estimation and control of systems with unknown covariance and multiplicative noise the problem of estimation and control for systems with multiplicative noise and unknown second order statistics is considered conditions are found for the existence of a solution based on game theoretic ideas the conditions for the existence of a saddle point for the time invariant filtering problem are necessary and sufficient whereas for all other cases only necessary the central idea of the solution is to convert the stochastic problem to a deterministic optimal control problem whose minimax point is sought with respect to the control filter and unknown statistics parameters the results that are derived show that the problem of estimation for systems with unknown covariances depends on the costate matrix which in turn is a function of the performance measure thus the filter loses one of its best known properties that of independence of the performance functional this property holds not only for the classical kalman filter but also for multiplicative systems

polynomial filtering for linear discrete time non gaussian systems in this work we propose a new filtering approach for linear discrete time non gaussian systems that generalizes a previous result concerning quadratic filtering a de santis a germani and m raimondi ieee trans automat control 40 1995 pp 1274 1278 a recursive nu th order polynomial estimate of finite memory delta is achieved by defining a suitable extended state which allows one to solve the filtering problem via the classical kalman linear scheme the resulting estimate will be the mean square optimal one among those estimators that take into account nu polynomials of the last delta observations numerical simulations show the effectiveness of the proposed method

existence and regularity of isometries we use local harmonic coordinates to establish sharp results on the regularity of isometric maps between riemannian manifolds whose metric tensors have limited regularity e g are holder continuous we also discuss the issue of local flatness and of local isometric embedding with given first and second fundamental form in the context of limited smoothness

minimax linear observers and regulators for stochastic systems with uncertain second order statistics the problem of minimax design of linear observers and regulators for linear time varying multivariable stochastic systems with uncertain models of their second order statistics is treated in this paper general classes of allowable covariance matrices and means of the process and observation noises and of the random initial condition are considered a game formulation of the problem is adopted and it is shown that the optimal filter for the least favorable set of covariances is minimax robust for each of the filtering situations analyzed conditions satisfied by the saddle point solutions are given and their utility for finding the worst case covariances is illustrated by way of several examples of uncertainty classes of practical interest

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

limits and degenerations of unitary conformal field theories in the present paper degeneration phenomena in conformal field theories are studied for this purpose a notion of convergent sequences of cfts is introduced properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of cfts which as familiar from large volume limits of non linear sigma models can be regarded as commutative degenerations of the corresponding quantum geometries as an application the large level limit of the a series of unitary virasoro minimal models is investigated in detail in particular its geometric interpretation is determined

minimax quadratic filtering of uncertain linear stochastic systems with partial fourth order information a minimax filtering problem for a class of uncertain linear stochastic systems is studied uncertainties involving fourth order moments of the noise distribution and of the initial state are considered under the hypothesis that the second order statistics are known hence the linear filter is available the quadratic minimax filter is found moreover it is shown that the minimax filter gives a worst case error variance even less than the exact error variance of the linear filter numerical simulations show that this improvement is meaningful also in cases of great uncertainty regarding the higher order statistics

a two dimensional version of the godunov scheme for scalar balance laws a godunov scheme is derived for two dimensional scalar conservation laws without or with source terms following ideas originally proposed by boukadida and leroux math comput 63 1994 pp 541 553 in the context of a staggered lax friedrichs scheme in both situations the numerical fluxes are obtained at each interface of a uniform cartesian computational grid just by means of the external waves involved in the entropy solution of the elementary two dimensional 2d riemann problems in particular all the wave interaction phenomena are overlooked this restriction of the wave pattern suffices for deriving the exact numerical fluxes of the staggered lax friedrichs scheme but it furnishes only an approximation for the godunov scheme we show that under convenient assumptions these flux functions are smooth and the resulting discretization process is stable under nearly optimal cfl restriction a well balanced extension is presented relying on the curl free component of the helmholtz decomposi

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

optimal filtering in stochastic discrete time systems with unknown inputs in this note we derive a recursive filtering algorithm for the linear discrete time dynamic system with indeterminate stochastic inputs the algorithm is based on the minimax optimal method of parameter estimation in the linear regression model with parameters of two different types unknown and stochastic with partially known characteristics

symmetric hyperbolic systems in algebras of generalized functions and distributional limits we study existence uniqueness and distributional aspects of generalized solutions to the cauchy problem for first order symmetric or hermitian hyperbolic systems of partial differential equations with colombeau generalized functions as coefficients and data the proofs of solvability are based on refined energy estimates on lens shaped regions with spacelike boundaries we obtain several variants and also partial extensions of previous results in oberguggenberger 1989 lafon and oberguggenberger 1991 and hormann 2004 26 23 16 and provide aspects accompanying related recent work in oberguggenberger 2009 garetto and oberguggenberger 2011 28 10 9

a boundary of the set of the riemannian manifolds with bounded curvatures and diameters on etudie la structure de la fermeture de l ensemble constitue par des classes d isometrie de varietes de riemann a courbures et diametres bornes

minimax control of linear stochastic systems with noise uncertainty the problem of linear quadratic gaussian control of multivariable linear stochastic systems with uncertain second order statistical properties is considered uncertainty is modeled by allowing process and observation noise spectral density matrices to vary arbitrarily within given classes and a minimax control formulation is applied to the quadratic objective functional general theorems proving the existence and characterization of saddle point solutions to this problem are presented and the relationship of these results to earlier results on minimax state estimation are discussed to illustrate the analytical results the specific example of regulating a double integrator plant is treated in detail

robust filtering and prediction for linear systems with uncertain dynamics a game theoretic approach we examine the existence and behavior of game theoretic solutions for robust linear filters and predictors our basic uncertainty class includes m th order time varying discrete time systems with uncertain dynamics uncertain initial state covariance and uncertain nonstationary input and observation noise covariance our results include recursive kalman filter predictor realizations for the resulting robust procedures our approach is based on saddle point theory we emphasize the notion of a least favorable prior distribution for the uncertain parameter values to obtain a worst case design technique in this paper we highlight the role such distributions with finite support play in these decision models in particular we demonstrate that in these decision models the least favorable prior distribution is always discrete

the elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions one dimensional collisions between two isotropic solids in which the equations of physics lead to multiplications of distributions are considered based on this example a general physicomathematical method to be adapted to each particular case is proposed to resolve the ambiguity inherent in such products this can be achieved with the aid of a new mathematical theory of generalized functions which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point this tool has recently been applied in various situations in continuum mechanics in which the equations of physics lead to heuristic products of distributions one obtains new algebraic formulas in the simplest cases and new numerical schemes in more general cases the key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory and have no analog in classical analysis so that in general a resolution cannot be obtained from formal calculations

collapsing riemannian manifolds to ones of lower dimensions on considere l effondrement d une suite de varietes de riemann mi vers un espace de dimension inferieure x on etudie le cas ou l espace limite x est une variete de riemann et les courbures sectionnelles de mi sont bornees

noncausal minimax linear state estimation for systems with uncertain second order statistics this note considers the problem of minimax state estimation of the states of a linear time invariant system which is driven by and observed in the presence of noise processes with uncertain second order statistics when the process noise and observations are scalars the problem is shown to be equivalent to a scalar minimax estimation problem the existence of a minimax solution is thereby established and the minimax filter is shown to be a linear transformation of the minimax filter for the scalar problem

rationality of geometric signatures of complete 4 manifolds according to cg4 and cfg the complete manifolds with bounded sectional curvature and finite volume admit positive rank f structures near infinity in this paper we show that in dimension four if the manifolds also have bounded covering geometry near infinity then there exist f structures with special topological properties f structures with these properties cannot be constructed solely by means of the general methods in cg4 and cfg using these special properties we prove a conjecture of cheeger gromov on the rationality of the geometric signatures in the four dimensional case

hyperbolic systems with discontinuous coefficients generalized solutions and a transmission problem in acoustics where v v v g g g a and f are n x n matrices a is real valued and diagonal in the case of discontinuous coeffkients more precisely we wish to assume solely that a is a bounded measurable function and f is the distributional derivative of such a function the example of hurd and sattinger 3 p 1661 shows that in general such a system will not have solutions in the sense of distributions even if it is in the form of a conservation law observe that a priori 8 v will not be a function and so multiplication of distributions is involved in 0 1 our approach will be to solve the problem in the colombeau algebra y a differential algebra which contains the distributions and in addition has the algebra of infinitely differentiable functions as a sub algebra in section 1 we shall show that if a f g are generalized functions belonging to 9 lr2 and a belongs to 9 r then system 0 1 has a unique solution ve a r provided the generalized functions a and f satisfy some assumptions corresponding to a e l lr fe w l m lr distributional solutions are recovered from the generalized solutions via a partial projection 452 0022 247x 89 3 00

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

polynomial filtering of discrete time stochastic linear systems with multiplicative state noise the problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied for such models a technique of state augmentation is used leading to the definition of a general polynomial filter the theory is developed for time varying systems with nonstationary and non gaussian noises moreover the steady state polynomial filter for stationary systems is also studied numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques

first order hyperbolic pseudodifferential equations with generalized symbols abstract we consider the cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized resembling a representative of a colombeau generalized function such equations arise for example after a reduction decoupling of second order model systems of differential equations in seismology we prove existence of a unique generalized solution under log type growth conditions on the symbol thereby extending known results for the case of differential operators j math anal appl 160 1991 93 106 j math anal appl 142 1989 452 467

hyperbolic second order equations with non regular time dependent coefficients in this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients this means assuming that the coefficients are less regular than holder the characteristic roots are also allowed to have multiplicities for such equations we describe the notion of a very weak solution adapted to the type of solutions that exist for regular coefficients the construction is based on considering friedrichs type mollifiers of coefficients and corresponding classical solutions and their quantitative behaviour in the regularising parameter we show that even for distributional coefficients the cauchy problem does have a very weak solution and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist in concrete applications the dependence on the regularising parameter can be traced explicitly

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly