multiple optima and asymptotic approximations in the partial adjustment model abstract this paper examines statistical problems which arise in empirical applications of the partial adjustment model with autoregressive errors when the model is nearly nonidentified the results of monte carlo experiments show that the nls estimation criterion function is multipeaked with high probability when the model is nearly nonidentified in the cases examined the finite sample distributions of the nls estimators and the wald test statistics are poorly approximated by their asymptotic distributions the asymptotic approximation works better for the likelihood ratio lr test statistics but still can be unsatisfactory when the wald and lr tests are based on bootstrap critical values the size distortions are effectively eliminated

on largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces we apply topological methods to study the smallest non zero number lambda 1 in the spectrum of the laplacian on finite area hyperbolic surfaces for closed hyperbolic surfaces of genus two we show that the set s in mathcal m 2 lambda 1 s frac 1 4 is unbounded and disconnects the moduli space mathcal m 2 using this for genus g ge 3 we show the existence of eigenbranches that start as lambda 1 and eventually becomes frac 1 4

disappearance of cusp forms in special families in his 1954 g6ttingen lectures atle selberg sb posed the question of the asymptotic density of the poles of the eisenstein series for r a cofinite non cocompact subgroup of psl 2 r the question is known to be equivalent to the asymptotic count of the number of eigenvalues for the laplacebeltrami operator d acting in l2 h r where h is the upper halfplane with hyperbolic metric in general the spectral decomposition of d consists of the continuous band 1 4 oo with multiplicity the number of cusps of h r and a discrete component possibly in 1 4 0o spanned by eigenfunctions w1i wp2 see hj1 kb lxp for the eigenvalues aj dpj ajpj o consider the spectral counting function n r ij v n 0 see dips ps2 cor 2 3 luo lu recently established the unconditional result his argument does not involve the lindel6f hypothesis deshouillers et

persistence of embedded eigenvalues we consider conditions under which an embedded eigenvalue of a self adjoint operator remains embedded under small perturbations in the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m infinity we show that in favorable situations the set of small perturbations of a suitable banach space which do not remove the eigenvalue form a smooth submanifold of codimension in we also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite

seiberg witten invariants orbifolds and circle actions the main result of this paper is a formula for calculating the seiberg witten invariants of 4 manifolds with fixed point free circle actions this is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4 manifold and the moduli space of the quotient 3 orbifold two corollaries include the fact that b 1 4 manifolds with fixed point free circle actions are simple type and a new proof of the equality sw y 3 s 1 sw y 3 an infinite number of 4 manifolds with b 1 whose seiberg witten invariants are still diffeomorphism invariants is constructed and studied

the asymptotic behavior of green s functions for degenerating hyperbolic surfaces given a degenerating family of riemann surfaces of genus g 2 it is a natural and important question to study their spectral behaviors in particular green s functions during degeneration see 9 10 17 11 13 and 5 by the uniformization theorem any riemann surface of genus g 2 has a canonical metric of constant curvature 1 from now on we call a riemann surface with this canonical metric a hyperbolic surface and a degenerating family of hyperbolic surfaces is denoted by st l 0 the degeneration of st is caused by the pinching of several simple closed geodesics 71 l 7 1 on st that is the lengths of these geodesics go to zero as l 0 let at be the laplacian of st then the kernel of at consists of functions which are constant on each connected component of st and at is invertible on the orthogonal complement of the kernel the schwartz kernel of this inverse is called the green function of st and denoted by g z w t for z w sl we would like to study the asymptotic behavior of g z w i as l o in order to compare functions on s o and st we use the harmonic map of infinite energy so s 71 l 7 l constructed by wolf el4 intuitively speaking this map z opens up each node a pair of cusps of so into a simple closed pinching geodesic of sf at first we assume that there is only one pinching geodesic l 7 of length l on s if 7 separates st then so has two components se and so z of genus g gz respectively

the second moment of gl 4 gl 2 l functions at special points abstract in this paper we obtain upper bounds for the second moment of l u j 1 2 i t j where is a hecke maass form for s l 4 z and u j is taken from an orthonormal basis of hecke maass forms on s l 2 z with eigenvalue 1 4 t j 2 the bounds are consistent with the lindelof hypothesis previously these types of upper bounds are available for only g l n g l 2 where n 3

strong consistency of approximate maximum likelihood estimators with applications in nonparametrics wald s general analytic conditions that imply strong consistency of the approximate maximum likelihood estimators amles have been extended by le cam kiefer and wolfowitz huber bahadur and perlman all these conditions use the log likelihood ratio of the type log f x 0 f x 60 where oo is the true value of the parameter however these methods usually fail in the nonparametric case thus in this paper for each 0 00 we look at the log likelihood ratio of the type log f x 0 f x r 0 where o 0 is a certain parameter selected in a neighborhood v of 60 some general analytic conditions that imply strong consistency of the amle are given the results are shown to be applicable to several nonparametric families having densities e g concave distributions functions and increasing failure rate distributions in particular they can be applied to several censored data cases

harmonic differentials and infinite geodesic joining two punctures on a riemann surface let m be a hyperbolic riemann surface of finite volume the harmonic dual form to an infinite geodesic joining two punctures on m is obtained in two different ways first of all using the degeneration of hyperbolic eisenstein series it is made explicit in terms of these secondly generalizing the construction of kudla and millson to the case of an infinite geodesic joining two punctures we give an automorphic realization of this harmonic form

an inconsistent maximum likelihood estimate abstract an example is given of a family of distributions on 1 1 with a continuous one dimensional parameterization that joins the triangular distribution when 0 to the uniform when 1 for which the maximum likelihood estimates exist and converge strongly to 1 as the sample size tends to infinity whatever be the true value of the parameter a modification that satisfies cramer s conditions is also given

degenerescence de series d eisenstein hyperboliques in this work we deal with hyperbolic eisenstein series and more particularly given a degenerating family sl l 0 of riemann surfaces with their canonical hyperbolic metrics we work out the degeneration of hyperbolic eisenstein series associated to the pinching geodesics in sl our principal results are theorems 2 2 4 1 et 4 2

asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces letz s r be the selberg zeta function of a compact riemann surfacer we study the behavior ofz s r asr tends to infinity in the moduli space of stable curves the main result is an estimate forz s r valid fors in a neighborhood depending only on the genus ofs 1 our analysis gives an alternate proof of the belavin knizhnik double pole result 5

consistent maximum likelihood estimation with dependent observations the general non normal case and the normal case abstract in this paper we aim to establish intuitively appealing and verifiable conditions for the existence and weak consistency of ml estimators in a multi parameter framework assuming neither the independence nor the identical distribution of the observations the paper has two parts in the first part theorems 1 and 2 we assume that the joint density of the observations is known except for the values of a finite number of parameters to be estimated but we do not specify this distribution in the second part theorems 3 6 we do specify the distribution and assume joint normality but not independence of the observations some examples are also provided

on the resolvent of the laplacian on functions for degenerating surfaces of finite geometry we consider families y n of degenerating hyperbolic surfaces the surfaces are geometrically finite of fixed topological type let z n be the selberg zeta function of y n and let z d n be the contribution of the pinched geodesics to z n extending a result of wolpert s we prove that z n s z d n s converges to the zeta function of the limit surface for all s with mathrm re s 1 2 the technique is an examination of resolvent of the laplacian which is composed from that for elementary surfaces via meromorphic fredholm theory the resolvent delta n t 1 is shown to converge for all totin 1 4 infty we also use this property to define approximate eisenstein functions and scattering matrices

the asymptotics of the arakelov geen s function and faltings delta invariant we study the behavior of the arakelov green s function and faltings delta invariant on degenerating riemann surfaces

symplectic 4 manifolds with kodaira dimension zero we discuss the notion of the kodaira dimension for symplectic manifolds in dimension 4 in particular we propose and partially verify betti number bounds for symplectic 4 manifolds with kodaira dimension zero

simply connected symplectic 4 manifolds with b2 1 and c12 2 in this article we construct a new simply connected symplectic 4 manifold with b2 1 and c12 2 which is homeomorphic but not diffeomorphic to a rational surface by using rational blow down technique as a corollary we conclude that a rational surface mathbf c p 2 sharp 7 overline mathbf c p 2 admits an exotic smooth structure

deformation of compact clifford klein forms of indefinite riemannian homogeneous manifolds 1 1 let g be a lie group and h a closed subgroup of g if a discrete subgroup of g acts properly discontinuously and freely on g h then the double coset space g h carries naturally a manifold structure such that the quotient map g h g h is locally diffeomorphic the manifold g h is said to be a clifford klein form of g h if it is compact then is said to be a uniform lattice for g h a typical example is a compact riemann surface mg with genus g 2 which is biholomorphic to a compact clifford klein form of the poincare disk g h sl 2 r so 2 by the uniformization theorem it is important from geometric view point that a clifford klein form g h inherits any g invariant local geometric structure on g h such as indefinite riemannian metric complex structure symplectic structure causal structure and so on

double node neighborhoods and families of simply connected 4 manifolds with b 1 journal of the american mathematical society volume 19 number 1 pages 171 180 s 0894 0347 05 00500 x article electronically published on august 15 2005 double node neighborhoods and families of simply connected 4 manifolds with b 1 ronald fintushel and ronald j stern 1 introduction a basic question of 4 manifold topology is whether the complex projective plane cp 2 admits exotic smooth structures thus one is interested in knowing the smallest m for which cp 2 m cp admits an exotic smooth structure in the late 1980s dieter kotschick k proved that the barlow surface which was known to be homeomorphic to cp 2 8cp is not di eomorphic to it in the following years the subject of simply connected smooth 4 manifolds with b 1 languished because of a lack of suitable examples however largely due to a beautiful paper of jongil park p who found the rst examples of exotic simply connected 4 manifolds with b 1 and b 7 the past year has found renewed interest in this subject peter ozsvath and zoltan szabo os proved that park s manifold is minimal and andras stipsicz and szabo ss used a technique similar to park s to construct an exotic manifold with b 1 and b 6 the rational elliptic surface e 1 cp 2 9 cp admits in nitely many distinct smooth structures up to now it has not been known whether cp 2 m cp can have an in nite family of smooth structures when m 9 it is the purpose of this paper to introduce a new technique which we use to show that for m 6 7 8 cp 2 m cp does have an in nite family of smooth structures the construction of these examples to some extent follows the ideas of park p after performing suitable knot surgeries fs3 on e 1 the essential new idea is that after certain knot surgeries on e 1 one is able to nd an immersed sphere of self intersection 1 representing the pseudo section this is accomplished by studying double node neighborhoods as described below shortly after the appearance of a preliminary version of this article park stip sicz and szabo pss used the techniques described herein to give even better examples an in nite family of pairwise nondi eomorphic smooth 4 manifolds all homeomorphic to cp 2 5 cp in the nal section of this paper we give a quick construction of such a family of examples received by the editors january 13 2005 2000 mathematics subject classi cation primary 14j26 53d05 57r55 57r57 key words and phrases 4 manifold seiberg witten invariant knot surgery rational blowdown the rst author was partially supported by nsf grant dms0305818 and the second author by nsf grant dms0204041 c 2005 american mathematical society reverts to public domain 28 years from publication

surfaces with boundary their uniformizations determinants of laplacians and isospectrality let be a compact surface of type g n n 0 obtained by re moving n disjoint disks from a closed surface of genus g assuming 0 while osgood phillips and sarnak o p s3 showed the properness when g 0

stability of spectra of hodge de rham laplacians in this paper we investigate the stability of eigenspaces of the laplace operator acting on differential forms satisfying relative or absolute boundary conditions on a compact oriented riemannian manifold with boundary this includes in particular both neumann and dirichlet conditions for the laplace beltrami operator on functions more precisely our main result is that the gap between corresponding eigenspaces precise definition will be recalled below measured using the l norm converges to zero when smooth metrics g converge to g0 in the c 1 topology it is quite well known cf 3 or 14 that the eigenvalues of the laplacian vary continuously under c 0 continuous perturbations of the metric it is perhaps less well known but implicit in the work of cheeger 3 that eigenspaces vary continuously as subspaces of l2 when the metric is perturbed c 0 continuously we reprove this c 0 l2 stability in sect 4 for completeness and in order to be able to use certain notation conventions and partial results in the proof of c 1 l stability in sect 5 the second section of the paper contains a review of the hodge theory for the laplace operator with absolute and relative boundary conditions we also state here the sobolev embedding theorems and the basic a priori estimates for the square root d of the laplacian we need to work with d rather than d 2 since the coefficients of depend on the second derivatives of the metric tensor and we allow only c 1 continuous perturbations of the metric and do not assume any bounds on the second derivatives in the third section we review following kato 12 and osborn 16 general results from functional analysis concerning perturbation theory for compact operators on banach spaces that reduce proving convergence of eigenvalues and eigenspaces

new symplectic 4 manifolds with b 1 in 1995 dusa mcduff and dietmar salamon conjectured the existence of symplectic 4 manifolds x which satisfy b 1 k2 0 k 0 and which fail to be of lefschetz type this is equivalent to finding a symplectic homology t2 s2 manifold with nontorsion canonical class and a cohomology ring which is not isomorphic to the cohomology ring of t2 s2 they needed such examples to complete a list of possible symplectic 4 manifolds with b 1 in that same year tian jun li and ai ko liu working from a different point of view questioned whether there existed symplectic 4 manifolds with b 1 with seiberg witten invariants that did not depend on the chamber structure of the moduli space the purpose of this paper is to construct an infinite number of examples which satisfy both requirements

deformation of γ0 5 cusp forms we develop an algorithm for numerical computation of the eisenstein series on a riemann surface of constant negative curvature we focus in particular on the computation of the poles of the eisenst

decomposition spectrale de 1 formes differentielles sur une surface de riemann et series d eisenstein on etablit la formule de decomposition spectrale des 1 formes differentielles sur une surface de riemann non compacte m en introduisant le fibre en spheres de m on explicite egalement les series d eisenstein qui entrent en jeu on en donne quelques proprietes notamment dans le cas de la surface gamma 2 backslash h ou gamma 2 est le sous groupe de congruence principal d ordre 2

on the existence and uniqueness of the maximum likelihood estimate of a vector valued parameter in fixed size samples the maximum likelihood estimate is shown to exist and to be unique if a twice continuously differentiable likelihood function is constant on the boundary of the parameter space and if the hessian matrix is negative definite whenever the gradient vector vanishes the condition of constancy on the boundary cannot be completely removed cf tarone and gruenhage 1975 the theory is illustrated with several examples

tracking eigenvalues to the frontier of moduli space i convergence and spectral accumulation abstract we consider the spectral behavior of the laplace beltrami operator associated to a class of singular perturbations of a riemannian metric on a complete manifold the class of perturbations generalizes the well known opening node perturbation of teichmuller theory in particular we recover results of l ji and m zworski j func anal 114 1993 412 420 and s wolpert invent math 108 1992 91 129 from our more general methods

convergence of the normalized spectral counting function on degenerating hyperbolic riemann surfaces of finite volume abstract in this paper we study spectral asymptotics of degenerating families of hyperbolic riemann surfaces either compact or non compact but always of finite volume we prove that the second integral of the spectral counting function has an asymptotic expansion out to o l where l is the degeneration parameter the first term in the expansion is a diverging term which depends solely on the degeneration parameter and the counting parameters and the second term is the second integral of the spectral counting function of the limit surface

on selberg s small eigenvalue conjecture and residual eigenvalues we show that selberg s eigenvalue conjecture concern ing small eigenvalues of the automorphic laplacian for congruence groups is equivalent to a conjecture about the non existence of residual eigenvalues for a perturbed system we prove this using a combination of methods from asymptotic perturbation theory and number theory

non completeness of zograf takhtajan s kahler metric for teichmuller space of punctured riemann surfaces zograf and takhtajan introduced a new kahler metric on the teichmuller space tg n n 0 in calculating the first chern form of the quillen metric for families of operators the metric is described in terms of the eisenstein maass series we prove that it is incomplete and we also give an alternative proof of non completeness of the weil petersson metric for that we use the pinching family constructed by wolpert whose tangent vectors are always represented by using the relative poincare series associated with the pinched geodesic

grafting hyperbolic metrics and eisenstein series the family hyperbolic metric for the plumbing variety zw t and the non holomorphic eisenstein series e zeta 2 are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of riemann surfaces applications include an asymptotic expansion for the weil petersson metric and a local form of symplectic reduction

a nonnested approach to testing continuous time models against discrete alternatives abstract this paper considers a test of two competing models when one is formulated in continuous time as a system of stochastic differential equations and the other is formulated in discrete time as a system of nonlinear equations since the competing hypotheses are nonnested the appropriate formulae are derived for the implementation of the cox test this approach is illustrated with an application to the demand for food and related products in the united kingdom