physical space time and nonrealizable text cr structures space time views leading up to einstein s general relativity are described in relation to some of poincare s early ideas on the subject the basic geometry of twistor theory is introduced as it arises both from minkowski space time and the more general curved einstein models it is shown how this provides a cr structure this being in essence another of poincare s poioneering concepts in a natural way nonrealizable cr structure can arise and an example is presented due to c d hill g a j sparling and the author of a complex manifold with boundary which cannot be extended as a complex manifold beyond its c sup infinity boundary

parametrization of optimal fault detection filters this paper is concerned with the fault detection filter design problem for linear time invariant systems subject to disturbance and possible faults to make a tradeoff between fault sensitivity and disturbance sensitivity frameworks such as 2 and are considered it is shown that for and frameworks all optimal fault detection filters can be characterized by an envelope besieged by the special solutions serving as upper and lower bounds furthermore the explicit parametrization is given in terms of a free contractive system parameter this free parameter provides freedom to emphasize fault and disturbance sensitivities at different frequency ranges which can be used to construct new optimal fault detection filter the results are also extended to the optimization over finite frequency range for which the upper and or lower bound of the optimal solution set may not be rational on the other hand it is shown that the solution set of 2 problem is unique up to square inner system finally examples are given to illustrate the results

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

the fujita exponent for the cauchy problem in the hyperbolic space abstract it is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the euclidean space the main purpose of this paper is to study its effect on the positive solutions of cauchy problems with power nonlinearities existence and non existence results for local solutions are derived emphasis is put on their long time behavior and on fujita s phenomenon to have the same situation as for the cauchy problem in r n namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent we must introduce a weight depending exponentially on the time in this respect the situation is similar to problems in bounded domains with dirichlet boundary conditions important tools are estimates for the heat kernel in the hyperbolic space and comparison principles

non realizable cr structures the examples given by nirenberg in 5 and 6 show that not all three dimensional cr manifolds may be realized as real hypersurfaces in 1172 these examples are perturbations of the cr structure on the sphere in this paper we extend his construction and show that any strictly pseudo convex realizable cr structure of dimension 3 may be perturbed so as to obtain a non realizable one this is done in sect 1 in sect 2 we show non realizability for higher dimensional structures provided the levi form is non degenerate and has only one positive eigenvalue in our last section we summarize cartan s approach to cr geometry from this viewpoint it was important to decide if nirenberg s example was really limited to perturbations of the sphere we have made this paper self contained and have provided proofs of several known results for instance lemma 1 l and the first part of lemma 1 5 were proved by lewy 2 and lemma 1 2 is the well known observation that a strictly pseudo convex hypersurface is convex in some coordinate system

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

travelling fronts and entire solutions of the fisher kpp equation in ℝn this paper is devoted to time global solutions of the fisher kpp equation in n where f is a c 2 concave function on 0 1 such that f 0 f 1 0 and f 0 on 0 1 it is well known that this equation admits a finite dimensional manifold of planar travelling fronts solutions by considering the mixing of any density of travelling fronts we prove the existence of an infinite dimensional manifold of solutions in particular there are infinite dimensional manifolds of nonplanar travelling fronts and radial solutions furthermore up to an additional assumption a given solution u can be represented in terms of such a mixing of travelling fronts

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

a geometric approach to the synthesis of failure detection filters a geometric formulation of beard s failure detection filter problem is stated using the concepts of c a invariant and unobservability subspaces the notions of output separable and mutually detectable families of subspaces introduced by beard are also clarified it is shown that mutual detectability is a necessary and sufficient condition for the existence of a detection filter with arbitrarily assignable spectrum moreover it is shown that the failure detection falter problem has a computationally simple solution when the failure events satisfy some mild restrictions finally the complete duality between a generalization of beard s detection filter problem and the restricted control decoupling problem is illustrated

steady state kalman filtering with an h error bound abstract an estimator design problem is considered which involves both l 2 least squares and h worst case frequency domain aspects specifically the goal of the problem is to minimize an l 2 state estimation error criterion subject to a prespecified h constraint on the state estimation error the h estimation error constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on the l 2 state estimation error the principal result is a sufficient condition for characterizing fixed order i e full and reduced order estimators with bounded l 2 and h estimation error the sufficient condition involves a system of modified riccati equations coupled by an oblique projection i e idempotent matrix when the h constraint is absent the sufficient condition specializes to the l 2 state estimation result given in 2

inversion based residual generation for robust detection and isolation of faults by means of estimation of the inverse dynamics in linear dynamical systems in this article the idea of inversion based direct input reconstruction for robust detection separation and estimation of multiple simultaneous faults in the presence of persistent disturbances which proved to be non separable by conventional decoupling methods in linear dynamical systems is presented in particular it is shown how in a specific filtering structure a residual generator relying on the inverse representation of the system by means of estimation of the inverse dynamics can be designed providing robust fault decoupling in the development of the method three solution approaches are contrasted with each other in the first approach the traditional h filter providing the classical disturbance attenuated non decoupling solution is presented then in a novel solution an h detection filter on the inverse system is designed providing detection residuals by exact fault decoupling the estimation of the inverse dynamics is obtained by means of the optimal filter thus ensuring h disturban

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

the freidlin gartner formula for general reaction terms abstract we devise a new geometric approach to study the propagation of disturbance compactly supported data in reaction diffusion equations the method builds a bridge between the propagation of disturbance and of almost planar solutions it applies to very general reaction diffusion equations the main consequences we derive in this paper are a new proof of the classical freidlin gartner formula for the asymptotic speed of spreading for periodic fisher kpp equations extension of the formula to the monostable combustion and bistable cases existence of the asymptotic speed of spreading for equations with almost periodic temporal dependence derivation of multi tiered propagation for multistable equations

transition semi wave solutions of reaction diffusion equations with free boundaries abstract in this paper we define the transition semi wave solution cf definition 1 1 of the following reaction diffusion equation with free boundaries 1 u t u x x f t x u t r x h t u t h t 0 t r h t u x t h t t r in the homogeneous case i e f t x u f u under the hypothesis f u c 1 0 1 f 0 f 1 0 f 1 0 f u 0 for u 1 we prove that the semi wave connecting 1 and 0 of 1 is unique provided it exists furthermore we prove that any bounded transition semi wave connecting 1 and 0 is exactly the semi wave in the cases where f is kpp fisher type and almost periodic in time space i e f t x u u c t u resp u a x u with c t resp a x being almost periodic applying totally different method we also prove any bounded transition semi wave connecting the unique almost periodic positive solution of u t u c t u resp u x x u a x u 0 and 0 is exactly the unique almost periodic semi wave of 1 finally we provide an example of the heterogeneous equation to show the existence of the transition semi wave without any global mean speeds

blow up of solutions to semilinear parabolic equations on riemannian manifolds with negative sectional curvature abstract on riemannian manifolds with negative sectional curvature we study finite time blow up and global existence of solutions to semilinear parabolic equations where the power nonlinearity is multiplied by a time dependent positive function h t we show that depending on the behavior at infinity of h either every solution blows up in finite time or a global solution exists if the initial datum is small enough in particular if h 1 we have global existence for small initial data whereas for h t e t a fujita type phenomenon appears for certain values of 0 a key role will be played by the infimum of the l 2 spectrum of the operator on m

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

a liouville type result for quasi linear elliptic equations on complete riemannian manifolds we extend a liouville type result of d g aronson and h f weinberger and e n dancer and y du concerning solutions to the equation pu b x f u to the case of a class of singular elliptic operators on riemannian manifolds which include the laplacian and are the natural generalization to manifolds of the operators studied by j serrin and collaborators in euclidean setting in the process we obtain an a priori lower bound for positive solutions of the equation in consideration which complements an upper bound previously obtained by the authors in the same context

how travelling waves attract the solutions of kpp type equations we consider in this paper a general reaction diffusion equation of the kpp kolmogorov petrovskii piskunov type posed on an infinite cylinder such a model will have a family of pulsating waves of constant speed larger than a critical speed c the family of all supercritical waves attract a large class of initial data and we try to understand how we describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity the solution will converge to a whole set of translates of the same wave and we identify the convergence dynamics as that of an effective drift around which an effective diffusion process occurs in several nontrivial particular cases we are able to describe the dynamics by an effective equation

fault detection in a mixed h sub 2 h sub spl infin setting in this note we study the problem of fault detection in linear time invariant lti systems which are simultaneously effected by two classes of unknown inputs noises having fixed spectral densities and unknown finite energy disturbances this problem is formulated as a mixed h sub 2 h sub spl infin filtering problem necessary and sufficient conditions for local optimality are presented moreover it is shown that suboptimal solutions can be computed by solving a convex minimization problem with a set of linear matrix inequality lmi constrains a numerical example is given to illustrate the advantage of the mixed h sub 2 h sub spl infin approach as compared to existing techniques which are based on optimization of h sub 2 and h sub spl infin criteria

a game theoretic fault detection filter the fault detection process is approximated with a disturbance attenuation problem the solution to this problem for both linear time varying and time invariant systems leads to a game theoretic filter which bounds the transmission of all exogenous signals except the fault to be detected in the limit when the disturbance attenuation bound is brought to zero a complete transmission block is achieved by embedding the nuisance inputs into an unobservable invariant subspace since this is the same invariant subspace structure seen in some types of detection filters we can claim that the asymptotic game filter is itself a detection filter one can also make use of this subspace structure to reduce the order of the limiting game theoretic filter by factoring this invariant subspace out of the state space the resulting lower dimensional filter will then be sensitive only to the failure to be detected a pair of examples given at the end of the paper demonstrate the effectiveness of the filter for time invariant and time varying problems in both full order and reduced order forms

hyperkahler structures and infinite dimensional grassmannians abstract in this paper we describe an example of a hyperkahler quotient of a banach manifold by a banach lie group although the initial manifold is not diffeomorphic to a hilbert manifold not even to a manifold modelled on a reflexive banach space the quotient space obtained is a hilbert manifold which can be furthermore identified either with the cotangent space of a connected component gr res j j z of the restricted grassmannian or with a natural complexification of this connected component thus proving that these two manifolds are isomorphic hyperkahler manifolds moreover kahler potentials associated with the natural complex structure of the cotangent space of gr res j and with the natural complex structure of the complexification of gr res j are computed using kostant souriau s theory of prequantization

transition fronts for the fisher kpp equation this paper is concerned with transition fronts for reaction diffusion equations of the fisher kpp type basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich in the paper we describe the class of transition fronts and we study their qualitative dynamical properties in particular we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate we also classify the transition fronts in the class of measurable superpositions of standard traveling fronts

spreading speeds and traveling waves for abstract monostable evolution systems abstract this paper is devoted to the development of the theory of spreading speeds and traveling waves for abstract monostable evolution systems with spatial structure under appropriate assumptions we show that the spreading speeds coincide with the minimal wave speeds for monotone traveling waves in the positive and negative directions then we use this theory to study the spatial dynamics of a parabolic equation in a periodic cylinder with the dirichlet boundary condition a reaction diffusion model with a quiescent stage a porous medium equation in a tube and a lattice system in a periodic habitat

propagation and extinction of fronts for semilinear parabolic equations on riemannian manifolds abstract we study the cauchy problem for the semilinear heat equation on riemannian manifolds propagation and extinction of solutions are addressed supposing that the nonlinear forcing term is either of kpp type or of bistable type in particular we highlight the influence both of sectional curvatures and of ricci curvature on such phenomena

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

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

on the nonlinear diffusion equation of kolmogorov petrovsky and piscounov abstract recently bramson proved a theorem that classifies the initial data under which solutions of the k p p equation converge to the appropriate travelling waves in this paper a simplified proof is given by using maximal principles instead of his brownian motion approach the regularity condition on the forcing term is also weakened

dynamics of interfaces in the fisher kpp equation for slowly decaying initial data abstract this paper is concerned with the behavior of solutions of the fisher kpp equation when initial data decay slowly in space first we show by a comparison technique that the motion of an interface or a thin transition layer can be approximated as a level set of a first order pde of hamilton jacobi type then by the method of characteristics it is shown that various but not all motion of interfaces can be observed by taking initial data appropriately finally we study the large time behavior of solutions especially the existence of similarly expanding interfaces

failure detection and identification using the geometric concept of an unobservability subspace a solution is given to the problem of detecting and identifying control system component failures in linear time invariant systems conditions are developed for the existence of a causal linear time invariant processor that can detect and uniquely identify a component failure first for the case where components can fail simultaneously and then for the case where they fail only one at a time explicit design algorithms are provided when these conditions are satisfied in addition to time domain solvability conditions frequency domain interpretations of the results are given and connections are drawn with results already available in the literature

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions