asymptotics of complete kahler metrics of finite volume on quasiprojective manifolds let x be a quasiprojective manifold given by the complement of a divisor d with normal crossings in a smooth projective manifold x using a natural compactification of x by a manifold with corners x we describe the full asymptotic behavior at infinity of certain complete kahler metrics of finite volume on x when these metrics evolve according to the ricci flow we prove that such asymptotic behaviors persist at later times by showing that the associated potential function is smooth up to the boundary on the compactification x however when the divisor d is smooth with kx d 0 so that the ricci flow converges to a kahler einstein metric we show that this kahler einstein metric has a rather different asymptotic behavior at infinity since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary

generalization of selberg s frac 3 16 theorem and affine sieve an analogue of the well known frac 3 16 lower bound for the first eigenvalue of the laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non elementary subgroup l of sl 2 z the proof in the case that the hausdorff of the limit set of l is bigger than frac 1 2 is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting in the case that delta is less than frac 1 2 we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function these spectral gaps are then applied to sieving problems on orbits of such groups

wave decay on convex co compact hyperbolic manifolds for convex co compact hyperbolic quotients x gamma backslash mathbb h n 1 we analyze the long time asymptotic of the solution of the wave equation u t with smooth compactly supported initial data f f 0 f 1 we show that if the hausdorff dimension of the limit set is less than n 2 then u t c delta f e delta frac n 2 t gamma delta n 2 1 e delta frac n 2 t r t where c delta f in c infty x and r t mathcal o t infty we explain in terms of conformal theory of the conformal infinity of x the special cases delta in n 2 mathbb n where the leading asymptotic term vanishes in a second part we show for all epsilon 0 the existence of an infinite number of resonances and thus zeros of selberg zeta function in the strip n delta epsilon rm re lambda delta as a byproduct we obtain a lower bound on the remainder r t for generic initial data f

extending downward collapse from 1 versus 2 queries to m versus m 1 queries the top part of figure 1 1 shows some classes from the truth table bounded query and boolean hierarchies it is well known that if either of these hierarchies collapses at a given level then all higher levels of that hierarchy collapse to that same level this is a standard upward translation of equality that has been known for over a decade the issue of whether these hierarchies can translate equality downwards has proven vastly more challenging in particular with regard to figure 1 1 consider the following claim psigkmtt psigkmponett implies diffmsigk codiffmsigk bh sigmak this claim if true says that equality translates downwards between levels of the bounded query hierarchy and the boolean hierarchy levels that before the fact are immediately below them until recently it was not known whether ever held except for the degenerate cases m 0 and k 0 then hemaspaandra hemaspaandra and hempel siam j comput 28 1999 pp 383 393 proved that holds for all m for k 2 buhrman and fortnow j comput system sci 59 1999 pp 182 199 then showed that when k 2 holds for the case m 1 in this paper we prove that for the case k 2 holds for all values of m since there is an oracle relative to which for k 1 holds for all m fails see buhrman and fortnow our achievement of the k 2 case cannot be strengthened to k 1 by any relativizable proof technique the new downward translation we obtain also tightens the collapse in the polynomial hierarchy implied by a collapse in the bounded query hierarchy of the second level of the polynomial hierarchy

bounded queries approximations and the boolean hierarchy this paper investigates nondeterministic bounded query classes in relation to the complexity of np hard approximation problems and the boolean hierarchy nondeterministic bounded query classes turn out be rather suitable for describing the complexity of np hard approximation problems the results in this paper take advantage of this machine based model to prove that in many cases np approximation problems have the upward collapse property that is a reduction between np approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level for example if maxclique reduces to log n approximating maxclique using many one reductions then the traveling salesman problem tsp is equivalent to maxclique under many one reductions several upward collapse theorems are presented in this paper the proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes in fact these results depend on a surprising connection between the boolean hierarchy and nondeterministic bounded query classes

the boolean hierarchy ii applications the boolean hierarchy i structural properties j cai et al siam j comput 17 1988 pp 1232 252 explores the structure of the boolean hierarchy the closure of np with respect to boolean

the wave group on asymptotically hyperbolic manifolds abstract we show that the wave group on asymptotically hyperbolic manifolds belongs to an appropriate class of fourier integral operators then we use now standard techniques to analyze its regularized trace we prove that as in the case of compact manifolds without boundary the singularities of the regularized wave trace are contained in the set of periods of closed geodesics we also obtain an asymptotic expansion for the trace at zero

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

occupation times distribution for brownian motion on graphs considering a brownian motion on a general graph we study the joint law for the occupation times on all the bonds in particular we show that the laplace transform of this distribution can be expressed as the ratio of two determinants we give two formulations with arc or vertex matrices for this result and discuss a simple example

a downward collapse within the polynomial hierarchy downward collapse also known as upward separation refers to cases where the equality of two larger classes implies the equality of two smaller classes we provide an unqualified downward collapse result completely within the polynomial hierarchy in particular we prove that for k 2 if rm p sigma p k 1 rm p sigma p k 2 then sigma p k pi p k rm ph we extend this to obtain a more general downward collapse result

the laplace operator on a hyperbolic manifold ii eisenstein series and the scattering matrix on considere des varietes m isometriques au quotient de l espace hyperbolique a n dimensions h n par un groupe discret geometriquement fini t sans elements paraboliques et sans elements d ordre fini on demontre que la serie de eisenstein associee admet un prolongement mesomorphe au plan complexe et on etudie l operateur diffusion associe

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

the laplace operator on hyperbolic three manifolds with cusps of non maximal rank where f is a geometrically finite discrete group of orientation preserving isometries which may contain parabolic elements of non maximal rank we prove the existence of a continuum eigenfunction expansion and show that in the spectral parameter the eigenfun ctions have meromorphic continuations to 112 1 these eigenfunctions are eisen stein series for f the continuous spectrum of laplace operators on hyperbolic manifolds was first analysed by number theorists as a means of studying eisenstein series maass and roelke i m r considered the case of finite volume manifolds in two dimen sions increasingly more general hyperbolic manifolds were studied by faddeev if patterson pal lax and phillips lpi lp4 mandouvalos m1 m3 and others recently agmon l a mazzeo and melrose mm and perry p proved the meromorphic continuation of the eisenstein series for geometrically finite groups with fundamental domains of infinite volume provided the groups have no parabolic elements or all the parabolic elements have maximal rank in this case the boundary at infinity which roughly speaking labels the directions at infinity in which scattering can occur is a smooth compact manifold or in the presence of parabolic elements of maximal rank the union of a smooth compact manifold with a finite number of points in fact the work of mazzeo and melrose and recent work of agmon applies to more general manifolds which have a similar structure near infinity however none of the proofs of meromorphic continuation of eigenfun ctions apply when the group has parabolic elements of non maximal rank we will study the laplace operator on a three dimensional hyperbolic manifold when parabolic elements of non maximal rank are present in this case the research supported by the national science and engineering research council of canada research supported in part by n s f grant dms 8911242 research supported in part by n s f grant dms 8802668

fredholm realizations of elliptic symbols on manifolds with boundary we show that the existence of a fredholm element of the zero calculus of pseudodifferential operators on a compact manifold with boundary with a given elliptic symbol is determined up to stability by the vanishing of the atiyah bott obstruction it follows that up to small deformations and stability the same symbols have fredholm realizations in the zero calculus in the scattering calculus and in the transmission calculus of boutet de monvel

on truth table reducibility to sat we show that polynomial time truth table reducibility via boolean circuits to sat is the same as logspace truth table reducibility via boolean formulas to sat and the same as logspace turing reducibility to sat in addition we prove that a constant number of rounds of parallel queries to sat is equivalent to one round of parallel queries we give an oracle relative to which 2p is not equal to the class of predicates polynomial time truth table reducible to sat

on the occupation times of cones by brownian motion we study some features concerning the occupation timeat of a d dimensional conec by brownian motion in particular in the case wherec is convex we investigate the asymptotic behaviour ofp a1 u asu 0 when the brownian motion starts at the vertex ofc we also give the precise integral test which decides whether a s lim inft at tf t 0 or for a decreasing functionf

the spectrum of kleinian manifolds abstract we obtain the plancherel theorem for l 2 g where g is a classical simple lie group of real rank one and g is convex cocompact discrete subgroup and deduce its consequences for the spectrum of locally invariant differential operators on bundles over kleinian manifolds as the main tool we develop a geometric version of scattering theory which in particular contains the meromorphic continuation of the eisenstein series for this situation the central role played by invariant distribution sections supported on the limit set is emphasized

scattering poles for asymptotically hyperbolic manifolds for a class of manifolds x that includes quotients of real hyperbolic n 1 dimensional space by a convex co compact discrete group we show that the resonances of the meromorphically continued resolvent kernel for the laplacian on x coincide with multiplicities with the poles of the meromorphically continued scattering operator for x in order to carry out the proof we use shmuel agmon s perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations

asymptotics of radiation fields in asymptotically minkowski space we consider a non trapping n dimensional lorentzian manifold endowed with an end structure modeled on the radial compactification of minkowski space we find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes the rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the northern cap of the compactification for small perturbations of minkowski space that fit into our framework our asymptotic expansions yield a rate of decay that improves on the klainerman sobolev estimates

spectral analysis of second order elliptic operators on noncompact manifolds on demontre l estimation de mourre et les bornes qui y sont reliees pour une classe d operateurs elliptiques qui inclut des laplaciens pour des quotients cofinis et coinfinis d espaces hyperboliques

scattering theory for conformally compact metrics with variable curvature at infinity abstract we develop the scattering theory of a general conformally compact metric by treating the laplacian as a degenerate elliptic operator with non constant indicial roots on a compact manifold with boundary variability of the roots implies that the resolvent admits only a partial meromorphic continuation and the bulk of the paper is devoted to studying the structure of the resolvent poisson and scattering kernels for frequencies outside the region of meromorphy for low frequencies the scattering matrix is shown to be a pseudodifferential operator with frequency dependent domain in particular generalized eigenfunctions exhibit l 2 decay in directions where the asymptotic curvature is sufficiently negative we explicitly construct the resolvent kernel for generic frequency in this part of the continuous spectrum

asymptotics of scalar waves on long range asymptotically minkowski spaces abstract we show the existence of the full compound asymptotics of solutions to the scalar wave equation on long range non trapping lorentzian manifolds modeled on the radial compactification of minkowski space in particular we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions of the inhomogeneous equation in two appendices we show how these results apply to certain spacetimes whose null infinity is modeled on that of the kerr family in these cases the leading order logarithmic term in our asymptotic expansions at null infinity is shown to be nonzero

eta invariant and selberg zeta function of odd type over convex co compact hyperbolic manifolds abstract we show meromorphic extension and give a complete description of the divisors of a selberg zeta function of odd type z o associated to the spinor bundle on an odd dimensional convex co compact hyperbolic manifold h 2 n 1 as a byproduct we do a full analysis of the spectral and scattering theory of the dirac operator on asymptotically hyperbolic manifolds we show that there is a natural eta invariant d associated to the dirac operator d over a convex co compact hyperbolic manifold h 2 n 1 and that exp i d z o 0 thus extending millson s formula to this setting under some assumption on the exponent of convergence of poincare series for the group we also define an eta invariant for the odd signature operator and we show that for schottky 3 dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the zograf factorization formula relating two natural kahler potentials for weil petersson metric on schottky space

a second step towards complexity theoretic analogs of rice s theorem abstract rice s theorem states that every nontrivial language property of the recursively enumerable sets is undecidable borchert and stephan 1997 initiated the search for complexity theoretic analogs of rice s theorem in particular they proved that every nontrivial counting property of circuits is up hard and that a number of closely related problems are spp hard the present paper studies whether their up hardness result itself can be improved to spp hardness we show that their up hardness result cannot be strengthened to spp hardness unless unlikely complexity class containments hold nonetheless we prove that every p constructibly bi infinite counting property of circuits is spp hard we also raise their general lower bound from unambiguous nondeterminism to constant ambiguity nondeterminism

resolvent of the laplacian on geometrically finite hyperbolic manifolds for geometrically finite hyperbolic manifolds n 1 we prove the meromorphic extension of the resolvent of laplacian poincare series einsenstein series and scattering operator to the whole complex plane we also deduce the asymptotics of lattice points of in large balls of n 1 in terms of the hausdorff dimension of the limit set of

scattering phase asymptotics with fractal remainders for a riemannian manifold m g which is isometric to the euclidean space outside of a compact set and whose trapped set has liouville measure zero we prove weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate for axiom a geodesic flows this gives a polynomial improvement over the known remainders we also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis and provide similar asymptotics for hyperbolic quotients using the selberg zeta function

equidistribution of eisenstein series on convex co compact hyperbolic manifolds the quantum ergodic theorem due to schnirelman 4 colin de verdiere 1 and zelditch 5 says that on any compact riemannian manifold x whose geodesic flow isergodic one can find a full density sequence j of eigenvalues of the laplacian delta x such that the corresponding normalized eigenfunctions j are equidistributed i e for all f l 2 x we have lim j rightarrow infty int limits x f z vert psi j z vert 2 dv z int limits x f z dv z where dv is the normalized volume measure

unambiguous computation boolean hierarchies and sparse turing complete sets this paper studies for up two topics that have been intensely studied for np boolean hierarchies and the consequences of the existence of sparse turing complete sets unfortunately as is often the case the results for np draw on special properties of np that do not seem to carry over straightforwardly to up for example it is known for np and more generally for any class containing sigma and emptyset and closed under union and intersection that the symmetric difference hierarchy the boolean hierarchy and the boolean closure all are equal we prove that closure under union is not needed for this claim for any class cal k that contains sigma and emptyset and is closed under intersection e g up us and fewp the symmetric difference hierarchy over cal k the boolean hierarchy over cal k and the boolean closure of cal k all are equal on the other hand we show that two hierarchies the hausdorff hierarchy and the nested difference hierarchy which in the np case are equal to the boolean closure fail to be equal for the up case in some relativized worlds regarding sparse turing complete sets for up we prove that if up has sparse turing complete sets then the levels of the unambiguous polynomial hierarchy are simpler than one would otherwise expect they collapse one level in terms of their location in the promise unambiguous polynomial hierarchy we obtain related results under the weaker assumption that up has sparse turing hard sets

a mourre estimate and related bounds for hyperbolic manifolds with cusps of non maximal rank we consider the laplace operator on quotients of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries with parabolic subgroups of non maximal rank using methods developed by the first two authors we prove a mourre estimate and commutator estimates on the laplacian which imply absolute continuity of the spectrum and quantitative resolvent estimates these estimates will be used elsewhere to study the scattering matrix and eisenstein series and their meromorphic continuations and should be useful in studying trace formulas for these discrete groups

the laplace operator on a hyperbolic manifold i spectral and scattering theory using techniques of stationary scattering theory for the schrodinger equation we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the laplace beltrami operator on manifolds mn arising as the quotient of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries we consider manifolds mn of infinite volume in subsequent papers we will use the techniques developed here to analytically continue eisenstein series for a large class of discrete groups including some groups with parabolic elements

reasoning with power defaults this paper introduces power default reasoning pdr a framework for non monotonic reasoning based on the domain theoretic idea of modeling default rules with partial information in a higher order setting pdr lifts a non monotonic operator at the base syntactic level to a well behaved almost monotonic operator in the higher order space of the smyth power domain effectively a space of sets of models working in the model space allows us to prove the dichotomy theorem and the extension splitting theorem leading to a more well behaved logic and modulo the usual complexity conjectures a less complex logic than standard default logic specifically we prove that skeptical normal default inference is a problem complete for co np 3 in the boolean hierarchy for strict propositional logic and np 4 complete in general these results by changing the underlying semantics contrasts favorably with similar results of gottlob j logic comput 2 3 1992 397 425 who proves that standard skeptical default reasoning is 2p complete furthermore we show that the skeptical non monotonic consequence relation defined using our domain theoretic semantics obeys all of the laws for preferential consequence relations defined by kraus lehmann and magidor in particular we get the property of being able to reason by cases and the so called law of cautious monotony both of these laws fail for the standard propositional default logic of reiter artificial intelligence 13 1980 81 132 but hold in pdr as a consequence of the dichotomy theorem and the extension splitting theorem

the hausdorff dimension of the limit set of a geometrically finite kleinian group we consider in this paper groups of mobius transformations of l r tj zc such a group is called kleinian if it acts discontinuously somewhere in 1 the action of g extends to the n d dimensional hyperbolic space h i r zc and g is geometrically finite if there is a hyperbolic fundamental polyhedron with a finite number of faces for the action of g in h t for a more precise definition see 15 1 b we prove in this paper that the hausdor f dimension dim l g of the limit set l g of a geometrically finite kleinian group g of 1 is less than n theorem d our proof of this theorem is based on the following obser ation assume that g is of compact t3 pe i e if izi i h iui then i l g g is compact then there is an integer q such that if we divide any n cube q of r into q equal subcubes then at least one of these subcubes does not touch l g let l4 q be the family of these subcubes which touch l g then the n measures of q e le q do not add up to the n measure of q and we get

separating complexity classes with tally oracles abstract long and selman 1986 proved that for most familiar pairs of complexity classes separating the classes with a tally oracle is no easier than truly separating the classes refuting a claim in the literature we prove for the first time that for many familiar pairs of complexity classes separating the classes with a tally oracle is easier than truly separating the classes

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

occupation times for planar and higher dimensional brownian motion we consider a planar brownian motion starting from o at time t 0 and stopped at t denoting by t the time spent in a wedge of apex o and angle we develop a method to compute systematically the moments of t for general values we apply it to obtain analytically the second and third moments for a general wedge angle and also the fourth moment for the quadrant 2 we compare our results with numerical simulations finally with standard perturbation theory we establish a general formula for the second moment of an orthant occupation time

scattering and inverse scattering on ach manifolds we study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds we show the existence of the scattering operator as a meromorphic family of operators in the heisenberg calculus on the boundary which is a contact manifold with a pseudohermitian structure then we define the radiation fields as in the real asymptotically hyperbolic case and reconstruct the scattering operator from those fields as an application we show that the manifold including its topology and the metric are determined up to invariants by the scattering matrix at all energies

boundedness and growth for the massive wave equation on asymptotically anti de sitter black holes abstract we study the global dynamics of free massive scalar fields on general globally stationary asymptotically ads black hole backgrounds with dirichlet neumann or robin boundary conditions imposed on at infinity this class includes the regular kerr ads black holes satisfying the hawking reall bound r 2 a l we establish a suitable criterion for linear stability in the sense of uniform boundedness of and demonstrate how the issue of stability can depend on the boundary condition prescribed in particular in the slowly rotating kerr ads case we obtain the existence of linear scalar hair i e non trivial stationary solutions for suitably chosen robin boundary conditions

lower bounds and the hardness of counting properties rice s theorem states that all nontrivial language properties of recursively enumerable sets are undecidable borchert and stephan math logic quart 46 4 2000 489 504 started the search for complexity theoretic analogs of rice s theorem and proved that every nontrivial counting property of boolean circuits is up hard hemaspaandra and rothe theoret comput sci 244 1 2 2000 205 217 improved the up hardness lower bound to upo 1 hardness the present paper raises the lower bound for nontrivial counting properties from upo 1 hardness to fewp hardness i e from constant ambiguity nondeterminism to polynomial ambiguity nondeterminism furthermore we prove that no relativizable technique can raise this lower bound to fewp 1 ttp hardness we also prove a rice style theorem for np namely that every nontrivial language property of np sets is np hard and for a broad class of leaf language classes we prove a sufficient condition for the natural analog of rice s theorem to hold

oracles that compute values this paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in npmv the class of all multivalued partial functions that are computable nondeterministically in polynomial time concerning deterministic polynomial time reducibilities it is shown that a multivalued partial function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable via 2k 1 nonadaptive queries to npmv a characteristic function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable with k adaptive queries to np unless the boolean hierarchy collapses for every k k adaptive nonadaptive queries to npmv are different than k 1 adaptive nonadaptive queries to npmv nondeterministic reducibilities lowness and the difference hierarchy over npmv are also studied the difference hierarchy for partial functions does not collapse unless the boolean hierarchy collapses but surprisingly the levels of the difference and bounded query hierarchies do not interleave as is the case for sets unless the polynomial hierarchy collapses