tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

berezin toeplitz quantization and composition formulas abstract extending results in l a coburn the measure algebra of the heisenberg group j funct anal 161 1999 509 525 l a coburn on the berezin toeplitz calculus proc amer math soc 129 11 2001 3331 3338 we derive composition formulas for berezin toeplitz operators with i g unbounded symbols in the range of certain integral transforms the question whether a finite product of berezin toeplitz operators is an operator of this type again can be answered affirmatively in several cases but there are also well known counter examples we explain some consequences of such formulas to c algebras generated by toeplitz operators

connections on discrete fibre bundles a new approach to gauge fields on a discrete space time is proposed in which the fundamental object is a discrete version of a principal fibre bundle if the bundle is twisted the gauge fields are topologically non trivial automatically

deformation quantizations with separation of variables on a kahler manifold we give a simple geometric description of all formal differentiable deformation quantizations on a kahler manifoldm such that for each open subsetu m multiplication from the left by a holomorphic function and from the right by an antiholomorphic function onu coincides with the pointwise multiplication by these functions we show that these quantizations are in 1 1 correspondence with the formal deformations of the original kahler metrics onm

ck invariant manifolds for maps of banach spaces abstract in this work we give a unified proof for the existence under appropriate gap conditions of the standard invariant manifolds for a ck map of a banach space near a fixed point where k 1 is an integer

topology of lattice gauge fields non abelian gauge fields on a four dimensional hypercubic lattice with small action density tr u dot p for su 2 gauge fields are shown to carry an integer topological chargeq which is invariant under continuous deformations of the field a concrete expression forq is given and it is verified thatq reduces to the familiar chern number in the classical continuum limit

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

invariant pre foliations for non resonant non uniformly hyperbolic systems given an orbit whose linearization has invariant subspaces satisfying some non resonance conditions in the exponential rates of growth we prove existence of invariant manifolds tangent to these subspaces the exponential rates of growth can be understood either in the sense of lyapunov exponents or in the sense of exponential dichotomies these manifolds can correspond to slow manifolds which characterize the asymptotic convergence let x i i n be a regular orbit of a c 2 dynamical system f let s be a subset of its lyapunov exponents assume that all the lyapunov exponents in s are negative and that the sums of lyapunov exponents in s do not agree with any lyapunov exponent in the complement of s denote by e s xi the linear spaces spanned by the spaces associated to the lyapunov exponents in s we show that there are smooth manifolds w s xi such that f w s xi w s xi 1 and t xi w s xi e s xi we establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non resonance conditions these systems of invariant manifolds are not in general a foliation

non perturbative deformation quantization of cartan domains abstract we construct families of non commuting c algebras of quantized functions for bounded irreducible hermitian symmetric spaces for this procedure we use algebras of toeplitz operators defined with respect to a perturbation of the ordinary bergman metric we prove the deformation quantization conditions for these algebras

high power asymptotics of some weighted harmonic bergman kernels abstract for weights which are either radial on the unit ball or depend only on the vertical coordinate on the upper half space we describe the asymptotic behaviour of the corresponding weighted harmonic bergman kernels with respect to as this can be compared to the analogous situation for the holomorphic case which is of importance in the berezin quantization as well as in complex geometry

hermite polynomials and quasi classical asymptotics we study an unorthodox variant of the berezin toeplitz type of quantization scheme on a reproducing kernel hilbert space generated by the real hermite polynomials and work out the associated quasi classical asymptotics

the e8 lattice and quasicrystals geometry number theory and quasicrystals the authors study the quasiperiodic structures which can be derived from e8 this lattice suitably oriented leads to a 4d quasicrystal which has 3 3 5 symmetry they develop a modified version of the out and projection method the high dimensional lattice is first foliated into successive shells surrounding a vertex the shells are embedded into 7d s7 spheres they then use the so called hopf fibration of s7 to gather families of e8 shell sites into s3 fibres which contain 24 sites or a multiple of 24 which are symmetrically disposed in s3 this has two advantages first in the selection process a whole fibre is either selected or rejected which reduces the computation more importantly the selection process for a fiber amounts to simple arithmetical criteria the whole process leads to a shell by shell analysis of the 4d quasicrystal which can be brought onto a 2d 1d algorithm similar to the fibonacci chain construction they propose an arithmetic formula which gives the number of points on the shells this 4d quasicrystal has the interesting feature that it can be sliced into lower dimensional quasicrystals for example with icosahedral and tetrahedral symmetry in three dimensions

behavior of the bergman kernel at infinity we give a precise decay rate of the bergman kernel and metric at infinity on model domains characterized in terms of certain convex polyhedron

differentiability at the tip of arnold tongues for diophantine rotations numerical studies and renormalization group explanations we study numerically the regularity of arnold tongues corresponding to diophantine rotation numbers of circle maps at the edge of validity of kam theorem this serves as a good test for the numerical stability of two different algorithms we find empirically that arnold tongues are only finitely differentiable at the tip we also find several scaling properties of the sobolev norms of the conjugacy near the breakdown we also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the sobolev regularity we also uncover empirically some other patterns which require explanation

finite tycz expansions and csck metrics abstract let m g be a kahler manifold whose associated kahler form is integral and let l h m be a quantization hermitian line bundle in this paper we study those kahler manifolds m g admitting a finite tycz expansion namely those for which the associated kempf distortion function t m g is of the form t m g p f s p m s f s 1 p m s 1 f r p m r f j c m s r z we show that if the tycz expansion is finite then t m g is indeed a polynomial in m of degree n n dim c m and the log term of the szego kernel of the disc bundle d l vanishes where l is the dual bundle of l moreover we provide a complete classification of the kahler manifolds admitting finite tycz expansion either when m is a complex curve or when m is a complex surface with a csck metric which admits a radial kahler potential

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

division algebras with integral elements pairing two elements of a given division algebra furnished with a multiplication rule leads to an algebra of higher dimension restricted by 8 this fact is used to obtain the roots of so 4 and sp 2 from the roots or 1 of su 2 and the weights or 1 2 of its spinor representation the root lattice of so 8 described by 24 integral quaternions are obtained by pairing two sets of roots of sp 2 the root system of f4 is constructed in terms of 24 integral and 24 half integral quaternions the root lattice of e8 expressed as 240 integral octonions are obtained by pairing two sets of roots of f4 twenty four integral quaternions of so 8 forming a discrete subgroup of su 2 are shown to be the automorphism group of the root lattices of so 8 f4 and e8 the roots of maximal subgroups so 16 e7 su 2 e6 su 3 su 9 and su 5 su 5 of e8 are identified with a simple method subsets of the discrete subgroup of su 2 leaving maximal subgroups of e8 are obtained constructions of e8 root lattice with integral octonions in seven distinct ways are made magic squares of integral lattices of goddard nahm olive ruegg and schwimmer are derived possible physical applications are suggested

convex hypersurfaces and fourier transforms hilbert transforms and maximal functions along curves and surfaces spectral synthesis problems and the study of certain operators related to hyperbolic partial differential and pseudodifferential operators the problem of estimating such fourier transforms has a long history see for example hlawka 3 herz 2 littman 4 randol 9 10 svensson 15 sogge and stein 12 13 stein 14 marshall 5 etc the results of these investigations suggest a strong relation between the local curvature properties of the surface s and the decay of the function a we are mainly interested in how the geometry of s near a fixed point xo affects the decay of a6 for by using a partition of unity on s it is clearly enough to study integrals of the type

newton polyhedra and the bergman kernel the purpose of this paper is to study singularities of the bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities under some assumptions on a domain in n 1 the bergman kernel b z of takes the form near a boundary point p b z frac phi w rho rho 2 2 d f hbox log 1 rho m f 1 where w is some polar coordinates on a nontangential cone with apex at p and means the distance from the boundary here admits some asymptotic expansion with respect to the variables 1 m and log 1 as 0 on the values of df 0 mf and m are determined by geometrical properties of the newton polyhedron of defining functions of domains and the limit of as 0 on is a positive constant depending only on the newton principal part of the defining function analogous results are obtained in the case of the szego kernel

a forelli rudin construction and asymptotics of weighted bergman kernels let be a pseudoconvex domain in cn with smooth boundary two smooth defining functions for 0 such that log log are plurisubharmonic z a point at which log is strictly plurisubharmonic and m 0 an integer we show that as k the bergman kernels with respect to the weights k m have the asymptotic expansionk k m z z kn n z k z m j 0 bj z k j b0 det 2log zj zk for strongly pseudoconvex with real analytic boundary real analytic and log log strictly plurisubharmonic on we obtain also the analogous result for k k m x y for x y near the diagonal and discuss consequences for the asymptotics of the berezin transform and for the berezin quantization the proofs rely on fefferman s expansion for the bergman kernel in a certain forelli rudin type domain over as another application they also yield a generalization of the cited fefferman s expansion to a class of weighted bergman kernels

berezin transforms on pluriharmonic bergman spaces we show that perhaps surprisingly in several aspects the behaviour of the reproducing kernels of toeplitz operators and of the berezin transform on some weighted pluriharmonic bergman spaces is the same as in the holomorphic case

berezin toeplitz quantization on kaehler manifolds we study the berezin toeplitz quantization on kaehler manifolds we explain first how to compute various associated asymptotic expansions then we compute explicitly the first terms of the expansion of the kernel of the berezin toeplitz operators and of the composition of two berezin toeplitz operators as application we estimate the norm of donaldson s q operator

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

boundary invariants of pseudoconvex domains the purpose of this paper is to describe an invariant ax z which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in cn the motivation for defining the invariant ax z referred to as the multitype of z comes from the study of the boundary regularity properties of solutions of the d neumann problem on domains of finite type by a domain of finite type we mean one that satisfies the definition given by d angelo 5 namely a domain such that at any boundary point the maximum order of contact of one dimensional complex analytic varieties with the boundary is bounded in 4 the author showed how one can give a proof of global regularity of the a neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper in a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains before giving the definition of the multitype ax z we must introduce some

sub stable and weak stable manifolds associated with finitely non resonant spectral subspaces in this work we study c k delta 2 leq k leq infty 0 leq delta leq 1 maps of a banach space near a fixed point we show the existence and uniqueness of a class of c k delta local invariant sub manifolds of the stable manifold which correspond to a spectral subspace satisfying a finite non resonance condition of order l leq k and an overriding condition of order l leq k condition 3 of theorem 1 we study the dependence of these invariant manifolds on a parameter that lies in a banach space we also show that a c k delta local weak stable manifold that satisfies these two conditions is unique in the class of c k delta maps the uniqueness is due to the fact that our method does not require a cut off function an infinite dimensional banach space does not always admit smooth cut off functions

berezin transform on the harmonic fock space abstract we show that the berezin transform associated to the harmonic fock segal bargmann space on c n has an asymptotic expansion analogously as in the holomorphic case the proof involves a computation of the reproducing kernel which turns out to be given by one of horn s hypergeometric functions of two variables and an ad hoc determination of the asymptotic behaviour of the resulting integrals to which the ordinary stationary phase method is not directly applicable

a closed formula for the asymptotic expansion of the bergman kernel we prove a graph theoretic closed formula for coefficients in the tian yau zelditch asymptotic expansion of the bergman kernel the formula is expressed in terms of the characteristic polynomial of the directed graphs representing weyl invariants the proof relies on a combinatorial interpretation of a recursive formula due to m englis and a loi

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

toeplitz quantization on fock space for toeplitz operators t f t acting on the weighted fock space h t 2 we consider the semi commutator t f t t g t t fg t where t 0 is a certain weight parameter that may be interpreted as planck s constant hbar in rieffel s deformation quantization in particular we are interested in the semi classical limit begin equation tag lim limits t to 0 t f t t g t t fg t t end equation it is well known that t f t t g t t fg t t tends to 0 under certain smoothness assumptions imposed on f and g this result was recently extended to f g in textup buc c n by bauer and coburn we now further generalize to not necessarily bounded uniformly continuous functions and symbols in the algebra textup vmo cap l infty of bounded functions having vanishing mean oscillation on mathbb c n our approach is based on the algebraic identity t f t t g t t fg t h bar f t h g t where h g t denotes the hankel operator corresponding to the symbol g and norm estimates in terms of the weighted heat transform as a consequence only f or likewise only g has to be contained in one of the above classes for to vanish for g we only have to impose limsup t to 0 h g t t infty e g g in l infty c n we prove that the set of all symbols f in l infty mathbb c n with the property that lim t rightarrow 0 t t ft t g t t fg t lim t rightarrow 0 t g t t f t t gf t t 0 for all g in l infty mathbb c n coincides with textup vmo cap l infty additionally we show that lim limits t to 0 t f t t f infty holds for all f in l infty c n finally we present new examples including bounded smooth functions where does not vanish

some remarks on pathologies of renormalization group transformations for the ising model the results recently obtained by van enter fernandez and sokal on non gibbsianness of the measurev t b h arising from the application of a single decimation transformationt b with spacingb to the gibbs measure h of the ising model for suitably chosen large inverse temperature and nonzero external fieldh are critically analyzed in particular we show that if keeping fixed the same values of h andb one iterates a sufficiently large number of timesn the transformationt b one obtains a new measurev t b n h which is gibbsian and moreover very weakly coupled

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

weighted bergman kernels and quantization let be a bounded pseudoconvex domain in c n two positive functions on such that log log are plurisubharmonic and z a point at which log is smooth and strictly plurisubharmonic we show that as k the bergman kernels with respect to the weights k have an asymptotic expansion for x y near z where x y is an almost analytic extension of phi x x x and similarly for further if in addition is of finite type behave reasonably at the boundary and log log are strictly plurisubharmonic on we obtain also an analogous asymptotic expansion for the berezin transform and give applications to the berezin quantization finally for smoothly bounded and strictly pseudoconvex and a smooth strictly plurisubharmonic defining function for we also obtain results on the berezin toeplitz quantization

berezin transform of two arguments abstract we describe the asymptotic behavior of the berezin transform of two arguments which generalizes the standard notion of the berezin transform the asymptotic analysis we give is in the setting of the holomorphic and the harmonic fock spaces respectively

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

identification of berezin toeplitz deformation quantization we give a complete identification of the deformation quantization which was obtained from the berezin toeplitz quantization on an arbitrary compact kahler manifold the deformation quantization with the opposite star product proves to be a differential deformation quantization with separation of variables whose dassifying form is explicitly calculated its characteristic dass which dassifies star products up to equivalence is obtained the proof is based on the microlocal description of the szego kernel of a strictly pseudoconvex domain given by boutet de monvel and sjostrand

on asymptotic expansion of the harmonic berezin transform on the half space abstract we give a complete asymptotic expansion of the berezin transform associated to the bergman space of harmonic functions on the half space h x y r n 1 x r n y 0 that are square integrable with respect to the measure y d x d y 0 as

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

on canonical metrics on cartan hartogs domains the cartan hartogs domains are defined as a class of hartogs type domains over irreducible bounded symmetric domains the purpose of this paper is twofold firstly for a cartan hartogs domain omega b d 0 mu endowed with the canonical metric g mu we obtain an explicit formula for the bergman kernel of the weighted hilbert space mathcal h alpha of square integrable holomorphic functions on left omega b d 0 mu g mu right with the weight exp alpha varphi where varphi is a globally defined kahler potential for g mu for alpha 0 and furthermore we give an explicit expression of the rawnsley s varepsilon function expansion for left omega b d 0 mu g mu right secondly using the explicit expression of the rawnsley s varepsilon function expansion we show that the coefficient a 2 of the rawnsley s varepsilon function expansion for the cartan hartogs domain left omega b d 0 mu g mu right is constant on omega b d 0 mu if and only if left omega b d 0 mu g mu right is biholomorphically isometric to the complex hyperbolic space so we give an affirmative answer to a conjecture raised by m zedda

invariant manifolds associated to nonresonant spectral subspaces we show that if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain nonresonance conditions the map leaves invariant a smooth manifold tangent to this subspace this manifold is as smooth as the map when the smoothness is measured in appropriate scales but is unique amongcl invariant manifolds wherel depends only on the spectrum of the linearization or on some more general smoothness classes that we detail we show that if the nonresonance conditions are not satisfied a smooth invariant manifold need not exist and we also establish smooth dependence on parameters we also discuss some applications of these invariant manifolds and briefly survey related work

period doubling bifurcations for families of maps on ℝ n infinite sequences of period doubling bifurcations in one parameter families 1 pf of maps enjoy very strong universality properties this is known numerically in a multitude of cases and has been shown rigorously for certain 1 pf of maps on the interval these bifurcations occur in 1 pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 4 6692 in this paper we indicate the main steps of a proof that the same is true for 1 pf of analytic maps from n to n whose restriction to n is real

lattice gauge fields principal bundles and the calculation of topological charge su2 valued lattice gauge fields are studied on a 4 dimensional simplicial lattice ifu has sufficiently small plaquette products then there is a unique principal su2 bundle admitting transition functions defined on the intersections of adjacent dual cells which take values within 8 ofu an algorithm is explicitly given which associates an integer to everyu off a certain set of measure zero this algorithm only involves evaluation of 4 4 determinants and the solution of quadratic equations whenu is as above the integer produced is the second chern number of i e the topological charge ofu

the explicit forms and zeros of the bergman kernel function for hartogs type domains abstract we define the cartan hartogs domain which is the hartogs type domain constructed over the product of bounded hermitian symmetric domains and compute the explicit form of the bergman kernel for the cartan hartogs domain using the virtual bergman kernel as the main contribution of this paper we show that the main part of the explicit form of the bergman kernel is a polynomial whose coefficients are combinations of stirling numbers of the second kind using this observation as an application we give an algorithmic procedure to determine the condition that their bergman kernel functions have zeros

deformation estimates for the berezin toeplitz quantization deformation estimates for the berezin toeplitz quantization ofcn are obtained these estimates justify the description of ccr k as a first order quantum deformation of ap c0 where ccr is the usualc algebra of boson canonical commutation relations k is the full algebra of compact operators ap is the algebra of almost periodic functions andc0 is the algebra of continuous functions which vanish at infinity

a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps rigorous results abstract in this paper we prove rigorous results on persistence of invariant tori and their whiskers the proofs are based on the parameterization method of x cabre e fontich r de la llave the parameterization method for invariant manifolds i manifolds associated to non resonant subspaces indiana univ math j 52 2 2003 283 328 x cabre e fontich r de la llave the parameterization method for invariant manifolds ii regularity with respect to parameters indiana univ math j 52 2 2003 329 360 the invariant manifolds results proved here include as particular cases of the usual strong stable and strong unstable manifolds but also include other non resonant manifolds the method lends itself to numerical implementations whose analysis and implementation is studied in a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical algorithms preprint 2005 a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical implementation and examples preprint 2005 the results are stated as a posteriori results namely that if one has an approximate solution which is not degenerate then one has a true solution not too far from the approximate one this can be used to validate the results of numerical computations

asymptotic behaviour of reproducing kernels of weighted bergman spaces let q be a domain in c f a nonnegative and g a positive function on q such that 1 g is locally bounded a2 the space of all holomorphic functions on q square integrable with respect to the measure f0gda where da is the 2n dimensional lebesgue measure and k x y the reproducing kernel for a2 it has been known for a long time that in some special situations such as on bounded symmetric domains q with g 1 and f the bergman kernel function the formula lim k x x 1 0 1 f x ae 00 holds true this fact even plays a crucial role in berezin s theory of quantization on curved phase spaces in this paper we discuss the validity of this formula in the general case the answer turns out to depend on loosely speaking how well the function log f can be approximated by certain pluriharmonic functions lying below it for instance holds if log f is convex and hence can be approximated from below by linear functions for any function g counterexamples are also given to show that in general may fail drastically or even be true for some x and fail for the remaining ones finally we also consider the question of convergence of k0 x y 1 0 for x y which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of f for instance if f is not real analytic at some point then kc x y must have zeroes for all at sufficiently large