate the probability density function of the fokker planck equation the calculations are carried out for three examples 1 the bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems 2 the malthus verhulst model which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light 3 the black scholes equation used in financial market option pricing although the maximum entropy approach has several advantages it is not convergent at large times and so cannot be used to calculate the steady state solution

response operators for markov processes in a finite state space radius of convergence and link to the response theory for axiom a systems using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state markov processes the results can be used for studying empirically constructed e g from observations or through coarse graining of model simulations finite state approximation of statistical mechanical systems recent results concerning the convergence of the statistical properties of finite state markov approximation of the full asymptotic dynamics on the srb measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of axiom a system our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language plus providing bounds on the radius of convergence of the perturbative theory in particular we relate the convergence of the response theory to the rate of mixing of the unperturbed system one can use the formulas derived for finite state markov processes to recover previous findings obtained on the response of continuous time axiom a dynamical systems to perturbations by considering the generator of time evolution for the measure and for the observables a very basic low tech and computationally cheap analysis of the response of the lorenz 63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state markov processes to compute the system s response

approximations of frobenius perron operators via interpolation abstract let s 0 1 0 1 be a chaotic map and let p l1 0 1 l1 0 1 be the corresponding frobenius perron operator we propose a piecewise linear approximations method based on interpolation that can be efficiently used to compute a fixed density of p the convergence of the method for a class of mappings is proved and numerical results are also presented

constructive approximations of markov operators we construct piecewise linear markov finite approximations of markov operators defined on l1 0 1 n and we study various properties such as consistency stability and convergence for the purpose of numerical analysis of markov operators

probability density adjoint for sensitivity analysis of the mean of chaos sensitivity analysis especially adjoint based sensitivity analysis is a powerful tool for engineering design which allows for the efficient computation of sensitivities with respect to many parameters however these methods break down when used to compute sensitivities of long time averaged quantities in chaotic dynamical systems this paper presents a new method for sensitivity analysis of ergodic chaotic dynamical systems the density adjoint method the method involves solving the governing equations for the system s invariant measure and its adjoint on the system s attractor manifold rather than in phase space this new approach is derived for and demonstrated on one dimensional chaotic maps and the three dimensional lorenz system it is found that the density adjoint computes very finely detailed adjoint distributions and accurate sensitivities but suffers from large computational costs

high order approximation of the frobenius perron operator we formulate a general convergence theory for the finite dimensional projection approximation of the fixed point of a class of linear operators in particular we explore the relation with the standard framework of stability and consistency of the approximation imply convergence applications include the frobenius perron operator p s corresponding to a nonsingular measurable transformation from 0 1 to itself numerical experiments are also given to support the theoretical analysis

invariant densities and escape rates rigorous and computable approximations in the l norm abstract in this article we study piecewise linear discretization schemes for transfer operators perron frobenius operators associated with interval maps we show how these can be used to provide rigorous pointwise approximations for invariant densities of markov interval maps we also derive the order of convergence of the approximate invariant density to the real one in the l norm the outcome of this paper complements recent results on the formulae of escape rates of open dynamical systems keller and liverani 2009 7 in particular the novelty of our work over previous results on b v and l approximations is that it provides a method for explicit computation of the approximation error

some properties of the class of positive dunford pettis operators abstract we characterize banach lattices for which each positive dunford pettis operator is m weakly compact resp l weakly compact and we give some consequences

fluctuations in a markovian system of pairwise interacting particles analogously to the kac s master equation approach to the spatially homogeneous boltzmann equation we introduce a system of markov processes of many particles moving on a countable set with pairwise interaction and investigate the fluctuation around mckean s non linear limit process our model possibly admits simultaneous jumps of two particles which make impossible both such characterizations of the fluctuation in the limit and techniques based on the cameron martin formula as has previously been obtained or used for diffusion models and for mckean s two speed gas model we obtain a new description of the variance functional for the fluctuation and by applying it in the case of no simultaneous jumps give a new derivation of a formula of h tanaka 7

identifying stochastic basin hopping by partitioning with graph modularity abstract it has been known that noise in a stochastically perturbed dynamical system can destroy what was the original zero noise case barriers in the phase space pseudobarrier noise can cause the basin hopping we use the frobenius perron operator and its finite rank approximation by the ulam galerkin method to study transport mechanism of a noisy map in order to identify the regions of high transport activity in the phase space and to determine flux across the pseudobarriers we adapt a new graph theoretical method which was developed to detect active pseudobarriers in the original phase space of the stochastic dynamic previous methods to identify basins and basin barriers require a priori knowledge of a mathematical model of the system and hence cannot be applied to observed time series data of which a mathematical model is not known here we describe a novel graph method based on optimization of the modularity measure of a network and introduce its application for determining pseudobarriers in the phase space of a multi stable system only known through observed data

on large deviations for particle systems associated with spatially homogeneous boltzmann type equations we consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous boltzmann type equation as the number of particles tends to infinity these laws of large numbers were proved for the maxwellian molecules by h tanaka tal and for the hard spheres by a s sznitman szl in the present paper we investigate the corresponding large deviations the large deviation upper bound is obtained and using convex analysis a non variational formulation of the rate function is given our results hold for maxwellian molecules with a cutoff potential and for hard spheres

integral and nonnegativity preserving approximations of functions in this paper we consider the problem of approximating a function by continuous piecewise linear functions that preserve the integral and nonnegativity of the original function

a posteriori error estimates for markov approximations of frobenius perron operators abstract we present lower and upper error bounds under the bv norm for piecewise linear markov finite approximations to frobenius perron operators in terms of the bv norm of the residuals so that a posteriori error estimates can be given numerical results using the monte carlo implementation are presented to support the theoretical analysis

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

about positive dunford pettis operators on banach lattices abstract we give several characterizations of banach lattices on which each positive dunford pettis operator is compact as consequences we obtain new sufficient and necessary conditions under which a norm of a banach lattice is order continuous a positive weakly compact operator is compact and the dual operator of a positive dunford pettis operator is dunford pettis

nonlinear reflecting diffusion process and the propagation of chaos and fluctuations associated abstract a nonlinear diffusion satisfying a normal reflecting boundary condition is constructed and a result of propagation of chaos for a system of interacting diffusing particles with normal reflecting boundary conditions is proven then a gaussian limit for the fluctuation field which is defined in l 0 2 b of a wiener type space b is obtained the covariance of the gaussian limit is computed in terms of a hilbert schmidt operator on l 0 2 b

a modified piecewise linear markov approximation of markov operators in this paper we propose a modified piecewise linear markov method for approximating markov operators defined on l 1 0 1 we present numerical results with the new method as compared to those using the original piecewise linear markov method

on the approximation of invariant measures given a discrete dynamical system defined by the map x x the density of the absolutely continuous a c invariant measure if it exists is the fixed point of the frobenius perron operator defined on l1 x ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator t y li proved the convergence of the scheme for expanding maps of the interval g keller and m blank extended this result to piecewise expanding maps of the cube in n we show convergence of a variation of ulam s scheme for maps of the cube for which the frobenius perron operator is quasicompact we also give sufficient conditions on for the existence of a unique fixed point of the matrix approximation and if the fixed point of the operator is a function of bounded variation we estimate the convergence rate

approximation of stationary statistical properties of dissipative dynamical systems time discretization we consider temporal approximation of stationary statistical properties of dissipative infinite dimensional dynamical systems we demonstrate that stationary statistical properties of the time discrete approximations i e numerical scheme converge to those of the underlying continuous dissipative infinite dimensional dynamical system under three very natural assumptions as the time step approaches zero the three conditions that are sufficient for the convergence of the stationary statistical properties are 1 uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre compact in the phase space 2 convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval 0 1 uniformly with respect to initial data from the union of the global attractors 3 uniform continuity of the solutions to the continuous dynamical system on the unit time interval 0 1 uniformly for initial data from the union of the global attractors the convergence of the global attractors is established under weaker assumptions an application to the infinite prandtl number model for convection is discussed

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

computing the invariant measure and the lyapunov exponent for one dimensional maps using a measure preserving polynomial basis we consider a generalisation of ulam s method for approximating invariant densities of one dimensional maps rather than use piecewise constant polynomials to approximate the density we use polynomials of degree n which are defined by the requirement that they preserve the measure on n 1 neighbouring subintervals over the whole interval this results in a discontinuous piecewise polynomial approximation to the density we prove error results where this approach is used to approximate smooth densities we also consider the computation of the lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself together with using cubic polynomials in the density approximation this yields a very efficient method for computing highly accurate estimates of the lyapunov exponent we illustrate the theoretical findings with some examples

finite approximations of markov operators in this paper we survey some recent developments in the numerical analysis of markov operators and in particular frobenius perron operators associated with chaotic discrete dynamical systems

error estimates of the markov finite approximation of the frobenius perron operator in the practical application of the modern ergodic theory one of the main problems is related to the computation of the absolutely continuous invariant measures for nonsingular transformations on measure spaces see lasota and mackey l a simple example is constructed in li 2 to show that the round off error can completely dominate the calculation if one is to compute the invariant measure directly using a sequence of cesaro sums an alternative is to compute the invariant density function for the frobenius perron operator p l 0 1 l 0 1 associated with the nonsingular measurable transformations s 0 11 0 11 the frobenius perron operator p is defined by

on the norm convergence of a piecewise linear least squares method for frobenius perron operators abstract in this paper we prove both the l 1 norm and the bv norm convergence for a piecewise linear least squares approximation method that was recently proposed for computing stationary density functions of frobenius perron operators associated with piecewise c 2 and stretching mappings of the unit interval we also give a convergence rate analysis under the bv norm

solving fredholm integral equations via a piecewise linear maximum entropy method we propose a piecewise linear approximation method based on the maximum entropy principle to approximate a nonnegative solution of a fredholm integral equation numerically the theoretical analysis and numerical examples show that our method has a convergence rate of order 2 and it can get more accurate approximations with more moments used without ill condition of the classic maximum entropy approach the method can also be applied to solve fredholm integral equations with singular kernels

on the spectrum of frobenius perron operators abstract we prove more results on the spectrum of the frobenius perron operator p l1 l1 associated with a nonsingular transformation s x x on a finite measure space x

markov finite approximation of frobenius perron operator with doubly stochastic tridiagonal action matrix we extend the set of finite rank approximations of frobenius perron operators from projections to general finite rank operators by indicating an arbitrary action matrix we prove the convergence in the case for which the action matrix is double stochastic and tridiagonal and we find the rate of convergence

computing invariant measures of piecewise convex transformations lets 0 1 0 1 be a piecewise convex transformation satisfying some conditions which guarantee the existence of an absolutely continuous invariant probability measure we prove the convergence of a class of markov finite approximations for computing the invariant measure using a compactness argument forl1 spaces

computational methods for solving two parameter singularly perturbed boundary value problems for second order ordinary differential equations singularly perturbed two point boundary value problems spbvps for second order ordinary differential equations odes with two small parameters multiplying the derivatives are considered computational methods are suggested in this paper to solve this type of problems one of these methods is fitted operator method fom and the other is bondary value technique bvt in fom we suggest an exponentially fitted finite difference effd scheme throughout the domain of definition of the differential equation and the error estimates are derived in bvt the domain of definition of the differential equation is divided into three subintervals called inner and outer regions then we solve the two parameter spbvp tpspbvp over these regions as two point boundary value problems an exponentially fitted finite difference scheme is used in the inner regions and a classical finite difference scheme in the outer region the data for boundary conditions at the transition points are obtained using zero order asymptotic expansion approximation of the solution of the problem the bvt is well suited for parallel computing numerical examples are given to illustrate the method

a maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings let s 0 1 0 1 be a nonsingular transformation such that the corresponding frobenius perron operator p s l 1 0 1 l 1 0 1 has a stationary density f we propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f an advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply newton s method the jacobian matrices are positive definite and tri diagonal the numerical experiments show that the new maximum entropy method is more accurate than the markov finite approximation method which also uses piecewise linear functions provided that the involved moments are known this is supported by the convergence rate analysis of the method

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

the projection method for computing multidimensional absolutely continuous invariant measures we present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewisec2 expanding mappings on a bounded region rn the method is based on the galerkin projection principle for solving an operator equation in a banach space with the help of the modern notion of functions of bounded variation in multidimension we prove the convergence of the algorithm

markov finite approximation of frobenius perron operator the problem of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory 3 lasota and yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations s from 0 l to itself 4 later in 6 li and yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping the classical birkhoff s individual ergodic theorem says that if u is an ergodic invariant probability measure under s then for any measurable set a c 0 11 the time average lim 1 n c ax s x which measures the average time spent in a under iteration of s exists to be i for p almost all x where xa is the characteristic function of a 1 on a and 0 off a this seems to suggest that we may use the cesaro sum of the above form to calculate the invariant measure a simple and important example in 5 however shows that the computer round off error can completely dominate the calculation and make the implementation difficult in order to overcome this difficulty li 5 proposed a rigorous numerical procedure which can be implemented on a computer with negligible round off error problem a mapping s 0 l 0 l is called piecewise c2 if there is a partition 0 a a a 1 of 0 l such that for k 1 r the restriction sk of s on uk a is a c2 function which can be extended to the closed interval ak i ak as a c2 function s need not be continuous at the point ak for a c 0 11 we write s a for x s x e a throughout the paper we assume s is a nonsingular measurable transformation i e s is measurable and for any measurable subset a c 0 l with m a 0 m s a 0 here m denotes the lebesgue measure on 0 11 the operator ps l 0 1 l 0 1 defined by