a lax pair for a lattice modified kdv equation reductions to q painleve equations and associated lax pairs we present a new nonautonomous lax pair for a lattice nonautomous modified korteweg devries equation and show that it can be consistently extended multidimensionally a property commonly referred to as being consistent around a cube this nonautonomous equation is reduced to a series ofq discrete painlev e equations and lax pairs for the reduced equations are found a 2 2 lax pair is given for a qpiii with multiple parameters and also for versions of qpii and qpv all for the first time pacs number 02 30 ik

maximal parabolic regularity for divergence operators including mixed boundary conditions abstract we show that elliptic second order operators a of divergence type fulfill maximal parabolic regularity on distribution spaces even if the underlying domain is highly non smooth the coefficients of a are discontinuous and a is complemented with mixed boundary conditions applications to quasilinear parabolic equations with non smooth data are presented

some remarks on operator norm convergence of the trotter product formula on banach spaces abstract we extend the trotter kato chernoff theory of strong approximation of c 0 semigroups on banach spaces to operator norm approximation of analytic semigroups with error estimate as application we obtain a criterion for the operator norm convergence of the trotter product formula on banach spaces with error estimate n 1 log n provided one of the generators has a bounded h functional calculus for both results we present versions for approximation in operator ideal norms such as the trace norm or the hilbert schmidt norm finally we give some remarks on the operator norm convergence of the trotter product formula for semigroups acting on a scale of l p spaces

new approach to the correlation spectrum near intermittency a quantum mechanical analogy the correlation spectrum of fully developed one dimensional mappings is studied near and at a weakly intermittent situation using a suitable infinitematrix representation the eigenvalue equation of the frobenius perron operator is approximately reduced to the radial schrodinger equation of the hydrogen atom corrections are calculated by quantum mechanical perturbation theory analytical expressions for the spectral properties and correlation functions are derived and checked numerically compared to our previous work the accuracy of the present results is significantly higher owing to the controlled and systematic approximation scheme

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

weak type p p estimates for riesz transforms we consider the problem of l p boundedness of higher order riesz transforms associated to elliptic operators l of order 2m on as an application of the recently solved kato conjecture we show for all this generalizes the result of auscher and tchamitchian restricted to the case d 2m

the wiener test for higher order elliptic equations we deal with strongly elliptic differential operators of an arbitrary even order 2m with constant real coefficients and introduce a notion of the regularity of a boundary point with respect to the dirichlet problem which is equivalent to that given by n wiener in the case of m 1 it is shown that a capacitary wiener s type criterion is necessary and sufficient for the regularity if n 2m in the case of n 2m the same result is obtained for a subclass of strongly elliptic operators

sharp endpoint l p lp estimates for schrodinger groups let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type with a dimension n suppose that the heat operator e tl satisfies the generalized gaussian p 0 p 0 estimates of order m for some 1 le p 0 2 in this paper we prove sharp endpoint l p sobolev bound for the schrodinger group e itl that is for every p in p 0 p 0 there exists a constant c c n p 0 independent of t such that begin aligned left i l s e itl f right p le c 1 t s vert f vert p quad t in mathbb r quad s ge n big 1 over 2 1 over p big end aligned as a consequence the above estimate holds for all 1 p infty when the heat kernel of l satisfies a gaussian upper bound this extends classical results due to feffermann and stein and miyachi for the laplacian on the euclidean spaces mathbb r n we also give an application to obtain an endpoint estimate for l p boundedness of the riesz means of the solutions of the schrodinger equations

gaussian bounds for higher order elliptic differential operators with kato type potentials abstract let p d be a nonnegative homogeneous elliptic operator of order 2m with real constant coefficients on r n and v be a suitable real measurable function in this paper we are mainly devoted to establish the gaussian upper bound for schrodinger type semigroup e t h generated by h p d v with kato type perturbing potential v which naturally generalizes the classical result for schrodinger semigroup e t v as v k 2 r n the famous kato potential class our proof significantly depends on the analyticity of the free semigroup e t p d on l 1 r n as a consequence of the gaussian upper bound the l p spectral independence of h is concluded

on the lattice geometry and birational group of the six point multi ratio equation the inherent self consistency properties of the six point multi ratio equation allow it to be considered on a domain associated with a t shaped coxeter dynkin diagram this extends the kadomtsev pe

elliptic painleve equations from next nearest neighbor translations on the e_8 1 lattice the well known elliptic discrete painleve equation of sakai is constructed by a standard translation on the lattice given by nearest neighbor vectors in this paper we give a new elliptic discrete painleve equation obtained by translations along next nearest neighbor vectors this equation is a generic 8 parameter version of a 2 parameter elliptic difference equation found by reduction from adler s partial difference equation the so called q4 equation we also provide a projective reduction of the well known equation of sakai

eigenvalue spectrum of the frobenius perron operator near intermittency the spectral properties of the frobenius perron operator of one dimensional maps are studied when approaching a weakly intermittent situation numerical investigation of a particular family of maps shows that the spectrum becomes extremely dense and the eigenfunctions become concentrated in the vicinity of the intermittent fixed point analytical considerations generalize the results to a broader class of maps near and at weak intermittency and show that one branch of the map is dominant in the determination of the spectrum explicit approximate expressions are derived for both the eigenvalues and the eigenfunctions and are compared with the numerical results textcopyright 1996 the american physical society

characterizations of hardy spaces associated to higher order elliptic operators abstract in this paper the authors first show that the classical hardy space h 1 r n can be characterized by the non tangential maximal functions and the area integrals associated with the semigroups e t p and e t p respectively where p is an elliptic operator with real constant coefficients of homogeneous order 2m m 1 moreover the authors also prove that h 1 r n can be characterized by the riesz transforms m p 1 2 if and only if m is an odd integer in the main part of this paper the authors develop a theory of hardy space associated with l where l is a higher order divergence form elliptic operator with complex bounded measurable coefficients the authors set up a molecular hardy space h l 1 r n and give its characterizations by area integrals related to the semigroups e t l and e t l respectively finally authors give the h l 1 l 1 boundedness of riesz transforms square functions and maximal functions associated with l

a geometrical description of the discrete painleve vi and v equations we present a geometrical approach for the discrete painleve equations based on weyl groups the method relies on the bilinear formalism and assumes that the multidimensional function lives on the weight lattice of the appropriate affine weyl group the equations for the function a system of nonautonomous hirota miwa equations govern the evolution along the independent variable and the parameters of the equation the latter evolution induced by the schlesinger transformations in the present paper we analyse the case of the e 1 7 group using the geometrical description we derive the nonlinear discrete equations we find that in the case of the e 1 7 group these are the asymmetric q pvi and d pv that were recently proposed

equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form we study the heat kernel of higher order elliptic operators or systems under divergence form on rn ellipticity is in the sense of garding inequality we show that for homogeneous operators gaussian upper bounds and holder regularity of the heat kernel is equivalent to local regularity of weak solutions we also show stability of such bounds under l perturbations of the coefficients or under perturbations with bounded coefficients lower order terms such a criterion allows us to obtain heat kernel bounds for operators or systems with uniformly continuous or vmo coefficients

correlation properties of exactly solvable chaotic oscillators we derive exact expressions for the autocorrelation and cross correlation functions of wave forms generated by two exactly solvable chaotic systems for each system an analytic expression exists for chaotic solutions which we use to evaluate correlation integrals explicitly for some specific parameter values we calculate the mean and variance of the correlation functions averaged over all possible solutions the mean autocorrelation is shown to decay at a rate equal to the kolmogorov sinai entropy all derived results are shown to agree with numerical calculations of the corresponding quantities

singularity confinement and full deautonomisation a discrete integrability criterion abstract we present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector our method is based on the full deautonomisation procedure which consists in analysing non autonomous extensions of a given discrete system obtained by adding terms that are initially absent but whose presence does not alter the singularity pattern a justification for this approach is given through an algebro geometric analysis we also introduce the notions of early and late confinement while the former is a confinement that may exist already for the autonomous system the latter corresponds to a singularity pattern longer than that of the autonomous case late confinement will be shown to play an important role in the singularity analysis of systems with non trivial gauge freedom for which the existence of an undetected gauge in conjunction with a sketchy analysis might lead to erroneous conclusions as to their integrability an algebro geometric analysis of the role of late confinement in this context is also offered this novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping

tropical geometric interpretation of ultradiscrete singularity confinement using the interpretation of the ultradiscretization procedure as a non archimedean valuation we use results of tropical geometry to show how roots and poles manifest themselves in piece wise linear systems as points of non differentiability this will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems

a classification of two dimensional integrable mappings and rational elliptic surfaces we classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces while the qrt mappings can be restricted on each fiber there exist several classes of integrable mappings which exchange fibers we also show an equivalent condition when a generalized halphen surface becomes a halphen surface of index m

resonances for axiom bf a flows etant donne un flot d axiome a sur m et des fonctions lisses b c m r on montre que la fonction de correlation en temps bc pour un etat de gibbs a une transformee de fourier bc mesomorphe dans une bande on etudie les residus des poles de bc

quantum discrete dubrovin equations the discrete equations of motion for the quantum mappings of kdv type are given in terms of the sklyanin variables which are also known as quantum separated variables both temporal discrete time evolutions and spatial along the lattice at a constant time level evolutions are considered in the classical limit the temporal equations reduce to the classical discrete dubrovin equations as given in a previous publication nijhoff f w 2000 chaos solitons fractals 11 19 28 the reconstruction of the original dynamical variables in terms of the sklyanin variables is also achieved

autocorrelation function and spectral linewidth of spiral chaos in a physical experiment we present results of physical experiments where we measure the autocorrelation function acf and the spectral linewidth of the basic frequency of a spiral chaotic attractor in a generator with inertial nonlinearity both without and in the presence of external noise it is shown that the acf of spiral attractors decays according to an exponential law with a decrement which is defined by the phase diffusion coefficient it is also established that the evolution of the instantaneous phase can be approximated by a wiener random process

markov approximations and decay of correlations for anosov flows we develop markov approximations for very general suspension flows based on this we obtain a stretched exponential bound on time correlation functions for 3 d anosov flows that verify uniform nonintegrability of foliations these include contact anosov flows and geodesic flows on compact surfaces of variable negative curvature our bound on correlations is stable under small smooth perturbations

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

new airy type solutions of the ultradiscrete painleve ii equation with parity variables the q difference painleve ii equation admits special solutions written in terms of determinants whose entries are the general solution of the q airy equation an ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables then we obtain new airy type solutions of the ultradiscrete painleve ii equation with parity variables and the solutions have a richer structure than the known solutions

solution of the equivalence problem for the third painleve equation we find necessary conditions for a second order ordinary differential equation to be equivalent to the painleve iii equation under a general point transformation their sufficiency is established by reduction to known results for the equations of the form y f x y we consider separately the generic case and the case of reducibility to an autonomous equation the results are illustrated by the primary resonance equation

a study of the associated linear problem for q pv we consider the associated linear problem for a q analogue of the fifth painleve equation q pv we identify a lattice of connection preserving deformations in the space of the characteristic data for the linear problem with the lattice of translational backlund transformations for q pv hence show all translational backlund transformations possess a lax pair we show that the big q laguerre polynomials and a suitable generalization solve a special case of the linear problem and hence find solutions to q pv in terms of determinants of hankel matrices with entries consisting of rational or hypergeometric functions we show that we may specialize the connection preserving deformations to rational deformations of the weight and the recurrence relations

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

on decay of correlations in anosov flows there is some disagreement about the meaning of the phrase chaotic flow however there is no doubt that mixing anosov flows provides an example of such systems anosov systems were introduced and extensively studied in his classical memoir a among other things he proved the following fact known now as anosov alternative for flows either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an anosov diffeomorphism by a constant roof function if the first alternative holds gt is mixing with respect to every gibbs measure see pp2 now the natural question is what is the estimated rate of mixing this is certainly one of the simplest questions concerning correlation decay in continuous time systems nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure the easier case of anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of sinai ruelle and bowen b2 namely one uses markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing this method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations however the simplest example suspensions with locally constant roof functions never have such a property ri one can use the above observation to produce examples of axiom a flows with arbitrary slow correlation decay it became clear therefore that some additional geometric properties should be taken into account in recent work chernov chl ch2 has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature his method relies on the technique of markov approximation developed in chl the aim of this paper is to combine geometric considerations of chernov with the thermodynamic formalism approach the later method seems to be more appropriate than markov approximations since it gives simpler

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

discrete painleve equations with maximal periodicities we present the discrete painleve equations the coefficients of which have the maximal periodicity allowed by the geometric description in terms of affine weyl groups for each equation of the degeneration pattern obtained by sakai and which starts at the group we give the corresponding discrete painleve equation some of these systems were already known but here we proceed in a systematic way filling in all the gaps in particular we derive the equations with maximal periodicity associated to the affine weyl group which is the richest system in the sakai classification and which has yet to be studied in detail

correlation analysis of dynamical chaos abstract we study correlation and spectral properties of chaotic self sustained oscillations of different types it is shown that some classical models of stochastic processes can be used to describe behavior of autocorrelation functions of chaos the influence of noise on chaotic systems is also considered

affine weyl groups discrete dynamical systems and painleve equations a new class of representations of affine weyl groups on rational functions are constructed in order to formulate discrete dynamical systems associated with affine root systems as an application some examples of difference and differential systems of painleve type are discussed

sharp spectral multipliers for operators satisfying generalized gaussian estimates abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that l generates a holomorphic semigroup e t l which satisfies generalized m th order gaussian estimates in this article we study singular and dyadically supported spectral multipliers for abstract self adjoint operators we show that in this setting sharp spectral multiplier results follow from plancherel or stein tomas type estimates these results are applicable to spectral multipliers for a large class of operators including m th order elliptic differential operators with constant coefficients biharmonic operators with rough potentials and laplace type operators acting on fractals

ladder operators for q orthogonal polynomials the q difference analog of the classical ladder operators is derived for those orthogonal polynomials arising from a class of indeterminate moments problem

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

bilinear equations and q discrete painleve equations satisfied by variables and coefficients in cluster algebras we construct cluster algebras the variables and coefficients of which satisfy the discrete mkdv equation the discrete toda equation and other integrable bilinear equations several of which lead to q discrete painleve equations these cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation period property we will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation

explicit estimates on the fundamental solution of higher order parabolic equations with measurable coefficients for some positive constants ci and all t 0 and x y 0 such estimates for uniformly elliptic operators have been proved under a variety of local regularity assumptions 14 17 and more recently for operators with measurable coefficients provided 2m n 8 this has been shown to be true in the limit case 2m n 3 11 while there are examples that show that 1 is not in general true when 2m n and h has measurable coefficients 10 in 6 estimates are obtained for singular or degenerate operators with measurable coefficients with the euclidean distance being replaced by one that reflects the behavior of the coefficients near the singularities we refer to 9 for a detailed account of much of the recent progress on the spectral theory of higher order elliptic operators the problem of sharpening such bounds was addressed recently the sharpness of estimates is checked by comparison against short time asymptotics of k t x y these were proved by tintarev 19 who showed following work of evgrafov and postnikov 12 that if 0 r if the coefficients a are smooth and if the symbol a x satisfies the strong doi 10 1006 jdeq 2000 3940 available online at http www idealibrary com on

spectral theory of singular elliptic operators with measurable coefficients we study higher order elliptic operators with measurable coefficients acting on euclidean domains the coefficients may have degeneracies or singularities on the boundary or at infinity we prove gaussian type bounds on the fundamental solution of the associated semigroup these bounds are expressed in terms of a distance d x y that reflects the singularity or degeneracy of the coefficients the estimates are then used to extend the semigroup to other l p spaces and to prove that the l p spectrum is p independent 1998 academic press

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure