transmutation techniques and observability for time discrete approximation schemes of conservative systems in this article we consider abstract linear conservative systems and their time discrete counterparts our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time discrete one as an application we show how observability properties for the time continuous model yield uniform with respect to the time step observability results for its time discrete approximation counterparts provided the initial data are suitably filtered the main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time discrete models besides using a reverse representation formula we also prove that this estimate on the time of uniform observability for the time discrete models is sharp we then conclude with some general comments and open problems

2d grushin type equations minimal time and null controllable data abstract we study internal null controllability for degenerate parabolic equations of grushin type g x x 2 x 2 y y 2 0 in the rectangle x y 1 1 0 1 previous works proved that null controllability holds for weak degeneracies small and fails for strong degeneracies large moreover in the transition regime and with strip shaped control domains a positive minimal time is required in this paper we work with controls acting on two strips symmetric with respect to the degeneracy we give the explicit value of the minimal time and we characterize some initial data that can be steered to zero in time t when the system is not null controllable their regularity depends on the control domain and the time t we also prove that with a control that acts on one strip touching the degeneracy line x 0 then grushin type equations are null controllable in any time t 0 and for any degeneracy 0 our approach is based on a precise study of the observability property for the one dimensional heat equations satisfied by the fourier coefficients in variable y this precise study is done through a transmutation process on the resulting one dimensional wave equations by lateral propagation of energy method

simply connected 4 manifolds with a given boundary let m be a closed oriented connected 3 manifold for each bilinear symmetric pairing zn l our goal is to calculate the set vl m of all oriented homeomorphism types of compact 1 connected oriented 4manifolds with boundary m and intersection pairing isomorphic to zn l for each pair zn l which presents h m we construct a double coset space bl m and a function ct vl m bl m the set bt m is the quotient of the group of all link pairing preserving isomorphisms of the torsion subgroup of h1 m by two naturally occuring subgroups when zn l is an even pairing we construct another double coset space bl m a function 6l vl m b fl m and a projection p2 bl m bl m such that p2 c l ct our main result states that when zn l is even the function c l is injective as is the function c x a vl m bl m x z 2 when z l is odd here a is a kirby siebenmann obstruction to smoothing it follows that the sets vl m are finite and of an order bounded above by a constant depending only on h1 m we also show that when hi m q 0 and z l is even ct p2 c 6l is injective it seems likely that via the functions cl x a and 6l the sets bl m x z 2 and bl m calculate vl m when zn l is respectively odd and even we verify this in several cases most notably when h1 m is free abelian the results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1connected 4 manifolds extending a given homeomorphism of their boundaries the theory developed is then applied to show that there is an m 0 depending only on h1 m such that for any self homeomorphism f of m fm extends to a self homeomorphism of any 1 connected compact 4 manifold with boundary m introduction in m freedman s fundamental paper fr he classified closed simply connected oriented 4 manifolds up to orientation preserving homeomorphism he showed that modulo the kirby siebenmann invariant these manifolds are in bijective correspondence through their intersection pairings with the set of unimodular bilinear symmetric pairings over z see theorem 1 5 of fr and corollary 2 2 3 of q in this paper we begin a classification of simply connected 4 manifolds with connected boundary our results with regard to this problem are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between 1 connected 4 manifolds extending a given homeomorphism of their boundaries received by the editors september 19 1985 1980 mathematics subject classification 1985 revision primary 57n15 research supported by an nserc canada post doctoral fellowship 1986 american mathematical society 0002 9947 86 1 00 25 per page

note on the cost of the approximate controllability for the heat equation with potential abstract we prove the approximate controllability for the heat equation with potential with a cost of order ec e when the target is in h 0 1 with a precision in l 2 norm also a quantification estimate of the unique continuation for initial data in l 2 of the heat equation with potential is established

on the observability of time discrete conservative linear systems abstract we consider various time discretization schemes of abstract conservative evolution equations of the form z a z where a is a skew adjoint operator we analyze the problem of observability through an operator b more precisely we assume that the pair a b is exactly observable for the continuous model and we derive uniform observability inequalities for suitable time discretization schemes within the class of conveniently filtered initial data the method we use is mainly based on the resolvent estimate given by burq and zworski in n burq m zworski geometric control in the presence of a black box j amer math soc 17 2 2004 443 471 electronic we present some applications of our results to time discrete schemes for wave schrodinger and kdv equations and fully discrete approximation schemes for wave equations

wave splitting and lattice dynamics a wave splitting approach used elsewhere to solve a black hole scattering problem is adapted to the formal solution of arbitrary self adjoint wave equations in 1 1 dimensions and yields potentially useful results in this more general case it is then shown that the self adjoint wave equation being solved and the non self adjoint linear wave equations satisfied by the one way component waves are naturally related to a pair of motions of the 1 1 dimensional toda lattice that together comprise a motion of the kac van moerbeke lattice this provides a partial explanation for the peculiar fact that every motion of the kac van moerbeke lattice can be viewed as two interpolated motions of the toda lattice

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

on the controllability of a fractional order parabolic equation the null controllability property of a 1 d parabolic equation involving a fractional power of the laplace operator delta alpha is studied the control is a scalar time dependent function g g t acting on the system through a given space profile f f x on the interior of the domain thus the control g determines the intensity of the space control f applied to the system the latter being given a priori we show that if alpha leq 1 2 and the shape function f is say in l 2 no initial datum belonging to any sobolev space of negative order may be driven to zero in any time this is in contrast with the existing positive results for the case alpha 1 2 and in particular for the heat equation that corresponds to alpha 1 this negative result exhibits a new phenomenon that does not arise either for finite dimensional systems or in the context of the heat equation on the contrary if more regularity of the shape function f is assumed then we show that there are initial data in any sobolev space h m that may be controlled once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which when increasing the regularity of the control profile the space of controllable data decreases these results show that in order for the control properties of the heat equation to be true the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when alpha 1 2 for the fractional powers of the dirichlet laplacian in 1 d the results we present here are in nature and with respect to techniques of proof similar to those on the control of the heat equation in unbounded domains in s micu and e zuazua trans amer math soc 353 2000 pp 1635 1659 and s micu and e zuazua portugal math 58 2001 pp 1 24 we also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models

linear wave equations as motions on a toda lattice a bijection is defined from the set of motions on the infinite toda lattice of strings to a set of sequences of linear wave equations in 1 1 dimensions the sequences being generated by a generalisation of the classical darboux map the bijection is applied to find probably all such wave equations for which characteristic initial data propagate without spreading

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

a generalized internal control for the wave equation in a rectangle abstract when is a rectangle in r 2 and a disk inside for each t 0 sufficiently large we establish the exact controllability of the evolution system y y h t x in 0 t y 0 on 0 t by means of a generalized control h t x localized on 0 t in other terms for any prescribed initial state of finite energy we find h t x a kind of analytic functional in x depending smoothly on t supported on 0 t such that y t y t 0

long time behavior of a coupled heat wave system arising in fluid structure interaction this paper is devoted to the analysis of the long time behavior of a coupled wave heat system in which a wave and a heat equation evolve in two bounded domains with natural transmission conditions at a common interface these conditions couple in particular the heat unknown with the velocity of the wave solution this model may be viewed as a simplified version of linearized models which arise in fluid structure interaction first we show the strong asymptotic stability of solutions to this system then based on the construction of ray like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface we show the lack of uniform decay in general domains further we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain finally in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface we also analyze the difficulty there is to extend this result to the more natural transmission conditions

a proof of the smale conjecture diff s3 0 4 the smale conjecture 9 is the assertion that the inclusion of the orthogonal group 0 4 into diff s3 the diffeomorphism group of the 3 sphere with the cw topology is a homotopy equivalence there are many equivalent forms of this conjecture some of which are listed in the appendix to this paper we shall prove

stable mapping class groups of 4 manifolds with boundary we give a complete algebraic description of the mapping class groups of compact simply connected 4 manifolds with boundary up to connected sum with copies of s 2 s 2

on discrete schrodinger equations and their two component wave equation equivalents an approach to inverse scattering problems for discrete schrodinger equations which are discrete three term recursions is presented by systematically transforming them into discrete two component wave propagation equations the wave propagation equations permit the immediate application of certain computationally efficient and physically insightful layer peeling algorithms for inverse scattering the mapping of three term recursions to two component evolution equations is one to many because the relation between the potential sequence parametrizing schrodinger equations and the reflection coefficient sequence determining local wave interaction is a nonlinear difference equation this mapping is examined in some detail and it is used to study both direct and inverse scattering problems associated with discrete schrodinger equations

attainable subspaces and the bang bang property of time optimal controls for heat equations in this paper we study two subjects on internally controlled heat equations with time varying potentials the attainable subspaces and the bang bang property for some time optimal control problems we present some equivalent characterizations on the attainable subspaces and provide a sufficient condition to ensure the bang bang property both the above mentioned characterizations and the sufficient condition are closely related to some function spaces consisting of some solutions to the adjoint equations it seems for us that the existing ways to derive the bang bang property for heat equations with time invariant potentials see for instance h o fattorini infinite dimensional linear control systems the time optimal and norm optimal problems elsevier amsterdam 2005 f gozzi and p loreti siam j control optim 37 1999 pp 1195 1221 v mizel and t seidman siam j control optim 35 1997 pp 1204 1216 and g wang siam j control optim 47 2008 pp 1701 1720 do not work

self adjoint acoustic equations with progressing wave solutions the simplest wave equations are those whose general solutions comprise progressing waves the author constructs a comparatively large possibly exhaustive family of self adjoint acoustic equations in 1 1 dimensions that are simple in this sense

the stable mapping class group of simply connected 4 manifolds we consider mapping class groups m 0diff m fix m of smooth compact simply connected oriented 4 manifolds m bounded by a collection of 3 spheres we show that if m contains cp 2 or cp 2 as a connected summand then all dehn twists around 3 spheres are trivial and furthermore m is independent of the number of boundary components by repackaging classical results in surgery and handlebody theory from wall kreck and quinn we show that the natural homomor phism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with cp 2 cp 2 we next consider the 3 1 dimensional cobordism 2 category c of 3 spheres 4 manifolds as above and enriched with isotopy classes of diffeomorphisms as 2 morphisms we identify the homotopy type of the classifying space of this cat egory as the hermitian algebraic k theory of the integers we also comment on versions of these results for simply connected spin 4 manifolds finally we observe that a related 4 manifold operad detects infinite loop spaces

simple toda lattice motions and their linear wave equations a structure preserving bijection b m to w from the set m of toda lattice motions to a natural partition w of the set of linear wave equations in 1 1 dimensions is applied to obtain some simple and related results about both m and w the toda lattice motions derived emphasise the existence of qualitatively distinct toda lattices depending on the overall sign in the defining dynamical equations the wave equations that arise illustrate a duality on the set of linear wave equations that is induced by the known duality on m generated by interpolating elements of m into motions of kac van moerbeke lattices

solvable systems of wave equations and non abelian toda lattices relates equivalence classes of coupled systems of n linear wave equations to motions of an n n matrix dynamical systems the two dimensional non abelian toda lattice in particular the correspondence is shown to relate those coupled systems of wave equations with progressing wave general solutions to motions of the finite non abelian toda lattice with free ends generalizing a known result for the n 1 case some non trivial motions of such toda lattices are found and the corresponding coupled wave equations and their progressing wave general solutions are given other consequences of the correspondence and possible application of the progressing waves are discussed

sharp observability estimates for heat equations the goal of this article is to derive new estimates for the cost of observability of heat equations we have developed a new method allowing one to show that when the corresponding wave equation is observable the heat equation is also observable this method allows one to describe the explicit dependence of the observability constant on the geometry of the problem the domain in which the heat process evolves and the observation subdomain we show that our estimate is sharp in some cases particularly in one space dimension and in the multi dimensional radially symmetric case our result extends those in fattorini and russell arch rational mech anal 43 272 292 1971 to the multi dimensional setting and improves those available in the literature namely those by miller j differ equ 204 1 202 226 2004 siam j control optim 45 2 762 772 2006 atti accad naz lincei cl sci fis mat natur rend lincei 9 mat appl 17 4 351 366 2006 and tenenbaum and tucsnak j differ equ 243 1 70 100 2007 our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation in contrast to the standard application of transmutation methods which uses a reverse representation of the heat solution in terms of the wave one we shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side we will also comment on the applications of our techniques to controllability properties of heat type equations