supervised learning of perceptron and output feedback dynamic networks a feedback analysis via the small gain theorem this paper provides a time domain feedback analysis of the perceptron learning algorithm and of training schemes for dynamic networks with output feedback it studies the robustness performance of the algorithms in the presence of uncertainties that might be due to noisy perturbations in the reference signals or due to modeling mismatch in particular bounds are established on the step size parameters in order to guarantee that the resulting algorithms will behave as robust filters the paper also establishes that an intrinsic feedback structure can be associated with the training schemes the feedback configuration is motivated via energy arguments and is shown to consist of two major blocks a time variant lossless i e energy preserving feedforward path and a time variant feedback path the stability of the feedback structure is then analyzed via the small gain theorem and choices for the step size parameter in order to guarantee faster convergence are deduced by using the mean value theorem simulation results are included to demonstrate the findings

staggered grids hybrid dual spectral element method for second order elliptic problems application to high order time splitting methods for navier stokes equations abstract this paper is a presentation of a mixed spectral element method based on staggered grids strategy used for the discretization of second order elliptic problems set under hybrid dual formulation a numerical analysis is performed proving the expected optimality and some numerical tests are described to confirm the predicted convergence rates the second part deals with a high order time splitting spectral scheme for the simulation of unsteady incompressible navier stokes flows we show how to use such a solver as a filtering tool to obtain divergence free discrete velocity some computational examples are reported which quantify the improvements induced by such a filter we have observed a strengthening of the stability and of the accuracy

a feedback approach to the steady state performance of fractionally spaced blind adaptive equalizers this paper proposes a new approach to the analysis of the steady state performance of constant modulus algorithms cma which are among the most popular adaptive schemes for blind equalization a major feature of the proposed feedback approach is that it bypasses the need for working directly with the weight error covariance matrix in so doing approximate expressions for the steady state mean square error of several cm algorithms are derived including cma2 2 cma1 2 normalized cma and a new normalized cma variant with less bias a comparison among the various algorithms is also performed along with several simulation results the conclusions confirm the superior performance of cma2 2

correlation function of fields in one dimensional bose gas correlation function of fields is presented as a fredholm minor at finite coupling constant in one dimensional bose gas

classical functional bethe ansatz for sl n separation of variables for the magnetic chain it is desirable for one to be able to construct separation variables for systems with sl n r matrices or lax pairs this article relates and extends the two known methods of construction the sklyanin s functional bethe ansatz fba commun math phys 150 181 191 1992 was previously only known for sl 2 and sl 3 and associated r matrices the algebraic geometric method of adams harnad and hurtubise commun math phys 155 385 413 1993 has been shown to work for sl n but up until now only for linear poisson brackets in this article sklyanin s program is advanced by giving the fba for certain systems with sl n r matrices this is achieved by constructing rational functions a u and b u of the matrix elements of t u so that in the generic case the zeros xi of b u are the separation coordinates and the pi a xi provide their conjugate momenta it is shown that the separation variables thus defined are the same as those given as the simultaneous solutions of several equations in two

on integrable models related to the osp 1 2 gaudin algebra the osp 1 2 gaudin algebra is defined and integrable models described by it are considered the models include the osp 1 2 gaudin magnet and the dicke model related to it detailed discussion of the simplest cases of these models is presented the effect of the presence of fermions on the separation of variables is indicated

quadrics on real riemannian spaces of constant curvature separation of variables and connection with gaudin magnet the integrable systems are considered which are connected with separation of variables in real riemannian spaces of constant curvature an isomorphism is given for these systems with hyperbolic gaudin magnet using this isomorphism the complete classification of separable coordinate systems is provided by means of the corresponding l operators for the gaudin magnet

a time domain feedback analysis of filtered error adaptive gradient algorithms this paper provides a time domain feedback analysis of gradient based adaptive schemes a key emphasis is on the robustness performance of the adaptive filters in the presence of disturbances and modeling uncertainties along the lines of h sup spl infin theory and robust filtering the analysis is carried out in a purely deterministic framework and assumes no prior statistical information or independence conditions it is shown that an intrinsic feedback structure can be associated with the varied adaptive schemes the feedback structure is motivated via energy arguments and is shown to consist of two major blocks a time variant lossless i e energy preserving feedforward path and a time variant feedback path the configuration is further shown to lend itself to analysis via a so called small gain theorem thus leading to stability and robustness conditions that require the contractivity of certain operators choices for the step size parameter in order to guarantee faster rates of convergence are also derived and simulation results are included to demonstrate the theoretical findings in addition the time domain analysis provided in this paper is shown to extend the so called transfer function approach to a general time variant scenario without any approximations

on the convergence of blind adaptive equalizers for constant modulus signals this paper studies the behavior of the error sequence of stop anti go variants of two adaptive blind equalizers namely cma2 2 and sato s 1975 algorithm it is shown that for transmitted signals with constant modulus spl gamma the equalizer output can be made to lie within the circle of radius spl gamma spl radic c infinitely often for some value of c that is only slightly larger than one

h sup spl infin optimality of the lms algorithm we show that the celebrated least mean squares lms adaptive algorithm is h sup spl infin optimal the lms algorithm has been long regarded as an approximate solution to either a stochastic or a deterministic least squares problem and it essentially amounts to updating the weight vector estimates along the direction of the instantaneous gradient of a quadratic cost function we show that the lms can be regarded as the exact solution to a minimization problem in its own right namely we establish that it is a minimax filter it minimizes the maximum energy gain from the disturbances to the predicted errors whereas the closely related so called normalized lms algorithm minimizes the maximum energy gain from the disturbances to the filtered errors moreover since these algorithms are central h sup spl infin filters they minimize a certain exponential cost function and are thus also risk sensitive optimal we discuss the various implications of these results and show how they provide theoretical justification for the widely observed excellent robustness properties of the lms filter

linear estimation in krein spaces i theory the authors develop a self contained theory for linear estimation in krein spaces the derivation is based on simple concepts such as projections and matrix factorizations and leads to an interesting connection between krein space projection and the recursive computation of the stationary points of certain second order or quadratic forms the authors use the innovations process to obtain a general recursive linear estimation algorithm when specialized to a state space structure the algorithm yields a krein space generalization of the celebrated kalman filter with applications in several areas such as h sup spl infin filtering and control game problems risk sensitive control and adaptive filtering

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

parameter identification for uncertain plants using h methods abstract we demonstrate the effective use of h filtering and cost to come methods for parameter identification in deterministic uncertain plants that are linear in the unknown parameters but nonlinear otherwise the cost to come method is an approach that has been used earlier to solve linear and nonlinear h optimal control and filtering problems it consists of constructing a cost to come function which assists in the design of an optimal observer scheme the method is used here in the design of a parameter identification scheme for uncertain plants where measurements on the state of the system are available but not on its derivative two approaches are adopted in both of which the parameter estimation problem is formulated as an h filtering problem one of the approaches uses a more standard prefiltering of the past states input and disturbance signals the other approach is a novel design method which leads to a new class of identification schemes it involves two subproblems fsdi full state derivative information problem where it is assumed that both the state and its derivative are available to the parameter estimator and npfsi noise perturbed fsi problem where the estimator is assumed to measure a noise perturbed measurement of the state in the latter problem we use singular perturbation methods to prove asymptotic convergence of the performance of the identifier to that of the unperturbed case thus providing an asymptotically optimal solution to the fsi full state measurement problem to illustrate both approaches several simulation studies on a numerical example are provided

decorated star triangle relations and exact integrability of the one dimensional hubbard model the exact integrability of the one dimensional hubbard model is demonstrated with the help of a novel set of triangle relations the decorated star triangle relations the covering two dimensional statistical mechanical model obeys the star triangle or yang baxter relation a conjecture is presented for the eigenvalues of the transfer matrix

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

an l 2 stable feedback structure for nonlinear adaptive filtering and identification this paper proposes a feedback structure for the design of l2 stable algorithms for nonlinear adaptive filtering and identification and establishes explicit connections between classical schemes in iir modeling and more recent results in h theory in particular two algorithms due to feintuch and to landau as well as the so called pseudo linear regression and gauss newton algorithms are discussed within the framework proposed here additional examples and simulation results are included to illustrate the applicability of the approach to several nonlinear scenarios

symmetries and reduced systems of equations for three boson and four boson interactions a reduced system of equations is found for three boson and four boson interactions the invariants and surfaces in the space of parameters connected with these systems are constructed it is shown that at these surfaces the eigenvectors can be presented in power form the connection between the ansatz and polynomial extensions of lie algebras is briefly discussed the computational procedures corresponding to the original equations and the method of polynomial algebras are compared

polynomial approximation of divergence free functions etude de la meilleure approximation d une fonction sans divergence par un polynome algebrique ou trigonometrique sans divergence etablissement d une estimation optimale

linear estimation in krein spaces ii applications we have shown that several interesting problems in h sup spl infin filtering quadratic game theory and risk sensitive control and estimation follow as special cases of the krein space linear estimation theory developed in part i we show that all these problems can be cast into the problem of calculating the stationary point of certain second order forms and that by considering the appropriate state space models and error gramians we can use the krein space estimation theory to calculate the stationary points and study their properties the approach discussed here allows for interesting generalizations such as finite memory adaptive filtering with varying sliding patterns

filtering and smoothing in an h sup infinity setting the problems of filtering and smoothing are considered for linear systems in an h sup infinity setting i e the plant and measurement noises have bounded energies are in l sub 2 but are otherwise arbitrary two distinct situations for the initial condition of the system are considered the initial condition is assumed known in one case while in the other the initial condition is not known but the initial condition the plant and measurement noise are in some weighted ball of r sup n xl sub 2 finite horizon and infinite horizon cases are considered necessary and sufficient conditions are presented for the existence of estimators both filters and smoothers that achieve a prescribed performance bound and algorithms that result in performance within the bounds are developed in case of smoothers the optimal smoother is also presented the approach uses basic quadratic optimization theory in a time domain setting as a consequence of which both linear time varying and time invariant systems can be considered with equal ease in the smoothing problem for linear time varying systems one considers only the finite horizon case

h robustness of adaptive filters against measurement noise and parameter drift recently the issue of robustness of adaptive filtering algorithms has been investigated using the h paradigm in particular in the constant parameter case the celebrated normalized lms algorithm has been shown to coincide with the central h filter ensuring the minumum achievable disturbance attenuation level in this paper the problem is re examined by taking into account the robust performance of three classical algorithms normalized lms kalman filter central h filter with respect to both measurement noise and parameter drift it turns out that normalized lms does not guarantee any finite level of h robustness on the other hand it is shown that striving for the minimum achievable attenuation level leads to a trivial nondynamic estimator with poor h performance this motivates the need for a design approach balancing h and h performance criteria in this regard an illustrative example is presented showing that the central h filter is best suited to achieve a satisfactory tradeoff between the two performances

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

analysis of stability and performance of adaptation algorithms with time invariant gains adaptation laws that track parameters of linear regression models are investigated the considered class of algorithms apply linear time invariant filtering on the instantaneous gradient vector and includes least mean squares lms as its simplest member the asymptotic stability and steady state tracking performance for prediction and smoothing estimators is analyzed for parameter variations described by stochastic processes with time invariant statistics the analysis is based on a novel technique that decomposes the inherent feedback of adaptation algorithms into one time invariant loop and one time varying loop the impact of the time varying feedback on the tracking error covariance can be neglected under certain conditions and the performance analysis then becomes straightforward performance analysis in the presence of a non negligible time varying feedback is performed for algorithms that use scalar measurements convergence in mean square error mse and the mse tracking performance is investigated assuming independent consecutive regression vectors closed form expressions for the tracking mse are thereafter derived without this independence assumption for a subclass of algorithms applied to finite impulse response fir models with white inputs this class includes wiener lms adaptation