reconstruction of closely spaced small inclusions in this paper we establish an explicit asymptotic formula for the steady state voltage perturbations caused by closely spaced small conductivity inhomogeneities based on this new formula we design a very effective numerical method to identify the location and some geometric features of these inhomogeneities from a finite number of boundary measurements the viability of our approach is documented by numerical examples

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

on kalman filtering for detectable systems with intermittent observations we consider the problem of kalman filtering when observations are available according to a bernoulli process it is known that there exists a critical probability pc such that if measurements are available with probability greater than pc then the expected prediction covariance is bounded for all initial conditions otherwise it is unbounded for some initial conditions we show that when the system observation matrix restricted to the observable subspace is invertible the known lower bound on pc is in fact the exact critical probability this result is based on a novel decomposition of positive semidefinite matrices

design of networked control systems with packet dropouts this note is concerned with stability and controller design of networked control systems ncss with packet dropouts new ncs models are provided considering both single and multiple packet transmissions both sensor to controller s c and controller to actuator c a packet dropouts are modeled and their history behavior is described by different independent markov chains in term of the given models sufficient conditions for stochastic stability are derived in the form of linear matrix inequalities lmis and corresponding control laws are given numerical examples illustrate the effectiveness of the results

optimal estimation and control for lossy network stability convergence and performance in this paper we study the problems of optimal estimation and control i e the linear quadratic gaussian lqg control for systems with packet losses but without acknowledgment such acknowledgment is a signal sent by the actuator to inform the estimator of the incidence of control packet losses for such system which is usually called as a user datagram protocol udp like system the optimal estimation is nonlinear and its calculation is time consuming making its corresponding optimal lqg problem complicated we first propose two conditions 1 the sensor has some computation abilities and 2 the control command exerted to the plant is known to the sensor for a udp like system satisfying these two conditions we derive the optimal estimation by constructing the finite and infinite product probability measure spaces for the estimation error covariances eec we give the stability condition for the expected eec and show the existence of a measurable function to which the eec converges in distribution and propose some practical methods to evaluate the estimation performance finally the lqg controllers are derived and the conditions for the mean square stability of the closed loop system are established

a unified approach to some recurrence sequences via faa di bruno s formula abstract the well known faa di bruno formula for higher derivatives of a composite function plays an important role in combinatorics in this paper we obtain a unified approach to some recurrence sequences for the complete bell polynomials by using faa di bruno s formula furthermore we obtain an inverse relation of lah numbers of two types and by using this relation we obtain some new and interesting combinatorial identities

stability of mmse state estimators over lossy networks using linear coding this paper studies the state estimation problem for a stochastic discrete time system over a lossy channel where the packet loss is modeled as an independent and identically distributed binary process to counter the effect of random packet loss we propose a linear coding method to preprocess the measured output and prove that the coded output is information preserving when packet loss is void and is information enhancing when packet loss is present an optimal state estimator under the minimum mean square error mmse criterion is derived for the coded output when subject to packet loss the maximum packet loss rate for ensuring a stable estimator is then derived and shown to be very close to a well known lower bound also considered is a compressed linear coding method where the measured output is first compressed onto a lower dimensional space before encoding and it is shown that the similar packet rate condition for stability holds

kalman filtering with intermittent observations tail distribution and critical value in this paper we analyze the performance of kalman filtering for discrete time linear gaussian systems where packets containing observations are dropped according to a markov process modeling a gilbert elliot channel to address the challenges incurred by the loss of packets we give a new definition of non degeneracy which is essentially stronger than the classical definition of observability but much weaker than one step observability which is usually used in the study of kalman filtering with intermittent observations we show that the trace of the kalman estimation error covariance under intermittent observations follows a power decay law moreover we are able to compute the exact decay rate for non degenerate systems finally we derive the critical value for non degenerate systems based on the decay rate improving upon the state of the art

kalman filtering with intermittent observations weak convergence to a stationary distribution the paper studies the asymptotic behavior of discrete time random riccati equations rre arising in kalman filtering when the arrival of the observations is described by a bernoulli independent and identically distributed i i d process we model the rre as an order preserving strongly sublinear random dynamical system rds under a sufficient condition stochastic boundedness and using a limit set dichotomy result for order preserving strongly sublinear rds we establish the asymptotic properties of the rre the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution whose support exhibits fractal behavior for stabilizable and detectable systems stochastic boundedness and hence weak convergence holds for any nonzero observation packet arrival probability and in particular we can establish weak convergence at operating arrival rates well below the critical probability for mean stability the resulting invariant measure in that situation does not possess a first moment we apply the weak feller property of the markov process governing the rre to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points which in general is not dense in the set of positive semi definite matrices we use the explicit characterization of the support of the invariant distribution and the almost sure a s ergodicity of the sample paths to easily compute statistics of the invariant distribution a one dimensional example illustrates that the support is a fractured subset of the non negative reals with self similarity properties

on the spectrum of the poincare variational problem for two close to touching inclusions in 2d we study the spectrum of the poincare variational problem for two close to touching inclusions in r2 we derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero

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

on the eigenvalues of the electrostatic integral operator abstract a fundamental result in scattering and potential theory in r 3 states that the spectrum of the electrostatic integral operator lies in the interval 1 1 in the case of a sphere and a prolate spheroid it is known that the spectrum of the operator lies in the interval 1 0 a fundamental question arises whether the spectrum of the operator always lies in this interval or whether there exists a smooth surface for which the electrostatic integral operator has a positive eigenvalue in this paper this question is answered a surface is produced whereby the underlying integral operator has a positive eigenvalue

higher order quantum onsager coefficients from dynamical invariants functionals representing dynamical invariants under unitary quantum dynamics of open systems are used to derive onsager coefficients for entropy production in irreversible processes if the nonunitary time evolution is determined by quantum dynamical semigroups the procedure allows a derivation from first principles of the quantum analogue to the classical case

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

radial symmetry for elliptic boundary value problems on exterior domains by the alexandroff serrin method 2 14 of moving hyperplanes we obtain radial symmetry for the domain and the solutions of on an exterior domain subject to the overdetermined boundary conditions on at and in in particular the following conjecture from potential theory due to p gruber cf 11 8 is proved let or be a bounded smooth domain with a constant source distribution on and let be the induced single layer potential if is constant in then is a ball

on the magnetostatic integral operator for ellipsoids the magnetostatic integral operator in r3is of considerable interest in both scattering and potential theory as well as for its scalar pendant the electrostatic integral operator its eigenvalues and its eigenfunctions play a fundamental role in the theory of permanent magnetization while the eigenvalues of both operators are closely related explicit expressions for the eigenfunctions of the magnetostatic operator are known only in the case when the underlying surface is a sphere for the cases of ellipsoids and spheroids we present explicit expressions for the eigenvalues in terms of lame functions and associated legendre functions and expressions for eigenfunctions in terms of surface ellipsoidal and surface spheroidal harmonics respectively as an application the permanent magnetization of ellipsoids is discussed

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

evolution matrix in a coherence vector formulation for quantum markovian master equations of n level systems quantum markovian master equation in n dimensions are systematically transformed into vector form by using the lie algebra of the special unitary group su n density operators are represented through coherence vectors and the infinitesimal kossakowski generator of completely positive quantum dynamical semigroups appears as a real evolution matrix that is not completely diagonalisable in general a complete classification of the spectrum of the latter and its jordan canonical form together with all types of associated stationary states is given in detail the results of this analysis are particularly suited for computations in practical applications

electric field and potential calculation for a bioelectric current dipole in an ellipsoid computer based modelling in ellipsoidal geometry can be useful in electro encephalography and gastrography due to the resemblance of the human brain and stomach to an ellipsoid both theoretically and computationally the bioelectric current dipole model is important for the study of electrical activity in these two organs computing the electric potential and electric field e due to a current dipole located in an ellipsoid requires truncated series expansions involving ellipsoidal harmonics which are the products of lame functions a theoretical model for this has appeared in the literature only for expansions of order 2 in however this may be insufficient in forward electrogastrography while an analogous situation is encountered in some geodetic problems where similar expansions are required in this paper we propose a generalized model for computing and e numerically using harmonic expansions of arbitrary order and degree the implementation of such a procedure involves finding roots of lame polynomials of degree 5 or higher using an optimization technique for solving nonlinear systems of equations this process can allow one to make a knowledgeable decision concerning the optimal expansion size required for the physical problem under investigation without compromising the overall accuracy of the computation

an analysis of a feti dp algorithm on irregular subdomains in the plane in the theory for domain decomposition algorithms of the iterative substructuring family each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra this is an unrealistic assumption in particular if the subdomains result from the use of a mesh partitioner in which case they might not even have uniformly lipschitz continuous boundaries the purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality two geometric parameters characterizing john and uniform domains and the maximum number of edges of any subdomain a related purpose is to explore to what extent well known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains some of these results are valid for any john domain while an extension theorem which is needed in this study requires that the subdomains have complements which are uniform the results so far are complete only for problems in two dimensions details are worked out for a feti dp algorithm and numerical results support the findings some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains

moderate deviations of a random riccati equation the paper characterizes the invariant filtering measures resulting from kalman filtering with intermittent observations in which the observation arrival is modeled as a bernoulli process with packet arrival probability our prior work showed that for 0 the sequence of random conditional error covariance matrices converges weakly to a unique invariant distribution this paper shows that as approaches one the family 0 satisfies a moderate deviations principle with good rate function i 1 as 1 the family converges weakly to the dirac measure p concentrated on the fixed point of the associated discrete time riccati operator 2 the probability of a rare event an event bounded away from p under decays to zero as a power law of 1 as 1 and 3 the best power law decay exponent is obtained by solving a deterministic variational problem involving the rate function i for specific scenarios the paper develops computationally tractable methods that lead to efficient estimates of rare event probabilities under

kalman filtering with intermittent observations motivated by navigation and tracking applications within sensor networks we consider the problem of performing kalman filtering with intermittent observations when data travel along unreliable communication channels in a large wireless multihop sensor network the effect of communication delays and loss of information in the control loop cannot be neglected we address this problem starting from the discrete kalman filtering formulation and modeling the arrival of the observation as a random process we study the statistical convergence properties of the estimation error covariance showing the existence of a critical value for the arrival rate of the observations beyond which a transition to an unbounded state error covariance occurs we also give upper and lower bounds on this expected state error covariance

magnetoencephalography in ellipsoidal geometry an exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the 1d case of a sphere and the 2d case of a spheroid where the excitation field is due to an electric dipole within the corresponding homogeneous conductor in the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry the results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid using lame functions and ellipsoidal harmonics the basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior lame functions the laborious task of reducing the results to the spherical geometry is also included

a sum property of the eigenvalues of the electrostatic integral operator abstract the double layer potential integral operator and the electrostatic integral operator in r 3 occur while using integral equation methods in scattering and potential theory if the underlying surface is either a sphere a spheroid or a triaxial ellipsoid explicit expressions of the eigenvalues and eigenfunctions of order n n 0 are known the main result of this paper is the following the sum of all eigenvalues of order n n 0 each counted with respect to its multiplicity is 1 this is trivial for the sphere in the case of spheroids a proof is given for n n 0 for the triaxial ellipsoid this sum property is verified for n 0 1 2 3 whereas for n 3 some numerical results are provided

unfolding a quantum master equation into a system of real valued equations computationally effective expansion over the basis of su n generators dynamics of an open n state quantum system is often modeled with a markovian master equation describing the evolution of the system density operator by using generators of su n group as a basis the density operator can be transformed into a real valued coherence vector a generator of the dissipative evolution so called lindbladian can be expanded over the same basis and recast in the form of a real matrix together these expansions result is a nonhomogeneous system of n 2 1 real valued linear ordinary differential equations now one can e g implement standard high performance algorithms to integrate the system of equations forward in time while being sure in exact preservation of the trace norm and hermiticity of the density operator however when performed in a straightforward way the expansion turns to be an operation of the time complexity o n 10 the complexity can be reduced when the number of dissipative operators is independent of n which is often the case for physically meaningful models here we present an algorithm to transform quantum master equation into a system of real valued differential equations and propagate it forward in time by using a specific scalable model we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with n 10 3 states on a single node of a computer cluster

electroencephalography in ellipsoidal geometry the human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6 6 5 and 9 cm this is a genuine 3 d shape that reflects the anisotropic characteristics of the brain as a conductive body the direct electroencephalography problem in such anisotropic geometry is studied in the present work the results which are obtained through successively solving an interior and an exterior boundary value problem are expressed in terms of elliptic integrals and ellipsoidal harmonics both in jacobian as well as in cartesian form reduction of our results to spheroidal as well as to spherical geometry is included in contrast to the spherical case where the boundary does not appear in the solution the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field moreover the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location

combinatorial formulae for the derivatives of lame functions lame functions play a central role in the theory of ellipsoidal harmonics and have many varied applications in mathematical physics in this article a generalized approach to the computation of lame function derivatives of arbitrary order is derived and it is demonstrated how these derivatives can be expressed recursively in terms of the original lame functions by making use of combinatorial formulae discovered by di bruno girard and waring

counting nodal lines which touch the boundary of an analytic domain we consider the zeros on the boundary of a neumann eigenfunction of a real analytic plane domain we prove that the number of its boundary zeros is o where 2 we also prove that the number of boundary critical points of either a neumann or dirichlet eigenfunction is o it follows that the number of nodal lines of components of the nodal set which touch the boundary is of order this upper bound is of the same order of magnitude as the length of the total nodal line but is the square root of the courant bound on the number of nodal components in the interior more generally the results are proved for piecewise analytic domains

on the electro sensing of weakly electric fish in this paper we are interested in electro sensing inverse problems our objective is to understand the electro perception mechanism of weakly electric fish these species of fish have the ability to recognize the environment around them in complete darkness by generating a weak electrical field at different frequencies and perceiving the transdermal potential perturbation assuming that the target has a known conductivity profile the electro sensing inverse problem consists in recovering the shape and location of the target from measurements of the electric potential over the skin using an original spectral decomposition of the solution to the direct problem in terms of poincare vari ational eigenfunctions we retrieve the cauchy data of the electric potential over the fish skin corresponding to the case where we substitute the target by a perfect conductor with the same shape and position we then identify the target from the recovered cauchy data we derive uniqueness and stability estimates to the considered electro sensing inverse problem the numerical validation of our theoretical approach is realized by reconstructing different targets using synthetic data in dimension two the numerical experiments are conducted using gradient descent optimization algorithms

an improved stability condition for kalman filtering with bounded markovian packet losses in this paper we consider the peak covariance stability of kalman filtering subject to packet losses the length of consecutive packet losses is governed by a time homogeneous finite state markov chain we establish a sufficient condition for peak covariance stability and show that this stability check can be recast as a linear matrix inequality lmi feasibility problem compared with the literature the stability condition given in this paper is invariant with respect to similarity state transformations moreover our condition is proved to be less conservative than the existing results numerical examples are provided to demonstrate the effectiveness of our result

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

stability of kalman filtering with markovian packet losses we consider kalman filtering in a network with packet losses and use a two state markov chain to describe the normal operating condition of packet delivery and transmission failure based on the sojourn time of each visit to the failure or successful packet reception state we analyze the behavior of the estimation error covariance matrix and introduce the notion of peak covariance as an estimate of filtering deterioration caused by packet losses which describes the upper envelope of the sequence of error covariance matrices p t t 1 for the case of an unstable scalar model we give sufficient conditions for the stability of the peak covariance process in the general vector case and obtain a sufficient and necessary condition for the scalar case finally the relationship between two different types of stability notions is discussed

stability conditions for multi sensor state estimation over a lossy network this paper studies a networked state estimation problem for a spatially large linear system with a distributed array of sensors each of which offers partial state measurements a lossy communication network is used to transmit the sensor measurements to a central estimator where the minimum mean square error mmse state estimate is computed under a markovian packet loss model we provide necessary and sufficient conditions for the stability of the estimator for any diagonalizable system in the sense that the mean of the state estimation error covariance matrix is uniformly bounded in particular the stability conditions for the second order systems with an i i d packet loss model are explicitly expressed as simple inequalities in terms of the largest open loop pole and the packet loss rate

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

poincare s variational problem in potential theory one of the earliest attempts to rigorously prove the solvability of dirichlet s boundary value problem was based on seeking the solution in the form of a potential of double layer and this leads to an integral equation whose kernel is in general both singular and non symmetric c neumann succeeded with this approach for smoothly bounded convex domains and h poincare by a tremendous tour de force showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity in this work he was according to his own account guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain unfortunately a rigorous treatment of this problem was not possible with the tools available at that time as poincare was well aware so far as we know the only one to propose a rigorous treatment of poincare s problem was t carleman in the two dimensional case in his doctoral dissertation thanks to later developments especially concerning sobolev spaces and spectral theory of operators on hilbert space we can now give a complete general and rigorous account of poincare s variational problem and that is the main object of the present paper as a by product we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensions the basic properties of the analogue of the planar bergman schiffer singular integral equation we interpret poincare s variational principle as a non selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials we also prove a series of novel spectral analysis facts some of them conjectured by poincare related to the poincare neumann integral operator

schatten classes on compact manifolds kernel conditions abstract in this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to schatten classes a specific test for nuclearity is established as well as the corresponding trace formulae in the special case of compact lie groups kernel criteria in terms of locally and globally hypoelliptic operators are also given

a spectral property of the electrostatic integral operator for an ellipsoid the eigenfunctions of the electrostatic integral operator and its adjoint are governed by the lame polynomials it is shown that the arithmetic mean of the eigenvalues corresponding to the maximum number of linearly independent eigenfunctions of equal order is the same for the ellipsoid as for the sphere

a universal finite memory source an irreducible parameterization for a finite memory source is constructed in the form of a tree machine a universal information source for the set of finite memory sources is constructed by a predictive modification of an earlier studied algorithm context it is shown that this universal source incorporates any minimal data generating tree machine in an asymptotically optimal manner in the following sense the negative logarithm of the probability it assigns to any long typical sequence generated by any tree machine approaches that assigned by the tree machine at the best possible rate

formulas for higher derivatives of the riemann zeta function the functional equation for c s is used to obtain formulas for all derivatives t k s a closed form evaluation of t k 0 is given and numerical values are computed to

some extensions of faa di bruno s formula with divided differences the well known formula of faa di bruno s for higher derivatives of a composite function has played an important role in combinatorics in this paper we generalize the divided difference form of faa di bruno s formula and give an explicit formula for the n th divided difference of a multicomposite function more generally we establish the relationships of the bell polynomials with respect to multicomposite functions applying these to multicomposite functions we obtain some extensions of faa di bruno s formula

iterative solutions to the steady state density matrix for optomechanical systems we present a sparse matrix permutation from graph theory that gives stable incomplete lower upper preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems this reordering is efficient adding little overhead to the computation and results in a marked reduction in both memory and runtime requirements compared to other solution methods with performance gains increasing with system size either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance this reordering optimizes the condition number of the approximate inverse and is the only method found to be stable at large hilbert space dimensions this allows for steady state solutions to otherwise intractable quantum optomechanical systems

new spectral results for the electrostatic integral operator abstract in this paper two new fundamental results are established concerning the spectra of both the double layer potential and the electrostatic integral operators in r 3 while it is known that 0 lies in the spectrum of each operator its specific nature is not well understood for some geometries e g a sphere and a prolate spheroid it is known that 0 lies in their continuous spectra in this paper however specific examples are produced where 0 lies in the point spectrum of each operator thus it is established that the spectral nature of 0 depends upon the geometry of the underlying surface the second new result given is an important generalization of a recent paper concerning the possible refinement of a classical result of plemelj

stochastic comparison boundedness weak convergence and ergodicity of a random riccati equation with markovian binary switching this paper considers the dynamical behavior of a discrete time random riccati equation with markovian binary switching arising from a kalman filter with intermittent observations a number of properties of the random riccati equation are studied such as comparability monotonicity boundedness weak convergence and ergodicity a stochastic order method is used to compare random riccati equations and establish the monotonicity property in an expectation sense by using the idea of stopping times we present a sufficient condition for an expectation boundedness property moreover the contraction and nonexpansion the riemannian metric and the markov feller operator are applied to the analysis of weak convergence and ergodicity properties these properties allow us to establish a relation between time average and the limit of the expectation of covariance matrices we also revisit the independently and identically distributed case by making use of the comparison and boundedness results in this paper we present a variant of the existence theorem about a critical value in existing literature

stability of a random riccati equation with markovian binary switching this paper is concerned with the boundedness of the solution of a random riccati difference equation arising from kalman filtering with observation losses characterized by a markovian binary jump parameter a new sufficient condition for stability in the peak covariance sense is obtained which has a simpler form and is shown to be less conservative for systems with the observability index of two than existing works meanwhile we give some conditions under which the covariance matrix is bounded or unbounded in the usual sense then the equivalence between the peak covariance stability and the usual covariance stability is established for systems with the observability index of one and independent and identically distributed i i d observation losses

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m