beam summation theory for waves in fluctuating media part ii stochastic field representation in part i of this two part sequence we introduced a novel beam summation bs scheme for tracking waves through fluctuating media a key step has been the proof that the set of beam propagators constitutes a frame everywhere in the propagation domain this beam frame bf provides a self consistent framework for wave tracking where the field is expanded using the bf and the local scattering of each beam is reexpanded using the same beam set and expressed as beam to beam b2b scattering coefficients this theory is used here to derive a bs representation for the stochastic field for cases where the medium fluctuations are random with a given statistics the stochastic b2b scattering moments are therefore expressed in terms of the local spectral statistics of the medium projected onto a phase space window formed by the intersection of the excitation and the scattered beams since the medium statistics is typically smooth unlike a single realization the resulting stochastic b2b scattering matrix is compact and smooth and may actually be calculated analytically it is demonstrated via numerical examples and a comparison with monte carlo simulations that the formulation is not only computationally efficient but also provides a compact representation for the scattering phenomenology

ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

the homotopy type of four dimensional knot complements one of the more useful ways of studying knots is by studying various algebraic invariants of their complements this paper studies the general problem of characterizing smooth four dimensional knot complements and gives examples which show how certain algebraic invariants can fail to determine the k n o t complement for example two well known theorems in three manifold theory show that i the homotopy type of a knot exterior in s 3 is determined by its fundamental group and ii the homeomorphism type of a knot exterior in s 3 is determined by its fundamental group together with the conjugacy class of the z x7z subgroup of 1 coming from the boundary so that a boundary preserving homotopy equivalence between knot exteriors can be deformed to a homeomorphism we show how both of these properties can fail in dimension four for a knot in s 4 we have the following hierarchy of properties of their exteriors that is successively stronger statements which one might make about two knot exteriors x and x the exterior of a knot k in s 4 is s 4 k x 2 the complement of an open tubular neighborhood of k

phase space beam summation analysis of rough surface waveguide a gaussian beam summation gbs formulation is introduced for a doubly rough boundary waveguide wherein the coherent and incoherent scattered fields are decomposed into a discrete phase space summation of gaussian beams gb that emanate from the rough surfaces in all directions the scheme involves deterministic gb propagators and stochastic gb to gb gb2gb scattering matrices for the coherent and incoherent fields where each scattered beam is propagated inside the waveguide and is scattered again from the rough boundaries the gb2gb matrices are calculated from the statistical moments of the scattering amplitude which are given either analytically or empirically an analytical and numerical example for a waveguide with weak boundary roughness is presented and discussed the formulation reveals explicitly the phase space footprint of the stochastic multiple scattering process at the rough boundaries thus providing a cogent physical interpretation and an effective mathematical representation to the f

a phase space beam summation formulation for ultrawide band radiation a new discrete phase space gaussian beam gb summation representation for ultrawide band uwb radiation from an aperture source distribution is presented the formulation is based on the theory of the windowed fourier transform wft frames wherein we introduce a novel relation between the frequency and the frame overcompleteness with this procedure the discrete lattice of beams that are emitted by the aperture satisfies the main requirement of being frequency independent so that only a single set of beams needs to be traced through the medium for all the frequencies in the band it is also shown that a properly tuned class of iso diffracting id gaussian windows provides the snuggest frame representation for all frequencies thus generating stable and localized expansion coefficients furthermore due to the id property the resulting gbs propagators are fully described by frequency independent matrices whose calculation in the ambient environment need to be done only once for all frequencies consequently the theory may also be expressed directly in the time domain as will be presented elsewhere the localization implied by the new formulation is demonstrated numerically for an uwb focused aperture it is shown that the algorithm extracts the local radiation properties of the aperture source and enhances only those beams that conform with these properties i e those residing near the phase space lagrange manifold further localization is due to the fact the algorithm accounts only for beams that pass within a few beamwidths vicinity of the observation point it is thus shown that the total number of beams is much smaller than the landau pollak bound on the aperture s degrees of freedom

free involutions on s 1 s n topological free involutions on s 1 s n are classified up to conjugation weprovethatthisisthesameasclassifyingquotientmanifoldsuptohomeomorphism there are exactly four possible homotopy types of such quotients and surgery theory is used to classify all manifolds within each homotopy type

the algebraic characterization of the exteriors of certain 2 knots we study 2 knots with virtually solvable group by applying recent work of freedman to the 4 manifolds obtained by surgery on such knots in particular we show that gluck reconstruction is the only ambiguity in recovering the cappell shaneson knots as top locally flat knots from their groups alone

last exit decompositions and regularity at the boundary of transition probabilities the purpose of this paper is to give a probabilistic approach to studying the regularity at the boundary of the transition probabilities of certain hypoelliptic diffusions with boundary conditions the main tools are last exit decompositions of brownian motion the malliavin calculus the theory of excursions and the calculus of variations on brownian excursions

structure sets vanish for certain bundles over seifert manifolds let m i be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold k3 with an effective circle action of hyperbolic type assume that the fiber n in this bundle is a closed orientable manifold with noetherian integral group ring with vanishing projective class and whitehead groups and such that the structure set sjop n x dk 3 of topological surgery vanishes for sufficiently large k then the projective class and whitehead groups of m vanish and st0p m 3 x dk 3 0 if n k 3 or if at3 is closed and n 2 the unil groups of cappell are the main obstacle here and these results give new examples of generalized free products of groups such that unily vanishes in spite of the failure of cappell s sufficient condition a conjecture of long standing holds that a closed aspherical manifold is de termined up to homeomorphism by its fundamental group this claim is now known to be valid for aspherical manifolds of dimensions greater than four whose funda mental groups contain nilpotent subgroups of finite index 8 10 earlier work verified the conjecture for poly z fundamental groups subject to the same dimen sion restrictions 30 pp 228 231 this paper will show that the conjecture and related statements are valid for compact manifolds m which are total spaces of certain bundles over surfaces or 3 manifolds the fibers of these bundles are closed orientable manifolds n satisfy ing these two hypotheses

invariants of conformal laplacians the conformal laplacian d d d n 2 s 4 n 1 acting on functions on a riemannian manifold mn with scalar curvature s is a conformally invariant operator in this paper we will use d to construct new conformal invariants one of these is a pointwise invariant one is the integral of a local expression and one is a nonlocal spectral invariant derived from functional determinants we begin in 1 by describing the laplacian d and its green function in the context of conformal geometry we then derive a basic formula giving the variation in the heat kernel of d this formula is strikingly simpler than the corresponding formula for the ordinary laplacian given by ray and singer 15 the heat kernel of d has an asymptotic expansion k t x x 4 t n 2 ak x tk in 2 we prove that a n 2 2 is a pointwise conformal invariant of weight 2 i e it satisfies a n 2 2 x 2g 2a n 2 2 x g where g is the metric and is any smooth positive function in particular this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold the key to the proof is to consider the parametrix of the green s function which is obtained from the heat kernel by an integral transform one finds that a n 2 2 occurs as the coefficient of the first log term in this parametrix and its conformal invariance then follows directly from the conformal invariance of the green s function in 3 we show that an 2 is a global conformal invariant the calculations in 4 show that it is not a pointwise invariant the proof is a direct calculation of the invariant of an 2 using equation 1 10

densities for rough differential equations under hormander s condition we consider stochastic differential equations dy v y dx driven by a multidimensional gaussian process x in the rough path sense t lyons rev mat iberoamericana 14 1998 215 310 using malliavin calculus we show that y t admits a density for t 0 t provided i the vector fields v v 1 v d satisfy hormander s condition and ii the gaussian driving signal x satisfies certain conditions examples of driving signals include fractional brownian motion with hurst parameter h 1 4 the brownian bridge returning to zero after time t and the ornstein uhlenbeck process

on the orders of periodic diffeomorphisms of 4 manifolds this paper initiated an investigation on the following question sup pose a smooth 4 manifold does not admit any smooth circle actions does there exist a constant c 0 such that the manifold support no smooth zp actions of prime order for p c we gave affirmative results to this question for the case of holomorphic and symplectic actions with an interesting finding that the constant c in the holomorphic case is topological in nature while in the symplectic case it involves also the smooth structure of the manifold

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

smooth s cobordisms of elliptic 3 manifolds the main result of this paper states that a symplectic s cobordism of elliptic 3 manifolds is diffeomorphic to a product assuming a canonical contact structure on the boundary based on this theorem we conjecture that a smooth s cobordism of elliptic 3 manifolds is smoothly a product if its universal cover is smoothly a product we explain how the conjecture fits naturally into the program of taubes of constructing symplectic structures on an oriented smooth 4 manifold with b 1 from generic self dual harmonic forms the paper also contains an auxiliary result of independent interest which generalizes taubes theorem sw gr to the case of symplectic 4 orbifolds

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

sensitivity analysis using ito malliavin calculus and martingales and application to stochastic optimal control we consider a multidimensional diffusion process x alpha t 0 leq t leq t whose dynamics depends on a parameter alpha our first purpose is to write as an expectation the sensitivity nabla alpha j alpha for the expected cost j alpha mathbb e f x alpha t in order to evaluate it using monte carlo simulations this issue arises for example from stochastic control problems where the controller is parameterized which reduces the control problem to a parametric optimization one or from model misspecifications in finance previous evaluations of nabla alpha j alpha using simulations were limited to smooth cost functions f or to diffusion coefficients not depending on alpha see yang and kushner siam j control optim 29 1991 pp 1216 1249 in this paper we cover the general case deriving three new approaches to evaluate nabla alpha j alpha which we call the malliavin calculus approach the adjoint approach and the martingale approach to accomplish this we leverage ito calculus malliavin calculus and martingale arguments in the second part of this work we provide discretization procedures to simulate the relevant random variables then we analyze their respective errors this analysis proves that the discretization error is essentially linear with respect to the time step this result which was already known in some specific situations appears to be true in this much wider context finally we provide numerical experiments in random mechanics and finance and compare the different methods in terms of variance complexity computational time and time discretization error

hypoellipticity for filtering problems of partially observable diffusion processes we prove that under hormander s type conditions on the coefficients of the unobservable component of a partially observable diffusion process the filtering density is infinitely differentiable and can be represented as the integral of an infinitely differentiable kernel against the prior initial distribution these results are derived from more general results obtained for spdes one of the main novelties of the paper is the existence and smoothness of the kernel another one is that we allow the coefficients of our partially observable process to be just measurable with respect to the time variable and lipschitz continuous with respect to the observation variable

a phase space gaussian beam summation representation of rough surface scattering a gaussian beam gb summation representation for rough surface scattering is introduced in this scheme the coherent and incoherent scattered fields are described by a phase space summation of gbs that emanate from the rough surface at discrete set of points and directions it thus involves stochastic gb2gb scattering matrices for the coherent and incoherent fields and deterministic gb propagators it benefits from the simplicity and accuracy of the latter and can be used in applications involving propagation in complex scenarios comprising inhomogeneous media with rough surface boundaries the gb2gb matrices are calculated from the statistical moments of the scattering amplitude which are given either analytically or empirically an analytical and numerical example for weakly rough surface is presented and discussed applications to the more complicated propagation scenario of doubly rough surface waveguide with multiple reflection phenomena will be presented in a follow up publication

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

on the connection between the malliavin covariance matrix and hormander s condition abstract a celebrated theorem of hormander gives a sufficient condition for a second order differential operator to be hypoelliptic for operators with analytic coefficients this condition turns out to be also necessary but this is not true for general smooth coefficients on the other hand malliavin conceived a probabilistic approach to the same problem known as malliavin calculus in which a key role is played by the malliavin covariance matrix the aim of our paper is to give several characterizations of the malliavin covariance matrix which are equivalent to hormander s condition and consequently imply the hypoellipticity in this way the distance between hormander s condition and the hypoellipticity property is clearly pointed out in probabilistic terms

density estimates for a random noise propagating through a chain of differential equations we here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d the first one being forced by a non degenerate random noise and the n 1 other ones being degenerate the system formed by the n equations satisfies a suitable hormander condition the second equation feels the noise plugged into the first equation the third equation feels the noise transmitted from the first to the second equation and so on so that the noise propagates one way through the system when the coefficients of the system are lipschitz continuous we show that the density of the solution satisfies gaussian bounds with non diffusive time scales the proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem

hypoelliptic non homogeneous diffusions let be a time dependent second order operator written in usual or hormander form we study the regularity of the law of the associated non homogeneous time dependent diffusion process under hormander s like conditions coefficients are only holder continuous in time the main tool is malliavin calculus our results extend and correct previous ones 17 and related works 15 related topics like filtering theory killed or reflected processes parabolic hypoellipticity are also discussed

beam summation theory for waves in fluctuating media part i the beam frame and the beam domain scattering matrix we present a novel beam summation bs formulation for tracking wavefields in fluctuating media this formulation utilizes the ultrawideband phase space bs method which is structured upon a windowed fourier transform wft frame expansion of the sources and expresses the field as a discrete phase space sum of beam propagators this paper extends the frame concept proving that this beam set constitutes a frame not only in the source domain where it reduces to the conventional wft but everywhere in the propagation domain this beam frame provides a self consistent framework for tracking wavefields through scattering media in which the local interaction of the incident beams with the medium is expanded using the same set of beams the resulting beam to beam b2b scattering coefficients depend on the local spectral properties of the medium the overall b2b scattering matrix is therefore compact coupling only beams that are adjacent in phase space given the fluctuation properties the new formulation expresses the entire scattering problem in terms of coefficients dynamics in the phase space as demonstrated the formulation is computationally efficient and captures all the relevant phenomenology part ii extends this formulation for stochastic fields in random fluctuating medium characterized by the medium statistics

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2