from geometric quantization to moyal quantization it is shown how the moyal product of phase space functions and the weyl correspondence between symbols and operator kernels may be obtained directly using the procedures of geometric quantization applied to the symplectic groupoid constructed by doubling the phase space

semi classical properties of berezin toeplitz operators with c k symbol we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

on the asymptotic expansion of bergman kernel we study the asymptotic of the bergman kernel of the spin c dirac operator on high tensor powers of a line bundle

semiclassical spectral analysis of toeplitz operators on symplectic manifolds the case of discrete wells we consider toeplitz operators associated with the renormalized bochner laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold we study the asymptotic behavior in the semiclassical limit of low lying eigenvalues and the corresponding eigensections of a self adjoint toeplitz operator under assumption that its principal symbol has a non degenerate minimum with discrete wells as an application we prove upper bounds for low lying eigenvalues of the bochner laplacian in the semiclassical limit

on the composition of berezin toeplitz operators on symplectic manifolds we compute the second coefficient of the composition of two berezin toeplitz operators associated with the text spin c dirac operator on a symplectic manifold making use of the full off diagonal expansion of the bergman kernel

analytic bergman operators in the semiclassical limit using a new quantization scheme we construct approximate semi classical bergman projections on weighted l spaces with analytic weights and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols as a corollary we obtain new estimates for asymptotic expansions of the bergman kernel on c n and for high powers of ample holomorphic line bundles over compact complex manifolds

a formal model of berezin toeplitz quantization we give a new construction of symbols of the differential operators on the sections of a quantum line bundle l over a kahler manifold m using the natural contravariant connection on l these symbols are the functions on the tangent bundle tm polynomial on fibres for high tensor powers of l the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on tm corresponding to some pseudo kahler structure on tm surprisingly this star product is intimately related to the formal symplectic groupoid with separation of variables over m we extend the star product on tm to generalized functions supported on the zero section of tm the resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the bergman projection operator using this idempotent we define an algebra of toeplitz elements and show that it is naturally isomorphic to the algebra of berezin toeplitz deformation quantization on m

deformed noncommutative tori we recall a construction of non commutative algebras related to a one parameter family of deformed spheres and tori and show that in the case of tori the algebras can be completed into c algebras isomorphic to the standard non commutative torus as the former was constructed in the context of matrix or fuzzy geometries it provides an important link to the framework of non commutative geometry and opens up for a concrete way to study deformations of non commutative tori furthermore we show how the well known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite dimensional representations of the deformed algebras and their projective modules are described together with connections of constant curvature

szegő limit theorems for singular berezin toeplitz operators abstract we consider berezin toeplitz operators whose multipliers are compactly supported densities carried by a submanifold c n we compute asymptotically the moments of their spectral measures and we prove szeg limit theorems in cases when is isotropic or co isotropic from which weyl estimates follow we also obtain asymptotics of the schatten norms of such operators rescaled versions of these operators can be thought of as quantum mechanical mixed states and our results give the semi classical limit of their entropy

low energy spectrum of toeplitz operators with a miniwell in the semiclassical limit it is well known that the first eigenvector of a toeplitz operator concentrates on the minimal set of the symbol in this paper we give a more precise criterion for concentration in the case where the minimal set of the symbol is a submanifold in the spirit of the miniwell condition of helffer sjostrand

asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels variational methods for parameter estimation are an active research area potentially offering computationally tractable heuristics with theoretical performance bounds we build on recent work that applies such methods to network data and establish asymptotic normality rates for parameter estimates of stochastic blockmodel data by either maximum likelihood or variational estimation the result also applies to various sub models of the stochastic blockmodel found in the literature

on the preservation of local asymptotic normality under information loss on considere des variables non observables x j n j 1 k n n 1 2 et le cas ou l on peut voir d autres variables y j n qui sont moins informatives que les x j n on montre que si les x j n satisfont certaines conditions telles que des conditions lan alors les y j n satisfont les memes conditions

quantized nambu poisson manifolds and n lie algebras we investigate the geometric interpretation of quantized nambu poisson structures in terms of noncommutative geometries we describe an extension of the usual axioms of quantization in which classical nambu poisson structures are translated to n lie algebras at quantum level we demonstrate that this generalized procedure matches an extension of berezin toeplitz quantization yielding quantized spheres hyperboloids and superspheres the extended berezin quantization of spheres is closely related to a deformation quantization of n lie algebras as well as the approach based on harmonic analysis we find an interpretation of nambu heisenberg n lie algebras in terms of foliations of rn by fuzzy spheres fuzzy hyperboloids and noncommutative hyperplanes some applications to the quantum geometry of branes in m theory are also briefly discussed

quantization effect on the log likelihood ratio and its application to decentralized sequential detection it is well known that quantization cannot increase the kullback leibler divergence which can be thought of as the expected value or first moment of the log likelihood ratio in this paper we investigate the quantization effects on the second moment of the log likelihood ratio it is shown via the convex domination technique that quantization may result in an increase in the case of the second moment but the increase is bounded above by 2 e the result is then applied to decentralized sequential detection problems not only to provide simpler sufficient conditions for asymptotic optimality theories in the simplest models but also to shed new light on more complicated models in addition some brief remarks on other higher order moments of the log likelihood ratio are also provided

bounds for fidelity of semiclassical lagrangian states in kahler quantization we define mixed states associated with submanifolds with probability densities in quantizable closed kahler manifolds then we address the problem of comparing two such states via their fidelity firstly we estimate the sub fidelity and super fidelity of two such states giving lower and upper bounds for their fidelity when the underlying submanifolds are two lagrangian submanifolds intersecting transversally at a finite number of points in the semiclassical limit secondly we investigate a family of examples on the sphere for which we manage to obtain a better upper bound for the fidelity we conclude by stating a conjecture regarding the fidelity in the general case

frechet algebraic deformation quantization of the poincare disk starting from formal deformation quantization we use an explicit formula for a star product on the poincare disk n to introduce a frechet topology making the star product continuous to this end a general construction of locally convex topologies on algebras with countable vector space basis is introduced and applied several examples of independent interest are investigated as e g group algebras over finitely generated groups and infinite matrices in the case of the star product on n the resulting frechet algebra is shown to have many nice features it is a strongly nuclear kothe space the symmetry group su 1 n acts smoothly by continuous automorphisms with an inner infinitesimal action and evaluation functionals at all points of n are positive functionals

on the lower order terms of the asymptotic expansion of tian yau zelditch we compute the first three coefficients of the asymptotic expansion of tian yau zelditch we also prove that in general the fcth coefficient is a polynomial of the curvature and its derivative of weight k

gl and geometric quantization an axiomatic approach to the approximation of infinite dimensional algebras is presented examples illustrating the need for a rigorous treatment of this subject geometric quantization is employed to construct systematicallysu n approximations of diffeomorphism algebras which first appeared in the theory of relativistic membranes

spectral asymptotics via the semiclassical birkhoff normal form this article gives a simple treatment of the quantum birkhoff normal form for semiclassical pseudo differential operators with smooth coefficients the normal form is applied to describe the discrete spectrum in a generalised non degenerate potential well yielding uniform estimates in the energy e this permits a detailed study of the spectrum in various asymptotic regions of the parameters e h and gives improvements and new proofs for many of the results in the field in the completely resonant case we show that the pseudo differential operator can be reduced to a toeplitz operator on a reduced symplectic orbifold using this quantum reduction new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved in the case of polynomial differential operators a combinatorial trace formula is obtained

the manifold of compatible almost complex structures and geometric quantization let m be an integral symplectic manifold we study a family of hermitian vector bundles on the space mathcal j of almost complex structures on m compatible with whose fibers consist of nearly holomorphic sections of powers of a prequantum line bundle we obtain asymptotics of the curvature of a natural connection in these bundles these results together with toeplitz operator theory provide another proof of donaldson s result that the action of ham m on mathcal j is hamiltonian with moment map the scalar curvature we also give an example involving teichmuller space and discuss a relationship between parallel transport and the schrodinger equation

deformation quantization of almost kahler models and lagrange finsler spaces finsler and lagrange spaces can be equivalently represented as almost kahler manifolds endowed with a metric compatible canonical distinguished connection structure generalizing the levi civita connection the goal of this paper is to perform a natural fedosov type deformation quantization of such geometries all constructions are canonically derived for regular lagrangians and or fundamental finsler functions on tangent bundles

deformation quantization and geometric quantization of abelian moduli spaces the berezin toeplitz deformation quantization of an abelian variety is explicitly computed by the use of theta functions an sl 2n equivariant complex structure dependent equivalence e between the constant moyal weyl product and this family of deformations is given this equivalence is seen to be convergent on the dense subspace spanned by the pure phase functions the toeplitz operators associated to the equivalence e applied to a pure phase function produces a covariant constant section of the endomorphism bundle of the vector bundle of theta functions for each level over the moduli space of abelian varieties

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

quantization and the hessian of mabuchi energy let l x be an ample bundle over a compact complex manifold fix a hermitian metric in l whose curvature defines a kahler metric on x the hessian of mabuchi energy is a fourth order elliptic operator d d on functions which arises in the study of scalar curvature we quantize d d by the hessian p k pk of balancing energy a function appearing in the study of balanced embeddings p k pk is defined on the space of hermitian endomorphisms of h0 x lk endowed with the l2 inner product we first prove that the leading order term in the asymptotic expansion of p k pk is d d we next show that if aut x l is discrete then the eigenvalues and eigenspaces of p k pk converge to those of d d we also prove convergence of the hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature kahler metric as consequences of our results we prove that an estimate of phong and sturm is sharp and give a negative answer to a question posed by donaldson we also discuss some possible applications to the study of calabi flow 2012

the semiclassical structure of low energy states in the presence of a magnetic field we consider a compact riemannian manifold with a hermitian line bundle whose curvature is non degenerate the laplacian acting on high tensor powers the semiclassical regime of the bundle exhibits a cluster of low energy states we demonstrate that the orthogonal projectors onto these states are the fourier components of an operator with the structure of the szego projector i e a fourier integral operator of hermite type this result yields semiclassical asymptotics for the low energy eigenstates

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

generalized bergman kernels and geometric quantization in geometric quantization it is well known that if f is an observable and f a polarization on a symplectic manifold m then the condition xf leaves f invariant where xf denotes the hamiltonian vector field associated to f is sufficient to guarantee that one does not have to compute the bks kernel explicitly in order to know the corresponding quantum operator it is shown in this paper that this condition on f can be weakened to xf leaves f f invariant and the corresponding quantum operator is then given implicitly by formula 4 8 in particular when f is a positive kahler polarization all observables can be quantized directly and moreover an explicit formula for the corresponding quantum operator is derived theorem 5 8 applying this to the phase space r2n one obtains a quantization prescription which ressembles the normal ordering of operators in quantum field theory when we translate this prescription to the usual position representation of quantum mechanics the result i

entanglement entropy and berezin toeplitz operators we consider berezin toeplitz operators on compact kahler manifolds whose symbols are characteristic functions when the support of the characteristic function has a smooth boundary we prove a two term weyl law the second term being proportional to the riemannian volume of the boundary as a consequence we deduce the area law for the entanglement entropy of integer quantum hall states another application is for the determinantal processes with correlation kernel the bergman kernels of a positive line bundle we prove that the number of points in a smooth domain is asymptotically normal

convergence of the wick star product we construct a frechet space as a subspace of c omega mathbb c n where the wick star product converges and is continuous the resulting frechet algebra mathcal a is studied in detail including a representation of mathcal a in the bargmann fock space and a discussion of star exponentials and coherent states

an obstruction to quantization of the sphere in the standard example of strict deformation quantization of the symplectic sphere s2 the set of allowed values of the quantization parameter is not connected indeed it is almost discrete li recently constructed a class of examples including s2 in which can take any value in an interval but these examples are badly behaved here i identify a natural additional axiom for strict deformation quantization and prove that it implies that the parameter set for quantizing s2 is never connected

quantization of kahler manifolds ii we use berezin s dequantization procedure to define a formal product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous kahler manifold m we prove that this formal product isconvergent when m isa hermitian symmetric space 1993 american mathematical society

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

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

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

spectral asymptotics of semiclassical unitary operators we introduce axiomatically a semiclassical quantization for non necessarily self adjoint operators then we focus on unitary operators and prove that in the semiclassical limit the convex hull of the joint spectrum of a finite commuting family of semiclassical unitary operators converges to the convex hull of the joint image of the principal symbols which can be shown to be a subset of a d torus mathbb s 1 d this result covers in particular hbar pseudodifferential and berezin toeplitz operators part of the paper is devoted to the definition of this notion of convex hull for subsets of tori the proof of our result builds on recent results for semiclassical self adjoint operators and involves the inverse cayley transform for unitary operators

semiclassical quantization and spectral limits of ħ pseudodifferential and berezin toeplitz operators we introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system this gives a quick alternative solution to the isospectrality problem for quantum toric systems if the operators are uniformly bounded the convergence is uniform analogous results hold for non commuting operators

asymptotic distribution of p values in composite null models abstract we investigate the compatibility of a null model h 0 with the data by calculating a p value that is the probability under h 0 that a given test statistic t exceeds its observed value when the null model consists of a single distribution the p value is readily obtained and it has a uniform distribution under h 0 on the other hand when the null model depends on an unknown nuisance parameter one must somehow get rid of e g by estimating it to calculate a p value various proposals have been suggested to remove each yielding a different candidate p value but unlike the simple case these p values typically are not uniformly distributed under the null model in this article we investigate their asymptotic distribution under h 0 we show that when the asymptotic mean of the test statistic t depends on the posterior predictive p value of guttman and rubin and the plug in p value are conservative i e their asymptotic distributions are more concentrated around 1 2 than a uni

categorical perspective on quantization of poisson algebra we propose a generalization of quantization using a categorical approach for a fixed poisson algebra quantization categories are defined as subcategories of the r module category equipped with the structure of classical limits we then construct the generalized quantization categories including matrix regularization strict deformation quantization prequantization and poisson enveloping algebra it is shown that the categories of strict deformation quantization prequantization and matrix regularization with certain conditions are equivalent categories on the other hand the categories of poisson enveloping algebra are not equivalent to the other categories

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

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

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

quantization towards a comparison between methods in this paper it is shown that the procedure of geometric quantiztion applied to kahler manifolds gives the following result the hilbert space h consists roughly speaking of holomorphic functions on the phase space m and to each classical observable f i e a real function on m is associated an operator f on h as follows first multiply by f 1 4 drf dr being the laplace de rham operator on the kahler manifold m and then take the holomorphic part see g m tuynman j math phys 27 573 1987 this result is correct on compact kahler manifolds and correct modulo a boundary term md on noncompact kahler manifolds in this way these results can be compared with the quantization procedure of berezin math ussr izv 8 1109 1974 9 341 1975 commun math phys 40 153 1975 which is strongly related to quantization by products e g see c moreno and p ortega navarro amn inst h poincare sec a 38 215 1983 lett math phys 7 181 1983 c moreno lett math phys 11 3

inverse spectral theory for semiclassical jaynes cummings systems quantum semitoric systems form a large class of quantum hamiltonian integrable systems with circular symmetry which has received great attention in the past decade they include systems of high interest to physicists and mathematicians such as the jaynes cummings model 1963 which describes a two level atom interacting with a quantized mode of an optical cavity and more generally the so called systems of jaynes cummings type in this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of jaynes cummings type we prove assuming the bohr sommerfeld rules hold that if the joint spectrum of two of these systems coincide up to mathcal o hbar 2 then the systems are isomorphic

toeplitz operators and hamiltonian torus actions this paper is devoted to semi classical aspects of symplectic reduction consider a compact prequantizable kahler manifold m with a hamiltonian torus action in the seminal paper v guillemin s sternberg geometric quantization and multiplicities of group representations invent math 67 3 1982 515 538 guillemin and sternberg introduced an isomorphism between the invariant part of the quantum space associated to m and the quantum space associated to the symplectic quotient of m provided this quotient is non singular we prove that this isomorphism is a fourier integral operator and that the toeplitz operators of m descend to toeplitz operators of the reduced phase space we also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced toeplitz operator a result related to the riemann roch kawasaki theorem

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

berezin toeplitz operators a semi classical approach this article is devoted to the toeplitz operators 4 in the context of the geometric quantization 11 15 we propose an ansatz for their schwartz kernel from this we deduce the main known properties of the principal symbol of these operators and obtain new results we define their covariant and contravariant symbols which are full symbols and compute the product of these symbols in terms of the kahler metric this gives canonical star products on the kahlerian manifolds this ansatz is also useful to introduce the notion of microsupport

asymptotic properties of the quantum representations of the mapping class group we establish two results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat su n bundle with hitchin s connection first we prove that these projective representations lift to asymptotic representations second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion the leading order term of this expansion agrees with the formula derived heuristically by e witten in quantum field theory and the jones polynomial

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n

generalized lorenz type systems exact general solutions of the lorenz system of differential equations with arbitrary parameters are unknown we first present exact particular solutions of this system for special values of the parameters in terms of jacobian elliptic functions but no sensitive dependence on the initial conditions leading to chaotic behavior is exhibited by these solutions we next consider two generalizations of the lorenz system involving quadratic and cubic terms and obtain their exact general solutions in terms of bessel and modified bessel functions exhibiting sensitive dependence on initial conditions for certain parameter ranges finally a generalized system involving arbitrary powers is reduced to the general duffing equation with damping which can be solved exactly when some of the parameters are interconnected the exact solutions presented showing precisely how chaotic behavior occurs are useful for testing conjectures about such differential systems as well as testing the validity and accuracy of approximate computer solutions for which numerical errors may grow exponentially fast

filtered likelihood for point processes abstract point processes are widely used in finance and economics to model the timing of defaults market transactions unemployment spells births and a range of other events we develop and analyze likelihood estimators for the parameters of a marked point process and incompletely observed explanatory factors that influence the arrival intensity and mark distribution we establish an approximation to the likelihood and analyze the convergence and large sample properties of the associated estimators numerical results illustrate the behavior of our estimators

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure

berezin transform on harmonic bergman spaces on the real ball abstract we provide the full asymptotic expansion of the harmonic berezin transform on the unit ball in r n purely by means of transformations of hypergeometric functions and function s hypergeometrization

semi classical properties of geometric quantization with metaplectic correction the geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure we prove that its semi classical limit does not depend on the choice of the complex structure we show this in two ways first by introducing unitary identifications between the quantum spaces associated to the various complex polarizations and second by defining an asymptotically flat connection in the bundle of quantum spaces over the space of complex structures furthermore berezin toeplitz operators are intertwined by these identifications and have principal and subprincipal symbols defined independently of the complex structure the relation with the schrodinger equation and the group of prequantum bundle automorphisms is considered as well

geometric quantization of vector bundles and the correspondence with deformation quantization i repeat my definition for quantization of a vector bundle for the cases of the toeplitz and geometric quantizations of a compact kahler manifold i give a construction for quantizing any smooth vector bundle which depends functorially on a choice of connection on the bundle