kalman filtering with intermittent observations on the boundedness of the expected error covariance this paper addresses the stability of a kalman filter when measurements are intermittently available due to constraints in the communication channel between the sensor and the estimator we give a necessary condition and a sufficient condition with a trivial gap between them for the boundedness of the expected value of the estimation error covariance these conditions are more general than the existing ones in the sense that they only require the state matrix of the system to be diagonalizable and the sequence of packet losses to be a stationary finite order markov process hence we extend the class of systems for which these conditions are known in two directions namely by including degenerate systems and by considering network models more general than i i d and gilbert elliott we show that these conditions recover known results from the literature when evaluated for non degenerate systems under the assumption of i i d or gilbert elliott packet loss models

birkhoff formula for conformal compressions of symmetric cones in this paper we derive the birkhoff formula for conformal compressions on symmetric cones let v be a euclidean jordan algebra and let be the associated symmetric cone then admits a natural finsler metric contracted by any conformal compressions of we show that the lipschitz constant of a conformal compression of is equal to the hyperbolic tangent of one fourth of the diameter of the image this is the same relation which was obtained by birkhoff on positive reals and by liverani and wojtkowski on the space of positive definite real matrices

effective and big divisors on a projective symmetric variety abstract we describe the effective and the big cones of a projective symmetric variety moreover we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor on a projective symmetric variety when the divisor is g stable such criterion has an explicit geometric interpretation finally we describe the spherical closure of a symmetric subgroup

projective normality of complete toroidal symmetric varieties in r chirivi a maffei projective normality of complete symmetric varieties duke math j 122 1 2004 93 123 chirivi and maffei have proved that the multiplication of sections of any two line bundles generated by global sections on a wonderful symmetric variety is surjective we prove two criterions that allow us to reduce the same problem on a smooth complete toroidal symmetric variety to the analogous problem on the corresponding complete toric variety respectively on the corresponding open toric variety we have also studied in details some family of complete toroidal symmetric varieties in particular those of rank 2

weighted maximum likelihood for dynamic factor analysis and forecasting with mixed frequency data for the purpose of forecasting key macroeconomic or financial variables from a panel of time series variables we adopt the dynamic factor model and propose a weighted likelihood based method for parameter estimation the loglikelihood function is split into two parts that are weighted differently the first part is associated with the key variables while the second part is associated with the related variables which may contribute to the forecasting of key variables we derive asymptotic properties including consistency and asymptotic normality of the weighted maximum likelihood estimator we show that this estimator outperforms the standard likelihood based estimator in approximating the true unknown distribution of the data as well as in out of sample forecasting accuracy we verify the new estimation method in a monte carlo study and investigate the role of different weights in different settings in the context of forecasting gross domestic product growth this key variable is typically observed at a low quarterly frequency while the supporting variables are observed at a high monthly frequency we adopt a low frequency representation of the mixed frequency dynamic factor model and discuss the computational efficiencies of this approach in our empirical study for the u s economy we present improvements in nowcasting and forecasting accuracy when the weighted likelihood based estimation procedure is adopted

the positive definite solution of the nonlinear matrix equation xp a m b x1 1m we consider the nonlinear matrix equation xp a m b x1 1m where p1 is a positive integer m is an arbitrary nn matrix and a and b are hermitian positive semidefinite matrices an elegant estimate of the hermitian positive definite hpd solution is derived a fixed point iteration and an inversion free variant iteration for obtaining the hpd solution are proposed some numerical examples are presented to show the efficiency of the proposed two iterative methods

remarks on holder continuity for solutions of the p laplace evolution equations we study the interior holder regularity problem for the gradient of solutions of the p laplace evolution equations with the external forces misawa gave some conditions for the holder continuity of the gradient of solutions we show holder estimates of the solutions with weaker condition as for misawa

asymptotics for parametric garch in mean models abstract in this paper we develop an asymptotic theory for the quasi maximum likelihood estimator qmle of the parametric garch in mean model the asymptotics is based on a study of the volatility as a process of the model parameters the proof makes use of stochastic recurrence equations for this random function and uses exponential inequalities to localize the problem our results show why the asymptotics for this specification is quite complex although it is a rather standard parametric model nevertheless our theory does not yet treat all standard specifications of the mean function

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

regularity results for nonlinear parabolic obstacle problems with subquadratic growth abstract in 26 it was shown that the spatial gradient of the solution u to the parabolic obstacle problem with superquadratic growth is local holder continuous provided the obstacle is regular enough in this paper we extend this regularity result to the subquadratic case this means we establish the local holder continuity of the spatial gradient of the solution u to the parabolic obstacle problem with subquadratic growth more precisely we prove that d u c loc 0 2 for some 0 1 provided the coefficients and the obstacle are regular enough moreover we use the local holder continuity to prove the local lipschitz continuity of the solution u i e u c loc 0 1 1 2

non extinction and critical exponent for a polytropic filtration equation abstract this paper is concerned with the homogeneous dirichlet boundary value problem of a polytropic filtration equation with reaction sources we will show the non extinction property of non trivial solutions and determine the critical fujita exponent for the problem

kalman filtering with random coefficients and contractions the riccati transformation of linear filtering control theory is shown to be a contraction on the space of positive symmetric matrices this is used to describe the asymptotic behavior of the filter for systems with stochastic stationary parameters

local continuity and asymptotic behaviour of degenerate parabolic systems abstract we study the local holder continuity and the asymptotic behaviour of solution u u 1 u k of the degenerate system u t i m u m 1 u i for m 1 and i 1 k which describes the population densities of k species whose diffusions are determined by their total population density u u 1 u k for the local holder continuity we adopt the intrinsic scaling and iteration arguments of degiorgi moser and dibenedetto under some regularity conditions we also prove that the population density of i th species with the population m i converges in c s to the function m i m b m x t as t where b m is the barenblatt profile of the standard porous medium equation with l 1 mass m m 1 m k as a consequence of asymptotic behaviour it is shown that each density becomes a concave function after a finite time

boundedness of the gradient for a doubly nonlinear parabolic equation we prove the local boundedness of the gradient for positive solutions to a doubly nonlinear parabolic equation in the case when the standard lebesgue measure has been replaced by a doubling measure which supports a weak poincare inequality

kalman filtering with intermittent observations convergence for semi markov chains and an intrinsic performance measure this technical note shows that the stationary distribution for the covariance of kalman filtering with intermittent observations exists under mild conditions for a very general class of packet dropping models semi markov chain these results are proved using the geometric properties of riccati recursions with respect to a particular riemannian distance moreover the riemannian mean induced by that distance is always bounded therefore it can be used for characterizing the performance of the system for regimes where the moments of the covariance do not exist other interesting properties of that mean include the symmetry between covariance and information matrices averaging covariances or their inverse gives the same result and its interpretation in information geometry as the natural mean for the manifold of gaussian distributions

quasi maximum likelihood estimation in conditionally heteroscedastic time series a stochastic recurrence equations approach this paper studies the quasi maximum likelihood estimator qmle in a general conditionally heteroscedastic time series model of multiplicative form xt t zt where the unobservable volatility t is a parametric function of xt 1 x t p t 1 t q for some p q 0 and zt is standardized i i d noise we assume that these models are solutions to stochastic recurrence equations which satisfy a contraction random lipschitz coefficient property these assumptions are satisfied for the popular garch asymmetric garch and exponential garch processes exploiting the contraction property we give conditions for the existence and uniqueness of a strictly stationary solution xt to the stochastic recurrence equation and establish consistency and asymptotic normality of the qmle we also discuss the problem of invertibility of such time series models 1 introduction gaussian quasi maximum likelihood estimation that is likelihood estimation under the hypothesis of gaussian innovations is a popular method which is widely used for inference in time series models however it is often a nontrivial task to establish the consistency and asymptotic normality of the quasi maximum likelihood estimator qmle applied to specific models and therefore an in depth analysis of the probabilistic structure generated by the model is called for a classical example of this kind is the seminal paper by hannan 18 on estimation in linear arma time series in this paper we study the qmle for a general class of conditionally heteroscedastic time series models which includes garch asymmetric garch and exponential garch recall that a garch p q generalized autoregressive conditionally heteroscedastic of order p q process 4 i s def ined by

partial regularity for subquadratic parabolic systems under controllable growth conditions in this paper we prove a regularity result for weak solutions away from singular set of parabolic systems with subquadratic growth condition of the type tu z diva z u du b z u du for z x t where the structure function a satisfies standard ellipticity and growth condition with polynomial growth rate 2nn 2 p 2 the inhomogeneity b satisfies the controllable growth condition b z u du 1 u p 1 du p 1 1p where p n 2 pn 2 p if n 2 p here p can be any exponent if n 2 p the proof is based on the a caloric approximation lemma to parabolic systems with subquadratic growth

on estimation error outage for scalar gauss markov signals sent over fading channels measurements of a scalar linear gauss markov process are sent over a fading channel the fading channel is modeled as independent and identically distributed random variables with known realization at the receiver the optimal estimator at the receiver is the kalman filter in contrast to the classical kalman filter theory given a random channel the kalman gain and the error covariance become random then the probability distribution function of expected estimation error and its outage probability can be chosen for estimation quality assessment in this paper and in order to get the estimation error outage we provide means to characterize the stationary probability density function of the random expected estimation error furthermore and for the particular case of the i i d rayleigh fading channels upper and lower bounds for the outage probability are derived which provide insight and simpler means for design purposes we also show that the bounds are tight for the high snr regime and that the outage probability decreases linearly with the inverse of the average channel snr

self improving property of the fast diffusion equation abstract we show that the gradient of the m power of a solution to a singular parabolic equation of porous medium type also known as fast diffusion equation satisfies a reverse holder inequality in suitable intrinsic cylinders relying on an intrinsic calderon zygmund covering argument we are able to prove the local higher integrability of such a gradient for m n 2 n 2 1 our estimates are satisfied for a general class of growth assumptions on the non linearity in this way we extend the theory for m 1 see 16 in the list of references to the singular case in particular an intrinsic metric that depends on the solution itself is introduced for the singular regime

strict contractions of symmetric cones let v be a euclidean jordan algebra with an associative inner product hji and let be the corresponding symmetric cone then the linear automorphism group h of the symmetric cone acts transitively on it let t v i be the symmetric tube domain associated to the symmetric cone and let g t be the biholomorphic automorphism group of t there exists a conformal compactification m g t p of v with respect to a parabolic subgroup p a lie semigroup that is naturally related to the symmetric cone occurs as the semigroup of compressionsof in the lie group g t fg 2 g t j g g it turns out that admits an ol shanskii decomposition h expc for an ad h invariant convex cone c in the lie algebra g t of g t a triple decomposition h 8 9 and it determines a globally hyperbolic causal order on the homogeneous space g t h defined by ah bh if and only if a 1 b 2 the family of bilinear forms x u v hp x 1 ujvi defines a hinvariant riemannian metric on where the map p is the quadratic representation of the jordan algebra v in 8 koufany showed that the elements of are contractions of for the riemannian metric the main result of this paper is that the elements of the interior of called strict compressions are strict contractions of for the riemannian distance induced by

distortion outage analysis for joint space time coding and kalman filtering in this paper we consider the scenario of transmitting a first order gauss markov vector signal over a multi input multi output mimo rayleigh nonfrequency selective fading channel the signal is reconstructed at the receiver side with the help of a kalman filter in order to minimize the mean squared error orthogonal space time codes are utilized in order to increase the quality of estimation and mitigate the destructing effects of the fading channel as a criterion for estimation quality assessment we use the distortion outage probability we first obtain upper and lower bounds for the outage probability as a function of system parameters we then perform high snr analysis of the bounds through which we prove the achievability of the maximum diversity order for a n times k mimo fading channel in addition we obtain upper and lower bounds for the coding gain of the distortion outage probability in the high snr regime and outline the relation between system parameters and the coding gain

gaussian distributions on riemannian symmetric spaces statistical learning with structured covariance matrices the riemannian geometry of covariance matrices has been essential to several successful applications in computer vision biomedical signal and image processing and radar data processing for these applications an important ongoing challenge is to develop riemannian geometric tools which are adapted to structured covariance matrices this paper proposes to meet this challenge by introducing a new class of probability distributions gaussian distributions of structured covariance matrices these are riemannian analogs of gaussian distributions which only sample from covariance matrices having a preassigned structure such as complex toeplitz or block toeplitz the usefulness of these distributions stems from three features 1 they are completely tractable analytically or numerically when dealing with large covariance matrices 2 they provide a statistical foundation to the concept of structured riemannian barycentre i e frechet or geometric mean and 3 they lead to efficient statistical learning algorithms which realise among others density estimation and classification of structured covariance matrices this paper starts from the observation that several spaces of structured covariance matrices considered from a geometric point of view are riemannian symmetric spaces accordingly it develops an original theory of gaussian distributions on riemannian symmetric spaces of their statistical inference and of their relationship to the concept of riemannian barycentre then it uses this original theory to give a detailed description of gaussian distributions of three kinds of structured covariance matrices complex toeplitz and block toeplitz finally it describes algorithms for density estimation and classification of structured covariance matrices based on gaussian distribution mixture models

cohomology of regular embeddings we give in this paper an explicit description of the rational cohomology ring of a complete symmetric variety or regular compactification of an algebraic symmetric variety the motivation for this computation comes from the desire to describe an explicit schubert calculus for these varieties which have been considered since the work of chasles and schubert in several special cases the technique we use is the result of our attempts to understand better the work of jurkiewicz and danilov in the case of torus embeddings the combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology the key point is that the reisner stanley algebra of the torus embedding coincides with its equivariant cohomology ring we consider a class of spaces regular embeddings that contains both complete symmetric varieties and smooth torus embeddings by suitably generalizing the notion of reisner stanley algebra we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way

existence of weak solutions of doubly nonlinear parabolic equations abstract we deal with a cauchy dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space time cylinder for doubly nonlinear parabolic equations whose prototype is t u div u m 1 d u p 2 d u f with a non negative lebesgue function f on the right hand side where p 2 n n 2 and m 0 the central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f the constructed solution is obtained by a limit of approximations i e we use solutions of regularized cauchy dirichlet problems and pass to the limit to receive a solution for the original cauchy dirichlet problem

on the solution of the nonlinear matrix equation xn f x abstract we consider a class of nonlinear matrix equations x n f x 0 where f is a self map on the convex cone p k of k k positive definite real matrices it is shown that for n 2 the matrix equation has a unique positive definite solution depending continuously on the function f if f belongs to the semigroup of nonexpansive mappings with respect to the gl k r invariant riemannian metric distance on p k which contains congruence transformations translations the matrix inversion and in particular symplectic hamiltonians appearing in kalman filtering we show that the sequence of positive definite solutions varying over n 2 converges always to the identity matrix

beta negative binomial auto regressions for modelling integer valued time series with extreme observations the paper introduces a general class of heavy tailed auto regressions for modelling integer valued time series with outliers the specification proposed is based on a heavy tailed mixture of negative binomial distributions that features an observation driven dynamic equation for the conditional expectation the existence of a stationary and ergodic solution for the class of auto regressive processes is shown under general conditions the estimation of the model can be easily performed by maximum likelihood given the closed form of the likelihood function the strong consistency and the asymptotic normality of the estimator are formally derived two examples of specifications illustrate the flexibility of the approach and the relevance of the theoretical results in particular a linear dynamic equation and a score driven equation for the conditional expectation are studied the score driven specification is shown to be particularly appealing as it delivers a robust filtering method that attenuates the effect of outliers empirical applications to the series of narcotics trafficking reports in sydney and the euro pound sterling exchange rate illustrate the effectiveness of the method in handling extreme observations

regularity for parabolic systems of uhlenbeck type with orlicz growth abstract we study the local regularity of p caloric functions or more generally of caloric functions in particular we study local solutions of non linear parabolic systems with homogeneous right hand side where the leading terms have uhlenbeck structure of orlicz type this paper closes the gap of 23 where liebermann proved that if the gradient of a solution is bounded it is holder continuous the crucial step is a novel local estimates for the gradient of the solutions which generalize and improve the pioneering estimates of dibenedetto and friedman 12 10 for the p laplace heat equation

asymptotic behavior of recursive state estimations with intermittent measurements convergence rate and stationary distribution approximation are investigated in this paper for both the covariance matrix of the kalman filter and the pseudo covariance matrix of a robust recursive state estimator developed by us when the measurement dropping process is described by a markov chain and an associated plant is both controllable and observable it is proved that if the dropping probability is less than 1 these two matrices converge exponentially to a stationary distribution independent of its initial values moreover when they are initialized with the stabilizing solution of an associated algebraic riccati equation both of them are shown to converge to an ergodic process based on these results two approximations are derived for their stationary probability distributions as well as a bound of approximation errors it has been made clear that rather than the delta function suggested by kar and moura a series of delta functions give a more accurate approximation and replacement of these estimators update gain by a constant matrix usually deteriorates their steady estimation accuracy a numerical example is provided to illustrate the obtained theoretical results

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

diversity effects in kalman filtering over rayleigh fading channels a scalar first order gauss markov signal is sent over d parallel i i d rayleigh fading channels using uncoded transmission assuming perfect channel estimation at the receiver the optimal mmse estimator is the kalman filter with random instantaneous estimation error variance statistical characteristics of the instantaneous estimation error variance are studied to characterize the estimation quality one such characteristic is the estimation error outage which probabilistically depicts how often the value of the instantaneous estimation error variance exceeds a certain threshold in this paper we study the effect of channel diversity on the estimation error outage probability we show that for a certain range of outage thresholds the estimation error outage probability decreases with inverse of d th power of the average signal to noise ratio at the receiver side this means that analogous to the context of digital communication over fading channels our result implies a diversity order of d in estimating the input process

robust recursive state estimation with random measurement droppings a recursive state estimation procedure is derived for a linear time varying system with both parametric uncertainties and stochastic measurement droppings this estimator has a similar form as that of kalman filter with intermittent observations but its parameters should be adjusted when a plant output measurement arrives a new recursive form is derived for the pseudo covariance matrix of estimation errors based on a riemannian metric for positive definite matrices some necessary and sufficient conditions have been obtained for the strict contractiveness of this recursion it has also been proved that under some controllability and observability conditions as well as some weak requirements on measurement arrival probability the update gain of this recursive robust state estimator and the mean of its squared estimation errors converge in probability one respectively to a corresponding stationary distribution numerical simulation results show that estimation accuracy of the suggested procedure is more robust against parametric modelling errors than kalman filter

the obstacle problem for the porous medium equation we prove existence results for the obstacle problem related to the porous medium equation for sufficiently regular obstacles we find continuous solutions whose time derivative belongs to the dual of a parabolic sobolev space we also employ the notion of weak solutions and show that for more general obstacles such a weak solution exists the latter result is a consequence of a stability property of weak solutions with respect to the obstacle

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes strong consistency and asymptotic normality of the quasi maximum likelihood estimator qmle are given for a general class of multidimensional causal processes for particular cases already studied in the literature for instance univariate or multivariate garch arch arma garch processes the assumptions required for establishing these results are often weaker than existing conditions the qmle asymptotic behavior is also given for numerous new examples of univariate or multivariate processes for instance tarch or nlarch processes

bounded solutions to a quasilinear and singular parabolic equation with p laplacian abstract in this paper we study the existence of a positive local in time solution for the following singular nonlinear problem with homogeneous dirichlet boundary conditions t u p u u f x u u in 0 t q t u 0 on 0 t u 0 in q t u 0 x u 0 0 in where stands for a regular bounded domain of r n p u is the p laplacian defined by p u div u p 2 u 2 p 0 and t 0 the nonlinear term f r r n r is a caratheodory function satisfying the growth condition f x s a s q 1 b c p p q for a a x s r and m where a c m 0 and b 0 are some constants and q p p where p p n n p if p n and p if p n we prove for any initial nonnegative data u 0 l r with r 2 large enough the existence of at least one weak solution to p in the case 2 1 p 1 we prove the uniqueness of the solution and further regularity results for that we use some estimates based on logarithmic sobolev inequalities to get ultracontractivity of a closely related semi group of linear operators

a borel weil bott type theorem for group completions abstract we prove a character formula of cohomologies of line bundles on the wonderful completion of an adjoint semisimple algebraic group in the sense of de concini and procesi in our case we need the general construction of de concini and springer

higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth abstract the aim of this paper is to derive the self improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p x t nonstandard growth more precisely we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator

convergence of nonlinear observers on bbr n with a riemannian metric part i we study how convergence of an observer whose state lives in a copy of the given system s space can be established using a riemannian metric we show that the existence of an observer guaranteeing the property that a riemannian distance between system and observer solutions is nonincreasing implies that the lie derivative of the riemannian metric along the system vector field is conditionally negative moreover we establish that the existence of this metric is related to the observability of the system s linearization along its solutions moreover if the observer has an infinite gain margin then the level sets of the output function are geodesically convex conversely we establish that if a complete riemannian metric has a lie derivative along the system vector field that is conditionally negative and is such that the output function has a monotonicity property then there exists an observer with an infinite gain margin

boundary estimates for solutions of nonlinear degenerate parabolic systems on etablit la regularite partout jusqu a la frontiere pour des solutions faibles de systemes paraboliques du type u l t div du p 2 du l f l x t u du dans t l 1 2 m u t r m m n p max 1 2n n 2 est un ouvert borne de r n n 2 0 t t 0 t

cauchy problem for fast diffusion equations with a gradient term we study the cauchy problem for fast diffusion equation with a gradient term of the form ut um duq where 0 0 and 0 0 however as q m the optimality of the stipulated assumptions is left open

kahler geometry of horosymmetric varieties and application to mabuchi s k energy functional we introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties we develop kahler geometry on these varieties with applications to canonical metrics in mind as a generalization of the guillemin abreu donaldson geometry of toric varieties namely we associate convex functions with hermitian metrics on line bundles and express the curvature form in terms of this function as well as the corresponding monge ampere volume form and scalar curvature we then provide an expression for the mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by li zhou and zhu on group compactifications

nonexpansiveness of the resolvent average abstract we show that the resolvent average for positive definite matrices which interpolates between the harmonic and the arithmetic means contracts the thompson metric on the convex cone of positive definite matrices classical results depending on the nonexpansive property of the arithmetic average are considered in the context of the resolvent average in particular resolvent power mean approximation to the karcher mean and ferrante and levy like matrix equations are investigated

mean square stability for kalman filtering with markovian packet losses this paper studies the stability of kalman filtering over a network subject to random packet losses which are modeled by a time homogeneous ergodic markov process for second order systems necessary and sufficient conditions for stability of the mean estimation error covariance matrices are derived by taking into account the system structure while for certain classes of higher order systems necessary and sufficient conditions are also provided to ensure stability of the mean estimation error covariance matrices all stability criteria are expressed by simple inequalities in terms of the largest eigenvalue of the open loop matrix and transition probabilities of the markov process their implications and relationships with related results in the literature are discussed

mutual information of wireless channels and block jacobi ergodic operators shannon s mutual information of a random multiple antenna and multipath time varying channel is studied in the general case where the process constructed from the channel coefficients is an ergodic and stationary process which is assumed to be available at the receiver from this viewpoint the channel can also be represented by an ergodic self adjoint block jacobi operator which is close in many aspects to a block version of a random schrodinger operator the mutual information is then related to the so called density of states of this operator in this paper it is shown that under the weakest assumptions on the channel the mutual information can be expressed in terms of a matrix valued stochastic process coupled with the channel process this allows numerical approximations of the mutual information in this general setting moreover assuming further that the channel coefficient process is a markov process a representation for the mutual information offset in the large signal to noise ratio regime is obtained in terms of another related markov process this generalizes previous results from levy et al it is also illustrated how the mutual information expressions that are closely related to those predicted by the random matrix theory can be recovered in the large dimensional regime

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

degenerate parabolic equations with initial data measures we address the problem of existence of solutions to degenerate and nondegenerate parabolic equations under optimal assumptions on the initial data which are assumed to be measures the requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation and with the form of the nonlinear forcing term the latter depends on the space gradient of a power of the solution applications to related problems are also outlined

waiting time for a non newtonian polytropic filtration equation with convection abstract in this paper we study the waiting time phenomena for a class of non newtonian polytropic filtration equation with convection according to the nonlinearity in the diffusion and the convection terms some necessary and sufficient conditions on the initial data are given for the existence of waiting time

holder estimates for parabolic p x t laplacian systems in this paper we establish the local holder continuity of the spatial gradient of weak solutions to the parabolic p x t laplacian system begin array lll partial t u rm div left a x t du p x t 2 du right 0 end array more precisely we prove that begin array lll du in c rm loc 0 alpha alpha 2 quad rm for some alpha in 0 1 end array provided p and a are holder continuous

stabilization and random linear regulator problem for random linear control processes we study in a dynamical system context the random feedback stabilization problem for linear random control processes we are led to study the random linear regulator problem which we solve by considering the spectral theory of linear time dependent hamiltonian systems this is done with the aid of the concepts of exponential dichotomy and rotation number

brief paper optimal estimation of linear discrete time systems with stochastic parameters this paper considers optimal linear state estimation in the general case of linear discrete time systems with stochastic parameters which are statistically independent with respect to time the estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise it is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence uniqueness and stability of the time invariant estimator

performance of ensemble kalman filters in large dimensions contemporary data assimilation often involves more than a million prediction variables ensemble kalman filters enkf have been developed by geoscientists they are successful indispensable tools in science and engineering because they allow for computationally cheap low ensemble state approximation for extremely large dimensional turbulent dynamical systems the practical finite ensemble filters like enkf necessarily involve modifications such as covariance inflation and localization and it is a genuine mystery why they perform so well with small ensemble sizes in large dimensions this paper provides the first rigorous stochastic analysis of the accuracy and covariance fidelity of enkf in the practical regime where the ensemble size is much smaller than the large ambient dimension for enkfs with random coefficients a challenging issue overcome here is that enkf in huge dimensions introduces unavoidable bias and model errors that need to be controlled and estimated 2017 the authors communications on pure and applied mathematics is published by the courant institute of mathematics and wiley periodicals inc

stochastic stability of the discrete time extended kalman filter the authors analyze the error behavior for the discrete time extended kalman filter for general nonlinear systems in a stochastic framework in particular it is shown that the estimation error remains bounded if the system satisfies the nonlinear observability rank condition and the initial estimation error as well as the disturbing noise terms are small enough this result is verified by numerical simulations for an example system

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

gossip and distributed kalman filtering weak consensus under weak detectability the paper presents the gossip interactive kalman filter gikf for distributed kalman filtering for networked systems and sensor networks where intersensor communication and observations occur at the same time scale the communication among sensors is random each sensor occasionally exchanges its filtering state information with a neighbor depending on the availability of the appropriate network link we show that under a weak distributed detectability condition 1 the gikf error process remains stochastically bounded irrespective of the instability of the random process dynamics and 2 the network achieves weak consensus i e the conditional estimation error covariance at a uniformly randomly selected sensor converges in distribution to a unique invariant measure on the space of positive semidefinite matrices independent of the initial state to prove these results we interpret the filtering states estimates and error covariances at each node in the gikf as stochastic particles with local interactions we analyze the asymptotic properties of the error process by studying as a random dynamical system the associated switched random riccati equation the switching being dictated by a nonstationary markov chain on the network graph

local calderon zygmund estimates for parabolic minimizers abstract we establish local calderon zygmund estimates for parabolic minimizers u to certain variational inequalities the formal corresponding parabolic partial differential equation would be t u div f z u d u u f z u d u div f p 2 f h with p 2 n n 2 however since we do not impose any condition on the existence of the derivative u f the pde above may not have any meaning at all for proving the calderon zygmund estimates we thus remain solely on the level of parabolic minimizers precisely for q p we prove that d u l loc q if u is locally holder continuous and f l loc q and h l loc q for a q depending on q

the contraction rate in thompson s part metric of order preserving flows on a cone application to generalized riccati equations abstract we give a formula for the lipschitz constant in thompson s part metric of any order preserving flow on the interior of a possibly infinite dimensional closed convex pointed cone this shows that in the special case of order preserving flows a general characterization of the contraction rate in thompson s part metric given by nussbaum leads to an explicit formula as an application we show that the flow of the generalized riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its lipschitz constant by a matrix inequality we also show that the same flow is no longer a contraction in other invariant finsler metrics on this cone including the standard invariant riemannian metric this is motivated by a series of contraction properties concerning the standard riccati equation established by bougerol liverani wojtkowski lawson lee and lim we show that some of these properties do and that some other do not carry over to the generalized riccati equation

travelling waves in a convection diffusion equation abstract we study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1 d euclidean space the diffusion coefficient depends on the gradient in analogy with the p laplacian and may be degenerate unconditional stability is established with respect to initial data perturbations in l 1 r

holder zygmund estimates for degenerate parabolic systems abstract we consider energy solutions of the inhomogeneous parabolic p laplacian system t u div d u p 2 d u div g we show in the case p 2 that if the right hand side g is locally in l bmo then u is locally in l c 1 where c 1 is the 1 holder zygmund space this is the borderline case of the calderon zygmund theory we provide local quantitative estimates we also show that finer properties of g are conserved by du e g holder continuity moreover we prove a new decay for gradients of p caloric solutions for all 2 n n 2 p

finsler metrics on symmetric cones let v be a simple euclidean jordan algebra with an associative inner product langle cdot cdot rangle and let omega be the corresponding symmetric cone let mathcal j v be the compact symmetric space of all primitive idempotents of v we show that the function s a b defined by

a transport model with micro and macro structure abstract in this paper we consider solute transport through porous media where the chemical species undergoes a chemical process through the surface of the porous skeleton the problem is modeled by a system of differential equations for the macro concentration u u x t and the micro concentration u u x x t we prove existence and uniqueness and some properties of the set with positive concentration

continuity estimates for porous medium type equations with measure data abstract we consider parabolic equations of porous medium type of the form u t div a x t u d u in e t in some space time cylinder e t the most prominent example covered by our assumptions is the classical porous medium equation u t u m in e t with m 1 we establish a sufficient condition for the continuity of u in terms of a natural riesz potential of the right hand side measure as an application we come up with a borderline condition ensuring the continuity of u more precisely if l n 2 2 1 then u is continuous in e t