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

a quasi newton differential dynamic programming algorithm for discrete time optimal control we develop a quasi newton differential dynamic programming algorithm qddp for discrete time optimal control problems in the spirit of dynamic programming the quasi newton approximations are performed in a stagewise manner we establish the global convergence of the method and also show a superlinear convergence rate among other advantages of the qddp method second derivatives need not be calculated in theory the computational effort of each recursion grows proportionally to the number of stages n whereas with conventional quasi newton techniques which do not take advantage of the optimal control problem structure the growth is as n2 computational results are also reported

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

convergence rate analysis of the state increment dynamic programming method it is shown that if the discretization of the state space is sufficiently fine and if the limiting trajectory is an interior point of the admissible policies then the state increment dynamic programming technique sidp has linear convergence and the coefficient of convergence can be explicitly related to the control problem components in the course of this analysis it is seen that the sidp method is in essence a nonlinear generalization of the block gauss seidel algorithm for solving linear equations some consequences of this insight are suggested and a computational example comparing sidp with differential dynamic programming is offered

perturbing uniformly stable nonlinear scalar delay differential equations on considere l equation differentielle a retard perturbee x t f t x t g t x t ou g o s r s x r x on donne des conditions sur g qui assurent que la solution zero est uniformement stable sous certaines conditions

asymptotic theory for a class of nonautonomous delay differential equations this paper deals with asymptotic behavior of solutions of the nonlinear nonautonomous delay differential equation x t t r t t f t x s d t s su where xf t x 0 f t 0 0 t r t nondecreasing t s is nondecreasing and of bounded variation general sufficient conditions which are easy to verify are obtained for the solutions to be bounded and asymptotically stable locally and globally these results improve many existing ones principally by allowing i r t to be unbounded ii both discrete and distributed delays and iii the equation to be strongly nonlinear and nonautonomous various examples are given in the form of corollaries with a highly flexible integrand

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

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

global stability for a class of nonlinear nonautonomous delay equations it is easy to imagine that more realistic models for single species growths should take into account both the changing of the environment and the effects of time delay thus it is neces sary and interesting to study the following general nonlinear nonautonomous delay equation x t l x t i s

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

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

global attractivity in scalar delayed differential equations with applications to population models for the scalar delay differential equation x t 1 x t f t xt sufficient conditions for the global stability of the zero solution are given the behavior of global solutions on r is also addressed several applications to population models which improve known results are presented

on complemented subspaces of non archimedean kothe spaces abstract the generalized power series spaces d f a r are the most known and important examples of nuclear kothe spaces we prove that every kothe montel space over a non archimedean field k has a complemented closed subspace isomorphic to a generalized power series space d f a r it follows that every non normable frechet space e that is not isomorphic to the product of a banach space and the space k n has a closed subspace isomorphic to a generalized power series space d f a r

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

engineering applications of discrete time optimal control abstract many problems of design and operation of engineering systems can be formulated as optimal control problems where time has been discretisized this is also true even if time is not involved in the formulation of the problem but rather another one dimensional parameter this paper gives a review of some well known and new results in discrete time optimal control methods applicable to practical problem solving within engineering emphasis is placed on dynamic programming the classical maximum principle and generalized versions of the maximum principle for optimal control of discrete time systems several case studies will illustrate the applicability of the above mentioned methods

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

computational aspects of discrete time optimal control our opinion is that a little known technique called differential dynamicprogramming offers the potential of enormously expanding the scale of discrete time optimal control problems which are subject to numerical solution among the attractive features of this method are that no discretization of control or state space is used the memory requirements grow as m 2 and the computational requirements as m 3 with m being the dimension of the control variable the successive approximations converge globally under lenient smoothness assumptions and the convergence is quadratic if certain convexity assumptions hold the contribution of the present paper is to demonstrate the practical merit of differential dynamic programming by reporting computational solutions to problems having as many as forty control variables and no particularly convenient structure additionally we give a more algorithmically oriented presentation of the method than hitherto available extend the basic methodto the nonconvex case and give a proof of global convergence

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

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

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

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

the stagewise kuhn tucker condition and differential dynamic programming the intention of this work is to describe and examine a differential dynamic programming ddp algorithm for constrained discrete time optimal control this algorithm has performed successfully on a large scale reservoir control problem 11 the present paper gives conditions under which convergence to the stationary policy is assured the convergence demonstration hinges upon a notion which we refer to as the stagewise kuhn tucker condition strategies generated to satisfy this condition determine policies which satisfy the conventional kuhn tucker condition this observation may be of wider importance in discrete optimal control theory for the stagewise condition might be a convenient criterion for constructing strategies

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

a dynamic planning technique for continuous activities under multiple resource constrains the solution technique developed in this paper integrates the simplex search algorithm into the recursive calculations of dynamic programming in order to mitigate the curse of dimensionality since differentiation is not required as a means of optimization procedure it is expected that this technique will have an important impact on planning problems such as the multiple resource allocation problems which involve nondifferentiable and or highly nonlinear performance functions applications of the technique to very complex problems seem to confirm the anticipated polynomial average performance

optimal control of nonlinear groundwater hydraulics using differential dynamic programming optimal groundwater management models are based on the hydraulic equations of the aquifer system these equations relate the state variables of the groundwater system the head and the decision variables that control the magnitude location and timing of pumping or artificial recharge for the unconfined aquifer these management models are large scale nonlinear programming problems a differential dynamic programming ddp algorithm is used for unsteady nonlinear groundwater management problems due to the stagewise decomposition of ddp the dimensionality problems associated with embedding the hydraulic equations as constraints in the management model are significantly reduced in addition ddp shows a linear growth in computing effort with respect to the number of stages or planning periods and quadratic convergence several example problems illustrate the application of ddp to the optimal control of nonlinear groundwater hydraulics

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

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

global attractivity results for nonlinear delay differential equations abstract the aim of this paper is to give sufficient conditions for global attractivity of the zero solution of the nonlinear delay differential equationx t p t f x t an example which guarantees that our 3 2 stability condition is the best possible is also given

stability theorems for nonlinear scalar delay differential equations in this paper the stability on nonlinear scalar delay differential equations is studied and improved 3 2 stability theorems are presented

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

the application of optimal control methodology to nonlinear programming problems dynamic programming techniques have proven to be more successful than alternative nonlinear programming algorithms for solving many discrete time optimal control problems the reason for this is that because of the stagewise decomposition which characterizes dynamic programming the computational burden grows approximately linearly with the numbern of decision times whereas the burden for other methods tends to grow faster e g n3 for newton s method the idea motivating the present study is that the advantages of dynamic programming can be brought to bear on classical nonlinear programming problems if only they can somehow be rephrased as optimal control problems

integration of accelerometer data filter analysis and design using riccati solutions the adjustment and integration of noisy accelerometer data is formulated as a discrete time optimal control problem which requires the minimization of a quadratic performance measure subject to linear dynamics the algebraic riccati equation associated with this problem is solved by expressing the entries of the riccati matrix explicitly in terms of the weights in the quadratic performance criterion steady state feedback gains are thus easily computed as functions of the weights the frequency domain analysis of the adjustment procedure provides the basis for selecting appropriate weights

global attractivity of solutions of nonlinear delay differential equations with a forcing term abstract in this paper we give sufficient conditions for every solution of the delay differential equation x t p t f x t r t to converge to zero our results extend and improve several known results in the literature an application of our main results to an equation derived from mathematical biology is also given

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