on the geometry of the term structure of interest rates we present recently developed geometric methods for the analysis of finite dimensional term structure models of the interest rates this includes an extension of the frobenius theorem for fr echet spaces in particular this approach puts new light on many of the classical models such as the hull white extended vasicek and cox ingersoll ross short rate models the notion of a finite dimensional realization fdr is central for this analysis we motivate it classify all generic fdrs and provide some new results for the corresponding factor processes such as hypoellipticity of its generators and the existence of smooth densities furthermore we include finite dimensional external factors thus admitting a stochastic volatility structure

on local rigidity of partially hyperbolic affine ℤ k actions the following dichotomy for affine z k actions on the torus t d k d is an element of n is shown to hold i the linear part of the action has no rank one factors and then the affine action is

analytic solutions to the inverse problem of the newtonian potential let g be a homogeneous body whose shape in r3 is unknown we want to determine the figure of g from measurements of the newtonian potential created by this body on a spherical surface surrounding g we prove the existence of a local analytic solution to the stated problem

delay nonlinear boundary conditions as limit of reactions concentrating in the boundary abstract in this work we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary we construct a reaction diffusion problem with delay in the interior where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter goes to zero we analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary this convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary

recursive construction of operator product expansion coefficients we derive a novel formula for the derivative of operator product expansion ope coefficients with respect to a coupling constant the formula involves just the ope coefficients themselves but no further input and is in this sense self consistent furthermore unlike other formal identities of this general nature in quantum field theory such as the formal expression for the lagrangian perturbation of a correlation function our formula requires no further uv renormalization i e it is completely well defined from the start this feature is a result of a cancelation of uv and ir divergences between various terms in our identity our proof and an analysis of the features of the identity is given for the example of massive euclidean varphi 4 theory in 4 dimensional euclidean space it relies on the renormalization group flow equation method and is valid to arbitrary but finite orders in perturbation theory the final formula however makes neither explicit reference to the renormalization group flow nor to perturbation theory and we conjecture that it also holds non perturbatively our identity can be applied constructively because it gives a novel recursive algorithm for the computation of ope coefficients to arbitrary finite perturbation order in terms of the zeroth order coefficients corresponding to the underlying free field theory which in turn are trivial to obtain we briefly illustrate the relation of this method to more standard methods for computing the ope in some simple examples

discrete time adaptive control utilizing prior information we present schemes for discrete time adaptive control of a linear time invariant system which is partially known along with an analysis of their convergence properties

string structures and the path fibration of a group we use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class that is the obstruction to a loop group bundle lifting to a kac moody group bundle in the case that the loop group bundle is constructed by taking loops into a principal bundle we show that the classifying map is the holonomy around loops and give an explicit formula for the string class relating it to the pontrjangin class of the principal bunble

a separable frechet space of almost universal disposition abstract the gurari space is the unique separable banach space g which is of almost universal disposition for finite dimensional banach spaces which means that for every e 0 for all finite dimensional normed spaces e f for every isometric embedding e e g there exists an e isometric embedding f f g such that f e e we show that g n with a special sequence of semi norms is of almost universal disposition for finite dimensional graded frechet spaces the construction relies heavily on the universal operator on the gurari space recently constructed by garbulinska wegrzyn and the third author in addition we consider a non graded sequence of semi norms on g n with which the space g n is of almost universal disposition for finite dimensional frechet spaces with a fixed sequence of semi norms in both cases this yields in particular that g n is universal in the class of all separable frechet spaces

existence results in an inverse problem of potential theory soit g un corps physique dont position et forme dans r 3 sont inconnues on suppose un modele pour la densite de g on cherche a determiner la figure de g a partir de mesures de potentiels a sa surface g on suppose que cette surface est differentiablement une sphere

identification of partially known systems abstract the concepts of partially known system identification are introduced and the advantages of a method whereby process information can be utilized in a linear transfer function representation of the system are highlighted and discussed in particular an algorithm for identifying a single unknown physical system parameter associated with a linear system is given it is assumed that the system is polynomial as opposed to linear in the physical parameter the bond graph representation is used as the basis for deriving a system dependent algorithm for each partially known system the method is illustrated by simulated examples

locally robust identification of linear systems containing unknown gain elements with application to adapted iir lattice models abstract we consider the identification of stable linear systems whose unknown parameters may be interpreted as feedback gains by using the output error between the true system and a model of it containing adjustable parameters we develop a recursive algorithm for estimating the unknown parameters we describe a persistency of excitation condition on the system input which guarantees a robust local convergence property for the algorithm we then apply our results to the identification of the parameters of a tapped lattice model of a linear infinite impulse response iir plant considering the identification of a lattice rather than a direct form model of a linear system is attractive due to i the simplicity of its crucial stability check and maintenance procedure for the adapted iir parametrization and ii the numerical insensitivity properties of the lattice structure reproducible simulation evidence is presented that supports our results

the density manifold and configuration space quantization on developpe la structure geometrique differentielle d une variete de frechet de densites ceci fournit un cadre geometrique a la quantification reliee a la mecanique stochastique de nelson on etudie les structures riemanniennes et symplectiques de la variete densite et on deduit l equation de schrodinger a partir d un principe variationnel

invariant tori for the billiard ball map for an n dimensional domain q n 3 with a smooth bound ary which is strictly convex in a neighborhood of an elliptic closed geodesic the existence of a family of invariant tori for the billiard ball map with a positive measure is proved under the assumptions of nondegeneracy and n elementarity n 5 of the corresponding to poincare map moreover the conjugating diffeomorphism constructed is symplectic an analogous result is obtained in the case n 2 it is shown that the lengths of the periodic geodesics determine uniquely the invariant curves near the boundary and the billiard ball map on them up to a symplectic diffeomorphism let n be a strictly convex and compact domain in ir n n 2 2 with a boundary an of class coo in this paper we investigate the so called billiard ball map b near the boundary s an of the coball bundle l b an x e t an 1 1 i the billiard ball map is an exact symplectic map in the interior of l 6 which is singular on the boundary al the local behaviour of b near al was described by melrose using the equivalence theorem for glancing hypersurfaces proved by melrose 15 one can introduce local symplectic coordinates x in t an near any point p e s an so that b an gn i 2 o and the billiard ball map b assumes the form

the lie algebra of homeomorphisms of the circle we define and study an infinite dimensional lie algebrahomeo which is shown to be naturally associated to thetopologicallie grouphomeo of all orientation preserving homeomorphisms of the circle roughly we rely on the universal decorated teichmuller theory developed before as motivation to provide frechet coordinates on the homogeneous space given byhomeo modulo the group of real fractional linear transformations whose corresponding vector fields on the circle we then extend by the usual lie algebrasl2of real traceless two by two matrices in order to definehomeo surprisingly homeo turns out to be equal to the algebra of all vector fields on the circle which are piecewisesl2 in the obvious sense it is evidently important to consider the relationship between our new frechet coordinates and the usual trigonometric functions on the circle and we undertake here both natural infinitesimal calculations we finally apply some further previous work in order to give sufficient conditions on the fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory

rigidity of hamiltonian actions on poisson manifolds abstract this paper is about the rigidity of compact group actions in the poisson context the main result is that hamiltonian actions of compact semisimple type are rigid we prove it via a nash moser normal form theorem for closed subgroups of sci type this nash moser normal form has other applications to stability results that we will explore in a future paper we also review some classical rigidity results for differentiable actions of compact lie groups and export it to the case of symplectic actions of compact lie groups on symplectic manifolds

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

on parameter convergence in adaptive control it is well known that the parameter error as well as the model plant mismatch error in a model reference adaptive scheme tends exponentially to zero iff a certain sufficient richness condition holds for signals inside the time varying plant control loop in this paper we give conditions on the reference signal the exogenous input to the adaptive loop namely that it have as many spectral lines as there are unknown parameters in order to guarantee parameter convergence

smooth tame frechet algebras and lie groups of pseudodifferential operators for pseudodifferential operator algebras with globally defined 0 order symbols belonging to either the hormander class s or certain prescribed subspaces it is shown that these algebras possess important smooth tame structures that make them amenable to nonlinear functional analysis of nash moser type

persistence of excitation conditions for partially known systems abstract this note derives persistence of excitation conditions on the input for systems with partially known transfer functions the assumed transfer function model is p 0 s k t p s q 0 s k t q s where k is the unknown parameter vector p 0 s q 0 s are known scalar polynomials and p s and q s are known vector polynomials

a local solution to the navier stokes equations on manifolds via stochastic representation abstract we obtain a c k local existence result for the navier stokes equations on compact riemannian manifolds using the representation formula for the solution obtained by fang and luo

well posedness of the prandtl equation in sobolev spaces we develop a new approach to study the well posedness theory of the prandtl equation in sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the crocco transformation precisely we firstly investigate the linearized prandtl equation in some weighted sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation we apply the nash moser hormander iteration to obtain a well posedness theory of classical solutions to the nonlinear prandtl equation when the initial data is a small perturbation of a monotonic shear flow

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

stable manifolds in the method of averaging on considere l equation differentielle ef z t e ou f est t periodique en t et e 0 est un petit parametre et l equation moyennee f z i 1 t 0 t f z t o dt on suppose que l equation moyennee a un equilibre hyperbolique en z 0 a variete stable w soit e t la solution t periodique hyperbolique de ef z t e pres de z 0 on demontre un resultat sur la convergence reguliere de la variete stable de e t a w r quand e 0

geodesics distance and the cat 0 property for the manifold of riemannian metrics given a fixed closed manifold m we exhibit an explicit formula for the distance function of the canonical l 2 riemannian metric on the manifold of all smooth riemannian metrics on m additionally we examine the metric completion of the manifold of metrics with respect to the l 2 metric and show that there exists a unique minimal path between any two points this path is also given explicitly as an application of these formulas we show that the metric completion of the manifold of metrics is a cat 0 space

paper an approach to stable indirect adaptive control stable indirect adaptive control of minimum and nonminimum phase plants is established for cases where a priori bounds on the unknown plant parameters are known and where for each set of parameter values within these bounds the plant has no unstable pole zero cancellation by incorporating this partial parameter knowledge in the adaptive law and using a nonminimal representation of the plant it is shown that the adaptive closed loop control system can be written as an exponentially stable system driven by the identification error the stability of the adaptive control system is then shown using techniques similar to those known from the model reference adaptive control approach

differentiable vectors and unitary representations of frechet lie supergroups a locally convex lie group g has the trotter property if for every x 1 x 2 in mathfrak g begin aligned exp g t x 1 x 2 lim n rightarrow infty left exp g left frac t n x 1 right exp g left frac t n x 2 right right n end aligned holds uniformly on compact subsets of mathbb r all locally exponential lie groups have this property but also groups of automorphisms of principal bundles over compact smooth manifolds a key result of the present article is that if g has the trotter property pi g rightarrow mathrm gl v is a continuous representation of g on a locally convex space and v in v is a vector such that overline mathtt d pi x v frac d dt t 0 pi exp g tx v exists for every x in mathfrak g then the map mathfrak g rightarrow v x mapsto overline mathtt d pi x v is linear using this result we conclude that for a representation of a locally exponential frechet lie group g on a metrizable locally convex space the space of mathcal c k vectors coincides with the common domain of the k fold products of the operators overline mathtt d pi x for unitary representations on hilbert spaces the assumption of local exponentiality can be weakened to the trotter property as an application we show that for smooth resp analytic unitary representations of frechet lie supergroups g mathfrak g where g has the trotter property the common domain of the operators of mathfrak g mathfrak g overline 0 oplus mathfrak g overline 1 can always be extended to the space of smooth resp analytic vectors for g

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

levi decomposition for smooth poisson structures we prove the existence of a local smooth levi decomposition for smooth poisson structures and lie algebroids near a singular point this levi decomposition is a kind of normal form or partial linearization which was established in the formal case by wade wad97 and in the analytic case by the second author zun03 in particular in the case of smooth poisson structures with a compact semisimple linear part we recover conn s smooth linearization theorem con85 and in the case of smooth lie algebroids with a compact semisimple isotropy lie algebra our levi decomposition result gives a positive answer to a conjecture of weinstein wei00 on the smooth linearization of such lie algebroids in the appendix of this paper we show an abstract nashmoser normal form theorem which generalizes our levi decomposition result and which may be helpful in the study of other smooth normal form problems

semilinear p evolution equations in sobolev spaces abstract we prove local in time well posedness in sobolev spaces of the cauchy problem for semi linear p evolution equations of the first order with real principal part but complex valued coefficients for the lower order terms assuming decay conditions on the imaginary parts as x

percival variational principle for invariant measures and commensurate incommensurate phase transitions in one dimensional chains we prove for a one dimensional system of classical particles with potential energy u alpha gamma sum limits n left alpha v x n f x n 1 x n gamma right the existence of such a smooth function 0 0 that the system with potential energyu has the equilibrium state at the temperaturet 0 this is the incommensurate phase with the ratio of periods equal to the prescribed irrational number badly approximated by rational ones a simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing percival s energy functional

cubature on wiener space in infinite dimension we prove a stochastic taylor expansion for stochastic partial differential equations spdes and apply this result to obtain cubature methods i e high order weak approximation schemes for spdes

power series space representations of nuclear frechet spaces let e be a nuclear graded frechet space such that the norms satisfy inequalities 2 ik ckjl lk li ik l for all k let f beagradedfrechet space such that the dual extended real valued norms satisfy inequalities ii2 dkil lii lk l for all k and let a be a tame resp linearly tame linear map from f to e then there exists a tame resp linearly tame factorization of a through a power series space a 2 a in the case of a tame quotient map e ib tamely equivalent to a power series space of infinite type this applies in particular to the range of a tame resp linearly tame projection in a power series space a2o a in this case one does not need nuclearity it also applies to the tame spaces in the sense of the various implicit function theorems if they are nuclear they are tamely equivalent to power series spaces ao a the question whether certain frechet spaces have isomorphic representations as kothe sequence spaces has become more and more important on one hand by examples of mityagin zobin 28 djakov mityagin 2 and others it has become clear that in general even for the nuclear case a representation is not possible on the other hand the theory of operators in frechet spaces has developed tools which are more or less connected to such representations an important open problem is for instance whether complemented subspaces of nuclear k6the spaces always have a basis i e whether they are again isomorphic to k6the spaces see pelczyfiski 15 this is not even clear for the complemented subspaces of the space s or more generally of a power series space a a of infinite type except for special cases see 4 5 10 23 27 moreover in this case it is of interest to determine the continuity estimates of isomorphisms in particular to determine whether there are tame or linearly tame isomorphisms see 6 in the case of finite type power series spaces any complemented subspace is again a finite type power series space see mityagin 12 and mityagin henkin 13 the consequence of this for questions on the linearization of the basic constructions of complex analysis are presented in mityagin henkin 13 extensions and generalizations of the method of mityagin henkin have also been given in dubinsky 3 and krone 8 9 received by the editors march 20 1987 1980 mathematics subject classification 1 98 5 revision primary 46a45 46a 1 2 58c 1 5 1990 american mathematical society 0002 9947 90 1 00 25 per page

lie group structures on quotient groups and universal complexifications for infinite dimensional lie groups we characterize the existence of lie group structures on quotient groups and the existence of universal complexifications for the class of baker campbell hausdorff bch lie groups which subsumes all banach lie groups and linear direct limit lie groups as well as the mapping groups ckr m g cr m g m k 1 for every bch lie group g second countable finite dimensional smooth manifold m compact subset k of m and 0 r also the corresponding test function groups dr m g kckr m g are bch lie groups

physically based parameterizations for designing adaptive algorithms abstract structural properties are examined of systems with physical component values as parameters both state variable realizations and transfer function descriptions are investigated the transfer functions in particular are shown to be the ratios of polynomials with coefficients multilinear in the parameters these structures prove useful in formulating adaptive algorithms

local solvability of elliptic and curvature equations on compact manifolds we discuss the search of local constraints for a nonlinear elliptic operator on a compact manifold prove a new version of fredholm theorem for a class of operators with vanishing index and apply it to the local solbability of some prescribed curvature equations

global homotopy with ck estimates for a family of compact regular q pseudoconcave cr manifolds let m 0 be a compact regular q pseudoconcave cr submanifold of a complex manifold g and open image in new window a holomorphic vector bundle on g such that dim open image in new window for some fixed r q we prove a global homotopy formula with c k estimates for r cohomology of open image in new window on arbitrary cr submanifold m close enough to m 0

infinite dimensional second order ordinary differential equations via t2m abstract the vector bundle structure obtained on the second order acceleration tangent bundle t 2 m of a smooth manifold m by means of a linear connection on the base provides an alternative way for the study of second order ordinary differential equations on manifolds of finite and infinite dimension second order vector fields and their integral curves could provide a new way of representing and solving a wide class of evolutionary equations for states on frechet manifolds of sections that arise naturally as inequivalent configurations of a physical field the technique is illustrated by examples in the framework of banach and frechet spaces and on lie groups in particular discussing the case of autoparallel curves which include geodesics if the connection is induced by a riemannian structure

the dolbeault complex in infinite dimensions i infinite dimensional complex analysis was a popular subject in the sixties and seventies but in the last fifteen years enjoyed much less attention oddly enough this very same period began to see the emergence of examples of infinite dimensional complex manifolds in mathematical physics representation theory and geometry thus it appears to be a worthwhile undertaking to revisit infinite dimensional complex analysis and in particular to clarify fundamental properties of infinite dimensional complex manifolds the most fundamental questions at this point seem to be related to the solvability of the inhomogeneous cauchy riemann or 0 equations and more generally to the study of the dolbeault complex as an example in le3 we use a result on the a equation to describe the virasoro group as an infinite dimensional complex manifold up to now precious little has been known about the solvability of the infinite dimensional equation the available results pertain to solving the equation in domains in or over locally convex topological vector spaces and almost exclusively on the level of 0 1 forms postponing the precise definitions to sections 1 2 below we discuss those results that are of immediate relevance to this paper see d2 for more in li ligocka observes that h6rmander s proof of ehrenpreis theorem see e hor can be extended to infinite dimensions to solve the equation

string structures on loop bundles differential geometry and topology of principal loop bundles bundles of loop groups over loop spaces are investigated string structures defined as bundle extensions corresponding to the central extension of the structure group do not always exist various methods for deriving the obstruction to the existence of string structures are discussed

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

clebsch lagrange variational principle and geometric constraint analysis of relativistic field theories inspired by the clebsch optimal control problem we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry its special feature is that the lagrange multipliers couple to the configuration variables via the symmetry group action the resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system we discuss the hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting we show that the yang mills higgs action and the einstein hilbert action fit into this new framework after a 1 3 splitting moreover we recover the gaus constraint of yang mills higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints inspired by the clebsch optimal control problem we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry its special feature is that the lagrange multipliers couple to the configuration variables via the symmetry group action the resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system we discuss the hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting we show that the yang mills higgs action and the einstein hilbert action fit into this new framework after a 1 3 splitting moreover we recover the gaus constraint of yang mills higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

global tchebychev nets on complete two dimensional riemannian surfaces let s be a given surface a local tchebychev net on s is a coordinate chart of s such that the coordinate vector fields have unit magnitudes it models physical surfaces which are reinforced with two sets of inextensible fibers the angle between fibers of each set at any point is allowed to change in this paper s is taken to be a two dimensional simply connected noncompact complete riemannian manifold on which the gaussian curvature is assumed to have compact support the existence of a global tchebychev net on s is investigated

existence of invariant manifolds for stochastic equations in infinite dimension we provide a frobenius type existence result for finite dimensional invariant submanifolds for stochastic equations in infinite dimension in the spirit of da prato and zabczyk 5 we recapture and make use of the convenient calculus on frechet spaces as developed by kriegl and michor 16 our main result is a weak version of the frobenius theorem on frechet spaces as an application we characterize all finite dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates

reliability of perturbation theory in general relativity the relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one parameter families of solutions on the one hand the coefficients of the taylor expansion in the parameter of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations on the other hand it is the converse question does any solution of the perturbation equations come from taylor expanding some family of exact solutions which is of importance for the mathematical justification of the use of perturbation theory this converse question is called the one of the reliability of perturbation theory using and completing recent results on the characteristic initial value problem the local reliability of perturbation theory for general relativity in vacuum is proven very generally this result is then generalized to the einstein yang mills equations and therefore in particular to the einstein maxwell ones