stochastic order on metric spaces and the ordered kantorovich monad abstract in earlier work we had introduced the kantorovich probability monad on complete metric spaces extending a construction due to van breugel here we extend the kantorovich monad further to a certain class of ordered metric spaces by endowing the spaces of probability measures with the usual stochastic order it can be considered a metric analogue of the probabilistic powerdomain our proof of antisymmetry of the stochastic order on these spaces is more general than previously known results in this direction the spaces we consider which we call l ordered are spaces where the order satisfies a mild compatibility condition with the metric itself rather than merely with the underlying topology as we show this is related to the theory of lawvere metric spaces in which the partial order structure is induced by the zero distances we show that the algebras of the ordered kantorovich monad are the closed convex subsets of banach spaces equipped with a closed positive cone with algebra morphisms given by the short and monotone affine maps considering the category of l ordered metric spaces as a locally posetal 2 category the lax and oplax algebra morphisms are exactly the concave and convex short maps respectively in the unordered case we had identified the wasserstein space as the colimit of the spaces of empirical distributions of finite sequences we prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the finite sequences generalizing a recent result of lawson the proof holds on any metric space equipped with a closed partial order

boundary value problems for first order elliptic wedge operators we develop an elliptic theory based in l 2 of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum in particular we do not require the indicial roots to be constant along the base of the boundary fibration our theory includes as a special case the classical theory of elliptic boundary value problems for first order operators with and without the shapiro lopatinskii condition and can be thought of as a natural extension of that theory to the geometrically and analytically relevant class of wedge operators wedge operators arise in the global analysis on manifolds with incomplete edge singularities our theory settles in the first order case the long standing open problem to develop a robust elliptic theory of boundary value problems for such operators

etale groupoids eta invariants and index theory let g be a discrete finitely generated group let m t be a g equivariant fibration with fibers dieomorphic to a fixed even dimensional manifold with boundary z we assume that g m m g is a galois covering of a compact manifold with boundary letd y ya t be a g equivariant family of dirac type operators under the assumption that the boundary family is l 2 invertible we define an index class in k0 c 0 t zr g if in addition g is of polynomial growth we define higher indices by pairing the index class with suitable cyclic cocycles our main result is then a formula for these higher indices the structure of the formula is as in the seminal work of atiyah patodi and singer with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family under similar assump tions we extend our theorem to any g proper manifold with g an etale groupoid we employ this generalization in order to establish a higher atiyah patodi singer index formula on certain foliations with boundary fundamental to our work is a suitable generalization of melrose b pseudodierential calculus as well as the superconnection proof of the index theorem on g proper manifolds recently given by gorokhovsky and lott in 10

operator means of probability measures abstract let p be the complete metric space consisting of positive invertible operators on an infinite dimensional hilbert space with the thompson metric we introduce the notion of operator means of probability measures on p in parallel with kubo and ando s definition of two variable operator means and show that every operator mean is contractive for the wasserstein distance by means of a fixed point method we consider deformation of such operator means and show that the deformation of any operator mean becomes again an operator mean in our sense based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures

fourier integral operators on lie groupoids abstract as announced in 36 we develop a calculus of fourier integral g operators on any lie groupoid g for that purpose we study lagrangian conic submanifolds of the symplectic groupoid t g this includes their product transposition and inversion we also study the relationship between these lagrangian submanifolds and the equivariant families of lagrangian submanifolds of t g x t g x parametrized by the units x g 0 of g this allows us to select a subclass of lagrangian distributions on any lie groupoid g that deserve the name of fourier integral g operators g fios by construction the class of g fios contains the class of equivariant families of ordinary fourier integral operators on the manifolds g x x g 0 we then develop for g fios the first stages of the calculus in the spirit of hormander s work finally we illustrate this calculus in the case of manifolds with boundary

groupoids and pseudodifferential calculus on manifolds with corners we associate to any manifold with corners even with non embedded hyperfaces a non hausdorff longitudinally smooth lie groupoid on which we define a pseudodifferential calculus this calculus generalizes the b calculus of melrose defined for manifolds with embedded corners the groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension using tools from the theory of c algebras of groupoids we also obtain new proofs for the study of b calculus

valuations and convex subrings of a commutative ring with higher level preordering the purpose of this paper is to investigate general valuations and convex subrings of a commutative ring with higher level preordering by defining the compatibility between valuations and preorderings or orderings of higher level in a commutative ring we discuss the interplay between valuations and orderings of higher level it is shown that the convex subrings are exactly valuation subrings compatible with some ordering moreover the elements in the convex hull of a subring with respect to a given preordering are described in this paper the emphasis on manis valuations is removed

unbounded pseudodifferential calculus on lie groupoids abstract we develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids also called lie groupoids with compact basis we analyze these operators as unbounded operators acting on hilbert modules over c g and we show in particular that elliptic operators are regular we construct a scale of sobolev modules which are the abstract analogues of the ordinary sobolev spaces and analyze their properties furthermore we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense

a groupoid approach to pseudodifferential calculi we give an algebraic geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural r action specifically we show that a properly supported semiregular distribution on m m is the schwartz kernel of a classical pseudo differential operator if and only if it extends to a smooth family of distributions on the range fibres of the tangent groupoid which is homogeneous for the r action modulo smooth functions moreover we show that the basic properties of pseudo differential operators can be proven directly from this characterization finally we show that with the appropriate generalization of the tangent bundle the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds and in particular the heisenberg calculus

the transmission problem for quasi linear elliptic second order equations in a conical domain i ii abstract we investigate the behavior of weak solutions to the transmission problem for quasi linear elliptic divergence second order equations in a neighborhood of the boundary conical point in part i we study the problem for quasi linear equation with semi linear principal part in part 2 we study the problem for general quasi linear elliptic divergence second order equations

reduced forms and reduced witt rings of higher level the present paper and its successor s deal with an extension of the socalled reduced theory of quadratic forms over a formally real field k 5 7 lo this theory deals with the reduced witt ring wred k w k nil w k and its natural embedding into c x z the ring of integer valued continuous functions on the space x x k of orderings of k recently it has been found that there is a far reaching generalization of the notion of ordering namely the orderings of higher level 11 the orderings of level s where s divides a given natural number n are related to sums of 2nth powers in the same way as the usual orderings are to sums of squares their intersection is the set of all sums of 2nth powers in k 11 in 6 orderings of levels were replaced by signatures of this level these are those characters x i s of order 2s whose kernels are additively closed where k k o it is exactly this change in point of view which leads to a natural definition of reduced witt rings of higher level let x x k denote the set of all signatures of k whose order divides 2n so that x k x k x being a closed subspace of the compact character group of x the set x k is also a compact topological space each element a e i induces a continuous function a x c in c x c defined by x x a where c is given the discrete topology by analogy with the natural representation of the ordinary reduced witt ring as a subring of 477 0021 8693 85 3 00

barycentric maps for compactly supported measures abstract after outlining the way in which an intrinsic mean g g n on a complete metric space gives rise to a contractive barycentric map on some class of borel probability measures and some basic examples of this process we show how the resulting barycentric map gives rise to a general theory of integration of measurable functions into the space we apply this machinery to the cone of positive invertible elements of a c algebra equipped with the thompson metric to derive barycentric maps and their basic properties arising from the power means finally we derive basic results for the karcher barycenter including its approximation by the barycentric maps for power means and its satisfaction of the karcher equation

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

analysis of periodic schrodinger operators regularity and approximation of eigenfunctions let v be a real valued potential that is smooth everywhere on r3 except at a periodic discrete set s of points where it has singularities of the coulomb type z r we assume that the potential v is periodic with period lattice l we study the spectrum of the schrodinger operator h v acting on the space of bloch waves with arbitrary but fixed wavevector k let t r3 l let u be an eigenfunction of h with eigenvalue and let 0 be arbitrarily small we show that the classical regularity of the eigenfunction u is u h5 2 t in the usual sobolev spaces and u k3 2 m t s in the weighted sobolev spaces the regularity index m can be as large as desired which is crucial for numerical methods for any choice of the bloch wavevector k we also show that h has compact resolvent and hence a complete eigenfunction expansion the case of the hydrogen atom suggests that our regularity results are optimal we present two applications to the numerical approximation of eigenvalues using wave functions and usi

adjoints of elliptic cone operators we study the adjointness problem for the closed extensions of a general b elliptic operator a x diff m b m e 0 initially defined as an unbounded operator a c c m e x l 2 b m e x l 2 b m e inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the case where a is a symmetric semibounded operator is of particular interest and we give a complete description of the domain of the friedrichs extension of such an operator

ordered probability spaces abstract let c be an open cone in a banach space equipped with the thompson metric with closure a normal cone the main result gives sufficient conditions for borel probability measures on c with finite first moment for which in the stochastic order induced by the cone to be order approximated by sequences n n of uniform finitely supported measures in the sense that n n for each n and n n in the wasserstein metric this result is the crucial tool in developing a pathway for extending various inequalities on operator and matrix means which include the harmonic geometric and arithmetic operator means on the cone of positive elements of a c algebra to the space p 1 c of borel measures of finite first moment on c as an illustrative and important particular application we obtain the monotonicity of the karcher geometric mean on p 1 a for the positive cone a of a c algebra a

automorphisms and isomorphisms of real henselian fields let k and l be ordered algebraic extensions of an ordered field f suppose k and l are henselian with archimedean real closed residue class fields then k and l are shown to be f isomorphic as ordered fields if they have the same value group analogues to this result are proved involving orderings of higher level unordered extensions and when k and l are maximal valued fields transcendental extensions as a corollary generalized real closures at orderings of higher level are shown to be determined up to isomorphism by their value groups the results on isomorphisms are applied to the computation of automorphism groups of k and to the study of the fixed fields of groups of automorphisms of k if k is real closed and maximal with respect to its canonical valuation then these fixed fields are shown to be exactly those real closed subfields of k which are topologically closed in k generalizations of this fact are proved an example is given to illustrate several aspects of the problems considered here

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

n real valuations and the higher level version of the krull baer theorem abstract in comm algebra 29 2001 193 201 cimpric gave examples of division rings containing an ordering of level 2m but not of level m for m n his examples were quite complicated we give substantially simplified examples in section 2 in sections 3 and 4 we investigate this phenomenon using valuation theory we define almost real and n real valuations and study liftings of orderings from the residue division ring to the original division ring such liftings are not always possible as is the case in the commutative setting but we give a necessary and sufficient condition for a lifting to exist we also prove a suitable generalization of the baer krull theorem finally in the last section we use our examples and the theory developed to answer a question given by marshall and zhang j algebra 212 1 1999 190 207

existence and uniqueness of the karcher mean on unital c algebras abstract the karcher mean on the cone of invertible positive elements of the c algebra b e of bounded operators on a hilbert space e has recently been extended to a contractive barycentric map on the space of l 1 probability measures on in this paper we first show that the barycenter satisfies the karcher equation and then establish the uniqueness of the solution next we establish that the karcher mean is real analytic in each of its coordinates and use this fact to show that both the karcher mean and karcher barycenter map exist and are unique on any unital c algebra the proof depends crucially on a recent result of the author giving a converse of the inverse function theorem

regular representation of groupoid c algebras and applications to inverse semigroups the analogue of the left regular representation of a locally compact groupoid is constructed in the hausdoras well as in the non hausdorcase a necessary and su cient condition for a locally compact groupoid with a cocycle to be morita equi valent to a group action is obtained as an application the c algebras of a class of inverse semigroups is shown to be morita equivalent to crossed products of groups by abelian c

double layer potentials on three dimensional wedges and pseudodifferential operators on lie groupoids abstract let w be a three dimensional wedge and k be the double layer potential operator associated to w and the laplacian we show that 1 2 k are isomorphisms between suitable weighted sobolev spaces which implies a solvability result in weighted sobolev spaces for the dirichlet problem on w furthermore we show that the double layer potential operator k is an element in c g m 2 c where g is the action transformation groupoid m g with g 1 0 a b a r b r which is a lie group and m is a kind of compactification of g this result can be used to prove the fredholmness of 1 2 k where is a domain with edge singularities and k the double layer potential operator associated to the laplacian and

boundary contact problems for domains with conical singularities abstract we study boundary contact problems for elliptic equations and systems with interfaces that have conical singularities such problems represent continuous operators between weighted sobolev spaces and subspaces with asymptotics ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities we show regularity and asymptotics of solutions in weighted spaces and we construct parametrices the result will be illustrated by a number of explicit examples

log majorizations for the symplectic eigenvalues of the cartan barycenter abstract in this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are lipschitz for the cartan hadamard riemannian metric and establish log majorizations for the symplectic eigenvalues of the cartan barycenter of integrable probability borel measures this leads a version of jensen s inequality for geometric integrals of matrix valued integrable random variables

the holonomy groupoid of a singular foliation we construct the holonomy groupoid of any singular foliation in the regular case this groupoid coincides with the usual holonomy groupoid of winkelnkemper h e winkelnkemper the graph of a foliation ann glob anal geom 1 3 1983 51 75 the same holds in the singular cases of j pradines how to define the differentiable graph of a singular foliation c top geom diff cat xxvi 4 1985 339 381 b bigonnet j pradines graphe d un feuilletage singulier c r acad sci paris 300 13 1985 439 442 c debord local integration of lie algebroids banach center publ 54 2001 21 33 c debord holonomy groupoids of singular foliations j diff geom 58 2001 467 500 which from our point of view can be thought of as being almost regular in the general case the holonomy groupoid can be quite an ill behaved geometric object on the other hand it often has a nice longitudinal smooth structure nonetheless we use this groupoid to generalize to the singular case connes construction of the c algebra of the foliation we also outline the construction of a longitudinal pseudo differential calculus the analytic index of a longitudinally elliptic operator takes place in the k theory of our c algebra in our construction the key notion is that of a bi submersion which plays the role of a local lie groupoid defining the foliation our groupoid is the quotient of germs of these bi submersions with respect to an appropriate equivalence relation

chain signatures and higher level witt rings in many respects becker s theory of orderings of higher level is very much reminiscent of the familiar artin schreier theory of ordered fields a few examples of similarities are fields with higher level orderings have real closures 1 3 orderings of higher level can be used to study sums of powers in fields 3 41 there is a theory of witt rings for fields with higher level preorderings 7 81 however among all these similarities one very obvious difference stands out a field with a higher level ordering has nonisomorphic real closures becker showed that in order to describe the isomorphism classes of real closures a whole sequence of higher level orderings is required 1 p 162 theorem 121 subsequently this has been relined and reformulated through the introduction of chains of higher level orderings 121 and chain signatures 27 see alo 10 11 i in view of this situation one should expect that if results from the usual theory of ordered fields level 1 carry over to the case of orderings of higher level then quite different results will also be brought in by the chains of higher level orderings the purpose of this paper is to look for such results in the theory of higher level witt rings developed recently by becker and rosenberg 7 81 to keep things from getting too complicated this will be done only for the case where all levels are powers of 2 thus throughout the term higher level will be synonymous with level 2 for some n iv w k denotes the witt ring of the field k with respect to the higher level preordering k c k2 nonzero sums of 2 th powers of 9 just as the use of signatures s leads to homomorphisms w 7 chain signatures lead to homomorphisms

the transmission problem on a three dimensional wedge we consider the transmission problem for the laplace equation on an infinite three dimensional wedge determining the complex parameters for which the problem is well posed and characterizing the infinite multiplicity nature of the spectrum this is carried out in two formulations leading to rather different spectral pictures one formulation is in terms of square integrable boundary data the other is in terms of finite energy solutions we use the layer potential method which requires the harmonic analysis of a non commutative non unimodular group associated with the wedge

gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients in this paper we derive global w1 and piecewise c1 estimates for solutions to divergence form elliptic equations with piecewise holder continuous coefficients the novelty of these estimates is that even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients they are independent of the distance between these surfaces

analytic k theory on manifolds with corners let y be a compact manifold in 1 atiyah proposed an analytic delinition of the k homology groups k y defined abstractly as the dual theory to the atiyah hirzebruch k cohomology theory these ideas were developed by kasparov 15 and brown douglas and fillmore lo cycles for this analytic k homology theory are abstract elliptic operators i e bounded linear operators between hilbert spaces satisfying additional conditions derived from the properties of elliptic pseudo differential operators in particular elliptic pseudodifferential operators between sections of vector bundles define such cycles the homology class of the cycle depends only on the principal symbol and indeed only on the k cohomology class it represents on the cotangent bundle in fact it had already been shown in essence by atiyah that the resulting map

adiabatic groupoid crossed product by r and pseudodifferential calculus abstract let g be a lie groupoid recall that the associated adiabatic groupoid g ad is the groupoid obtained by the deformation to the normal cone construction corresponding to the inclusion g 0 g in the present paper we consider the crossed product g ga of the natural action of r on the adiabatic groupoid g ad associated with any lie groupoid g we construct an explicit morita equivalence between the exact sequence of order 0 pseudodifferential operators on g and a restriction of the natural exact sequence associated with g ga as an important intermediate step we express a pseudodifferential operator on g as an integral associated with a smoothing operator on the adiabatic groupoid g ad of g

singularities of solutions to semilinear boundary value problems soit p un operateur differentiel d ordre 2 strictement hyperbolique sur une region ouverte cr n a frontiere lisse non caracteristique etant donnee une solution u h s loc s n 1 2 de pu f x u on etudie la propagation des singularites microlocales h r pour s r 2n n 2 1 2 dans la situation generale ou le champ hamiltonien de p hp peut etre tangent a t o a un ordre infini ou fini arbitrairement eleve

on real closed rings of higher level abstract the purpose of this paper is to introduce the notion of real closed rings of higher level in the category of reduced commutative rings such real closed rings of higher level are just generalized real closed fields in case the rings considered are fields with the aid of the theory of totally integrally closed rings some important results on real closed rings of higher level are established the results in this paper involve real valuations and a related characterization of real closed rings of higher level

the wave equation on asymptotically de sitter like spaces in this paper we obtain the asymptotic behavior of solutions of the klein gordon equation on lorentzian manifolds x g which are de sitter like at infinity such manifolds are lorentzian analogues of the so called riemannian conformally compact or asymptotically hyperbolic spaces under global assumptions on the null bicharacteristic flow namely that the boundary of the compactification x is a union of two disjoint manifolds y and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to and to the other manifold as the parameter goes to we also define the scattering operator and show that it is a fourier integral operator associated to the bicharacteristic flow from y to y

normal solvability of elliptic boundary value problems on asymptotically flat manifolds abstract normal solvability is shown for a class of boundary value problems on riemannian manifolds with noncompact boundary using a concept of weighted pseudodifferential operators and weighted sobolev spaces together with lopatinski shapiro type boundary conditions an essential step is to show that the standard normal derivative defined in terms of the riemannian metric is in fact a weighted pseudodifferential operator of the considered class provided the metric is compatible with the symbols

complex quantum groups and a deformation of the baum connes assembly map we define and study an analogue of the baum connes assembly map for complex semisimple quantum groups that is drinfeld doubles of qdeformations of compact semisimple lie groups our starting point is the deformation picture of the baum connes assembly map for a complex semisimple lie group g which allows one to express the k theory of the reduced group c algebra of g in terms of the k theory of its associated cartan motion group the latter can be identified with the semidirect product of the maximal compact subgroup k acting on k via the coadjoint action in the quantum case the role of the cartan motion group is played by the drinfeld double of the classical group k whose associated group c algebra is the crossed product of c k with respect to the adjoint action of k our quantum assembly map is obtained by varying the deformation parameter in the drinfeld double construction applied to the standard deformation kq of k we prove that the quantum assembly map is an isomorphism thus providing a description of the k theory of complex quantum groups in terms of classical topology moreover we show that there is a continuous field of c algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them it follows in particular that the quantum assembly map contains the classical baum connes assembly map as a direct summand

a generalization of manifolds with corners abstract in conventional differential geometry one studies manifolds locally modelled on r n manifolds with boundary locally modelled on 0 r n 1 and manifolds with corners locally modelled on 0 k r n k they form categories man m a n b m a n c manifolds with corners x have boundaries x also manifolds with corners with dim x dim x 1 we introduce a new notion of manifolds with generalized corners or manifolds with g corners extending manifolds with corners which form a category m a n gc with man m a n b m a n c m a n gc manifolds with g corners are locally modelled on x p hom mon p 0 for p a weakly toric monoid where x p 0 k r n k for p n k z n k most differential geometry of manifolds with corners extends nicely to manifolds with g corners including well behaved boundaries x in some ways manifolds with g corners have better properties than manifolds with corners in particular transverse fibre products in m a n gc exist under much weaker conditions than in m a n c this paper was motivated by future applications in symplectic geometry in which some moduli spaces of j holomorphic curves can be manifolds or kuranishi spaces with g corners rather than ordinary corners our manifolds with g corners are related to the interior binomial varieties of kottke and melrose 20 and the positive log differentiable spaces of gillam and molcho 6

strong law of large numbers for the l1 karcher mean abstract sturm s strong law of large numbers in cat 0 spaces has been an influential tool to study the geometric mean or also called karcher barycenter of positive definite matrices it provides an easily computable stochastic approximation based on inductive means convergence of a deterministic version of this approximation has been proved by holbrook providing his nodice theorem for the karcher mean of positive definite matrices the karcher mean has also been extended to the infinite dimensional case of positive operators on a hilbert space by lawson lim and then to probability measures with bounded support by the second author however the cat 0 property of the space is lost and one defines the mean as the unique solution of a nonlinear operator equation on a convex banach finsler manifold the formulations of sturm s strong law of large numbers and holbrook s nodice approximation are natural and both conjectured to converge however all previous techniques of their proofs break down due to the banach finsler nature of the space in this paper we prove both conjectures by establishing the most general l 1 form of sturm s strong law of large numbers and holbrook s nodice theorem in the operator norm by developing a stochastic discrete time resolvent flow for the karcher barycenter using its wasserstein contraction property

pseudodifferential operators on manifolds with a lie structure at infinity we define and study an algebra 0 v m0 of pseudodifferential operators canonically associated to a noncompact riemannian manifold m0 whose geometry at infinity is described by a lie algebra of vector fields v on a compactification m of m0 to a compact manifold with corners we show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to 0 v m0 we also consider the algebra diff m0 of differential operators on m0 generated by v and c m and show that 0 v m0 is a microlocalization of diff m0 our construction solves a problem posed by melrose in 1990 finally we introduce and study semi classical and suspended versions of the algebra 0 v m0

boundary contact problems for domains with edge singularities abstract we study boundary contact problems for elliptic equations and systems with interfaces that have edge singularities such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics ellipticity is formulated in terms of a principal symbolic hierarchy containing interior transmission and edge symbols we construct parametrices show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary contact problems for the laplacian with jumping coefficients

adiabatic groupoid and secondary invariants in k theory abstract in this paper we define new k theoretic secondary invariants attached to a lie groupoid g the receptacle for these invariants is the k theory of c r g a d where g a d is the adiabatic deformation g restricted to the interval 0 1 our construction directly generalises the cases treated in 29 30 in the setting of the coarse geometry to more involved geometrical situations such as foliations moreover we tackle the problem of producing a wrong way functoriality between adiabatic deformation groupoid k groups associated to transverse maps this extends the construction of the lower shriek map in 6 furthermore we prove a lie groupoid version of the delocalized aps index theorem of piazza and schick finally we give a product formula for secondary invariants

on the full calculus of pseudo differential operators on boundary groupoids with polynomial growth abstract in this paper we enlarge the space of uniformly supported pseudo differential operators on some groupoids by considering kernels satisfying certain asymptotic estimates we show that such enlarged space contains the compact parametrix and the generalized inverse of uniformly supported operators with fredholm vector representation

valuations and higher level orders in commutative rings valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields see for example br in mw the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting while it seems desirable to extend these results further the techniques used for rings with many units will not work for general commutative rings at the same time there is a general theory of valuations in commutative rings see lm m and g which in ma was used to study orders and the reduced theory of quadratic forms over general commutative rings thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings in this paper we use valuation theory to study the space of orders and the reduced witt ring relative to a higher level preorder in a commutative ring as in ma we first localize our ring at a multiplicative set without changing the space of orders in order to make the valuation theory work better this is a standard idea from real algebraic geometry remarkably many of the notions methods and results for fields carry over to this new setting we define compatiblity between valuations and orders and preorders and the ring a t associated to a preorder t which turns out to be prufer ring as in the field case we define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders we can then apply this to show that under a certain finiteness condition the space of orders is equivalent to the space of orders of a preordered field

the stochastic order of probability measures on ordered metric spaces abstract the general notion of a stochastic ordering is that one probability distribution is smaller than a second one if the second attaches more probability to higher values than the first motivated by recent work on barycentric maps on spaces of probability measures on ordered banach spaces we introduce and study a stochastic order on the space of probability measures p x where x is a metric space equipped with a closed partial order and derive several useful equivalent versions of the definition we establish the antisymmetry and closedness of the stochastic order and hence that it is a closed partial order for the case of a partial order on a banach space induced by a closed normal cone with interior we also consider order completeness of the stochastic order for a cone of a finite dimensional banach space and derive a version of the arithmetic geometric harmonic mean inequalities in the setting of the associated probability space on positive invertible operators on a hilbert space

boutet de monvel operators on lie manifolds with boundary abstract we introduce and study a general pseudodifferential calculus for boundary value problems on a class of non compact manifolds with boundary so called lie manifolds with boundary this is accomplished by constructing a suitable generalization of the boutet de monvel calculus for boundary value problems the data consists of a compact manifold with corners m that is endowed with a lie structure of vector fields 2 v a so called lie manifold the manifold m is split into two equal parts x and x which intersect in an embedded hypersurface y x our goal is to describe a transmission boutet de monvel calculus for boundary value problems compatible with the structure of lie manifolds starting with the example of b vector fields we show that there are two groupoids integrating the lie structures on m and on y respectively these two groupoids form a bibundle or a groupoid correspondence and under some mild assumptions these groupoids are morita equivalent with the help of the bibundle structure and canonically defined manifolds with corners which are blow ups in particular cases we define a class of boutet de monvel type operators we then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem finally we consider appropriate fredholm conditions and construct the parametrices for elliptic operators in the calculus

on the index of elliptic operators on a wedge abstract we propose an analytical approach to the index theory of pseudo differential operators on a manifold with edges it results in an intermediate algebraic index formula the latter permits much more freedom in homotopies and in particular can be transformed to the topological formula

a representation theorem for higher level reduced witt rings in 6 becker and rosenberg defined the reduced witt ring wak of a formally real field k with respect to a higher level preorder tin k wak is a subring of the ring c x of continuous c valued functions on the space x of signatures of k which are trivial on t and is built up from higher level t forms over k this higher level reduced theory is a natural generalization of the reduced theory of quadratic forms over fields spaces of signatures were introduced in 18 to provide an abstract framework for the higher level reduced theory over fields along the lines of marshall s spaces of orderings 13 h treatment of the classical reduced theory of quadratic forms there are many advantages to this axiomatic approach it allows for a unified study of reduced witt rings over fields and skew fields powers has shown that becker and rosenberg s study of higher level reduced witt rings can also be carried out over skew fields 22 the absence of technical considerations such as valuation theory not only yields simpler proofs of known results but also new results about the theory over fields 183 the space of orderings version 151 of the result which we prove here has found application in real algebraic geometry and we hope that our abstract higher level result might also prove useful in this context the question which concerns us here is the so called representation problem given fe c x how can we tell if f f for some f w x this has been solved for the higher level reduced witt ring of a field by becker and rosenberg 6 see s also and in the case of spaces of orderings by marshall 15 both solutions being generalizations of the original representation theorems of becker and brlicker 4 5 31 and brown and marshall 9 4 21 in the context of the reduced quadratic form theory over fields under a 2 power assumption we simultaneously extend all of these representation theorems to our abstract higher level setting this entails studying fans and more importantly a generalization called quasifans we not only obtain a common proof of the representation 105 0021 8693 88 3 00