exponential stability of general tracking algorithms tracking and adaptation algorithms are from a formal point of view nonlinear systems which depend on stochastic variables in a fairly complicated way the analysis of such algorithms is thus quite complicated a first step is to establish the exponential stability of these systems this is of interest in its own right and a prerequisite for the practical use of the algorithm it is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time varying true ones in this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms this includes least mean squares recursive least squares and kalman filter based adaptation algorithms we show how stability of an averaged linear and deterministic equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes we also give explicit conditions for exponential stability of the most common algorithms the tracking performance of the algorithms is studied in a companion paper

symbol length in the brauer group of a field we bound the symbol length of elements in the brauer group of a field k containing a cm field for example any field containing an algebraically closed field or a finite field and solve the local exponent index problem for a cm field f in particular for a cm field f we show that every f central simple algebra of exponent p is similar to the tensor product of at most len p f t pm 1 1 symbol algebras of degree p we then use this bound on the symbol length to show that the index of such algebras is bounded by p 1 which in turn gives a bound for any algebra of exponent n via the primary decomposition finally for a field k containing a cm field f we show that every f central simple algebra of exponent p and degree p is similar to the tensor product of at most len p p k len p l symbol algebras of degree p where l is a cm edl a ps t 1 field

the period index problem for fields of transcendence degree 2 using geometric methods we prove the standard period index conjecture for the brauer group of a field of transcendence degree 2 over fp

global feedback stabilization of the angular velocity of a symmetric rigid body abstract in this paper we prove that the angular velocity equations of a symmetric rigid body can be globally stabilized by means of linear feedback when two control torques act on the body

generalized invariant subspaces for infinite dimensional systems in this paper some generalized invariant subspaces for infinite dimensional systems are investigated and then some sufficient conditions for parameter insensitive disturbance rejection problems with state feedback and with measurement output feedback to be solvable are studied

the period index problem for the brauer group of an algebraic surface in this paper we show that the period equals the index for elements of brauer groups of function fields of surfaces a key idea of the proof is that any azumaya algebra over a surface can be transformed into an azumaya algebra that is unobstructed

paper tracking error bounds of adaptive nonstationary filtering adaptive filtering with error gradient algorithm and constant step size is analyzed for a deterministic time variable optimum filtering vector the unrealistic assumption of independent observations is replaced by a bounded memory model largely justifiable in applications then the mean square tracking deviation msd between the optimum vector and the algorithm output is proved to include two contributions the stationary mode error characteristic of convergence accuracy which is proportional to the step size and the transient mode error reflecting the rapidity of tracking which is proportional to the squared ratio of the maximum optimum estimator increment to the step size this result agrees with the common intuition that there exists an optimum step size which compromises between convergence accuracy and tracking speed

asymptotic perturbed feedback linearisation of underactuated euler s dynamics a unifying methodology is introduced for smooth asymptotic stabilisation of underactuated rigid body dynamics under one and two degrees of actuation the methodology is based on the concept of generalised inversion and it aims to realise a perturbation from the unrealisable feedback linearising transformation a desired linear dynamics in a norm measure of the angular velocity components about the unactuated axes is evaluated along solution trajectories of euler s underactuated dynamical equations resulting in a linear relation in the control variables this relation is used to assess asymptotic stabilisability of underactuated rigid bodies with arbitrary values of inertia parameters and generalised inversion of the relation produces a control law that consists of particular and auxiliary parts the generalised inverse in the particular part is scaled by a dynamic factor such that it uniformly converges to the moore penrose inverse and the null control vector in the auxiliary part is chosen for asympto

locally optimum adaptive signal processing algorithms we propose a new analytic method for comparing constant gain adaptive signal processing algorithms specifically estimates of the convergence speed of the algorithms allow for the definition of a local measure of performance called the efficacy that can be theoretically evaluated by definition the efficacy is consistent with the fair comparison techniques used in signal processing applications using the efficacy as a performance measure we prove that the lms newton algorithm is optimum and is thus the fastest algorithm within a very rich algorithmic class furthermore we prove that regular lms is better than any of its variants that apply the same nonlinear transformation on the elements of the regression vector such as signed regressor quantized regressor etc for an important class of input signals simulations support all our theoretical conclusions

analysis of boundedness and convergence of online gradient method for two layer feedforward neural networks this paper presents a theoretical boundedness and convergence analysis of online gradient method for the training of two layer feedforward neural networks the well known linear difference equation is extended to apply to the general case of linear or nonlinear activation functions based on this extended difference equation we investigate the boundedness and convergence of the parameter sequence of concern which is trained by finite training samples with a constant learning rate we show that the uniform upper bound of the parameter sequence which is very important in the training procedure is the solution of an inequality regarding the bound it is further verified that for the case of linear activation function a solution always exists and moreover the parameter sequence can be uniformly upper bounded while for the case of nonlinear activation function some simple adjustment methods on the training set or the activation function can be derived to improve the boundedness property then for the convergence analysis it is shown that the parameter sequence can converge into a zone around an optimal solution at which the error function attains its global minimum where the size of the zone is associated with the learning rate particularly for the case of perfect modeling a strong global convergence result where the parameter sequence can always converge to an optimal solution is proved

moduli of twisted orbifold sheaves abstract we study stacks of slope semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope semistable sheaves on smooth projective surfaces

on proper coverings of artin stacks abstract we prove that every separated artin stack of finite type over a noetherian base scheme admits a proper covering by a quasi projective scheme an application of this result is a version of the grothendieck existence theorem for artin stacks

robust stabilization of the angular velocity of a rigid body the problem of robust control for the angular velocity of a rigid body subject to external disturbances is addressed it is shown that if the disturbances are matched there exists a control law attenuating the effect of the disturbances whereas in the case of non matched disturbances no such a feedback law generically exists hence a new concept of disturbance attenuation is introduced and it is proved that the aforementioned problem is solvable in this weaker sense explicit expressions of the control laws solving the proposed robust stabilization problems are given

a note on robust disturbance rejection problems for infinite dimensional systems in this paper for infinite dimensional systems characterized by unknown and unbounded system operators some sufficient conditions for robust disturbance rejection problems to be solvable are studied in the framework of the so called geometric approach

partial update lms algorithms partial updating of lms filter coefficients is an effective method for reducing computational load and power consumption in adaptive filter implementations this paper presents an analysis of convergence of the class of sequential partial update lms algorithms s lms under various assumptions and shows that divergence can be prevented by scheduling coefficient updates at random which we call the stochastic partial update lms algorithm spu lms specifically under the standard independence assumptions for wide sense stationary signals the s lms algorithm converges in the mean if the step size parameter spl mu is in the convergent range of ordinary lms relaxing the independence assumption it is shown that s lms and lms algorithms have the same sufficient conditions for exponential stability however there exist nonstationary signals for which the existing algorithms s lms included are unstable and do not converge for any value of spl mu on the other hand under broad conditions the spu lms algorithm remains stable for nonstationary signals expressions for convergence rate and steady state mean square error of spu lms are derived the theoretical results of this paper are validated and compared by simulation through numerical examples

identification of systems with regime switching and unmodeled dynamics this paper is concerned with persistent identification of systems that involve deterministic unmodeled dynamics and stochastic observation disturbances and whose unknown parameters switch values possibly large jumps that can be represented by a markov chain two classes of problems are considered in the first class the switching parameters are stochastic processes modeled by irreducible and aperiodic markov chains with transition rates sufficiently faster than adaptation rates of the identification algorithms in this case tracking real time parameters by output observations becomes impossible and we show that an averaged behavior of the parameter process can be derived from the stationary measure of the markov chain and can be estimated with periodic inputs and least squares type algorithms upper and lower error bounds are established that explicitly show impact of unmodeled dynamics in contrast the second class of problems represents systems whose state transitions occur infrequently an adaptive algorithm with variable step sizes is introduced for tracking the time varying parameters convergence and error bounds are derived numerical results are presented to illustrate the performance of the algorithm

linear regression and filtering under nonstandard assumptions arbitrary noise this note is devoted to parameter estimation in linear regression and filtering where the observation noise does not possess any nice probabilistic properties in particular the noise might have an unknown but bounded deterministic nature the basic assumption is that the model regressors inputs are random optimal rates of convergence for the modified stochastic approximation and least squares algorithms are established under some weak assumptions typical behavior of the algorithms in the presence of such deterministic noise is illustrated by numerical examples

stability of recursive stochastic tracking algorithms first the paper gives a stability study for the random linear equation x n 1 i a n x n it is shown that for a quite general class of random matrices a n of interest the stability of such a vector equation can be guaranteed by that of a corresponding scalar linear equation for which various results are given without requiring stationary or mixing conditions then these results are applied to the main topic of the paper i e to the estimation of time varying parameters in linear stochastic systems giving a unified stability condition for various tracking algorithms including the standard kalman filter least mean squares and least squares with forgetting factor

output regulation for linear distributed parameter systems this work extends the geometric theory of output regulation to linear distributed parameter systems with bounded input and output operator in the case when the reference signal and disturbances are generated by a finite dimensional exogenous system in particular it is shown that the full state feedback and error feedback regulator problems are solvable under the standard assumptions of stabilizability and detectability if and only if a pair of regulator equations is solvable for linear distributed parameter systems this represents an extension of the geometric theory of output regulation developed in francis 1997 and isidori and byrnes 1990 we also provide simple criteria for solvability of the regulator equations based on the eigenvalues of the exosystem and the system transfer function examples are given of periodic tracking set point control and disturbance attenuation for parabolic systems and periodic tracking for a damped hyperbolic system

analysis of mean square error and transient speed of the lms adaptive algorithm for the least mean square lms algorithm we analyze the correlation matrix of the filter coefficient estimation error and the signal estimation error in the transient phase as well as in steady state we establish the convergence of the second order statistics as the number of iterations increases and we derive the exact asymptotic expressions for the mean square errors in particular the result for the excess signal estimation error gives conditions under which the lms algorithm outperforms the wiener filter with the same number of taps we also analyze a new measure of transient speed we do not assume a linear regression model the desired signal and the data process are allowed to be nonlinearly related the data is assumed to be an instantaneous transformation of a stationary markov process satisfying certain ergodic conditions

learning curves for lms and regular gaussian processes uses methods due to guo ljung and wang 1997 to obtain explicit bounds on the error of the lms algorithm used in a linear prediction of a signal using previous values of that signal the signal is assumed to be a mean zero gaussian regular stationary random process the bounds are then used to construct learning curves for the lms algorithm in situations where the statistics of the process are only partially known

necessary and sufficient conditions for stability of lms guo and ljung 1995 established some general results on exponential stability of random linear equations which can be applied directly to the performance analysis of a wide class of adaptive algorithms including the basic lms ones without requiring stationarity independency and boundedness assumptions of the system signals the current paper attempts to give a complete characterization of the exponential stability of the lms algorithms by providing a necessary and sufficient condition for such a stability in the case of possibly unbounded nonstationary and non spl phi mixing signals the results of this paper can be applied to a very large class of signals including those generated from e g a gaussian process via a time varying linear filter as an application several novel and extended results on convergence and the tracking performance of lms are derived under various assumptions neither stationarity nor markov chain assumptions are necessarily required in the paper

the stability of variable step size lms algorithms variable step site lms vslms algorithms are a popular approach to adaptive filtering which can provide improved performance while maintaining the simplicity and robustness of conventional fixed step size lms here we examine the stability of vslms with uncorrelated stationary gaussian data most vslms described in the literature use a data dependent step size where the step size either depends on the data before the current time prior step size rule or through the current time posterior step size rule it has often been assumed that vslms algorithms are stable in the sense of mean square bounded weights provided that the step size is constrained to lie within the corresponding stability region for the lms algorithm for a single tap fitter we find exact expressions for the stability region of vslms over the classes of prior and posterior step sizes and show that the stability region for prior step size coincides with that of fixed step size but the region for posterior step size is strictly smaller than for fixed step size for the multiple tap case we obtain bounds on the stability regions with similar properties the approach taken here is a generalization of the classical method of analyzing the exponential stability of the weight covariance equation for lms although it is not possible to derive a weight covariance equation for general data dependent vslms the weight variances can be upper bounded by the solution of a linear time invariant difference equation after appropriately dealing with certain nonlinear terms for prior step size like fixed step size the state matrix is symmetric whereas for posterior step size the symmetry is lost requiring a more detailed analysis the results are verified by computer simulations

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

analysis of stochastic gradient tracking of time varying polynomial wiener systems this paper presents analytical and monte carlo results for a stochastic gradient adaptive scheme that tracks a time varying polynomial wiener 1958 system i e a linear time invariant lti filter with memory followed by a time varying memoryless polynomial nonlinearity the adaptive scheme consists of two phases 1 estimation of the lti memory using the lms algorithm and 2 tracking the time varying polynomial type nonlinearity using a second coupled gradient search for the polynomial coefficients the time varying polynomial nonlinearity causes a time varying scaling for the optimum wiener filter for phase 1 these time variations are removed for phase 2 using a novel coupling scheme to phase 1 the analysis for gaussian data includes recursions for the mean behavior of the lms algorithm for estimating and tracking the optimum wiener filter for phase 1 for several different time varying polynomial nonlinearities and recursions for the mean behavior of the stochastic gradient algorithm for phase 2 the polynomial coefficients are shown to be accurately tracked monte carlo simulations confirm the theoretical predictions and support the underlying statistical assumptions

on the learning mechanism of adaptive filters this paper highlights both analytically and by simulations some interesting phenomena regarding the behavior of ensemble average learning curves of adaptive filters that may have gone unnoticed among other results the paper shows that even ensemble average learning curves of single tap lms filters actually exhibit two distinct rates of convergence one for the initial time instants and another faster one for later time instants in addition such curves tend to converge faster than predicted by mean square theory and can converge even when a mean square stability analysis predicts divergence these effects tend to be magnified by increasing the step size two of the conclusions that follow from this work are 1 the mean square stability alone may not be the most appropriate performance measure especially for larger step sizes a combination of mean square stability and almost sure a s stability seems to be more appropriate 2 care is needed while interpreting ensemble average curves for larger step sizes the curves can lead to erroneous conclusions unless a large number of experiments are averaged at times of the order of tens of thousands or higher

cyclic algebras over p adic curves abstract in this paper we study division algebras over the function fields of curves over q p the first and main tool is to view these fields as function fields over nonsingular s which are projective of relative dimension 1 over the p adic ring z p a previous paper showed such division algebras had index bounded by n 2 assuming the exponent was n and n was prime to p in this paper we consider algebras of prime degree and hence exponent q p and show these algebras are cyclic we also find a geometric criterion for a brauer class to have index q

tracking a markov modulated stationary degree distribution of a dynamic random graph this paper considers a markov modulated duplication deletion random graph where at each time instant one node can either join or leave the network the probabilities of joining or leaving evolve according to the realization of a finite state markov chain two results are presented first motivated by social network applications the asymptotic behavior of the degree distribution is analyzed second a stochastic approximation algorithm is presented to track empirical degree distribution as it evolves over time the tracking performance of the algorithm is analyzed in terms of mean square error and a functional central limit theorem is presented for the asymptotic tracking error also a hilbert space valued stochastic approximation algorithm that tracks a markov modulated probability mass function with support on the set of nonnegative integers is analyzed

using stacks to impose tangency conditions on curves we define a deligne mumford stack xp r which depends on a scheme x an effective carrier divisor d c x and a positive integer r then we show that the abramovich vistoli mod uli stack of stable maps into xq r provides compactifications of the locally closed substacks of mgtn x 3 corresponding to relative stable maps 1 introduction this paper has two main purposes the first is to introduce the rth root construction which is a way of creating stack structure along a divisor given a scheme x an effective cartier divisor d c x and a positive integer r which is invertible on x it produces a deligne mumford stack xo r when r is not invertible the construction still makes sense and produces an artin stack which has finite diagonal over x since the output is in general an artin stack we give the definition and basic properties when x is an artin stack this simplifies the exposition since we need to apply the construction multiple times this construction was known to several mathematicians before we became aware of it but very little appeared in the literature aside from a brief mention in agv 3 5 3 secondly we compare twisted stable maps into xp r with ordinary stable maps into x the theory of twisted stable maps was developed by chen and ruan cr in the symplectic category and by abramovich and vistoli av in the algebraic category the latter showed that there is a deligne mumford stack xg n x 3 parametrizing morphisms x from twisted rc marked genus g nodal curves into a deligne mumford stack x such that 3 in the case where x is a scheme this is just the space of kontsevich stable maps mg n x 3 the theory of relative stable maps was developed by several symplectic ge ometers and the algebraic definition was worked out partially by andreas gath mann ga and in general by jun li li let x be a smooth complex projective veriety d c x a smooth divisor and 3 g b x such that d 3 0 here b x denotes the group of 1 cycles modulo algebraic equivalence choose a partition d 3 q qn where each qi is a nonnegative integer then a stable map c x from a smooth curve c having marked points x xn such that f c 3 and d y qixi ls a relative stable map the space of such mor phisms was compactified by ga and li and they used this to define relative gromov witten invariants

noncyclic division algebras over fields of brauer dimension one abstract let k be a complete discretely valued field of rank one with residue field q p it is well known that period equals index in br k we prove that when p 2 there exist noncyclic k division algebras of every 2 power degree divisible by four otherwise every k division algebra is cyclic this gives the first published example of a field whose brauer dimension and cyclic length are not equal

stabilizability of the angular velocity of a rigid body revisited abstract we prove that the jurdjevic quinn techniques for smooth stabilization can be applied to the equation of the angular velocity of a rigid body an explicit polynomial control is given for the symmetric case

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

fluctuation analysis of stochastic gradient identification of polynomial wiener systems this correspondence presents analytical results and monte carlo simulations for the fluctuation behavior of a stochastic gradient adaptive identification scheme this scheme identifies a polynomial wiener system linear fir filter followed by a static polynomial nonlinearity for noisy output observations the analysis includes 1 bounds and a recursion for the misadjustment matrix and 2 algorithm mean square stability regions a diagonal step size matrix for the adaptive coefficients is introduced to speed up convergence the theoretical predictions of the fluctuation analysis are supported by monte carlo simulations

recursive nonlinear system identification by a stochastic gradient algorithm stability performance and model nonlinearity considerations a parameter estimation problem in a class of nonlinear systems is considered where the input output relation of a nonlinear system is approximated by a polynomial model e g a volterra series a least mean squares lms type algorithm is utilized for the recursive estimation of the polynomial coefficients and its resulting mean square error mse convergence properties are investigated conditions for the algorithm stability in the mean square sense are established steady state mse bounds are obtained and the convergence rate is discussed in addition modeling accuracy versus steady state performance is examined it is found that an increase of the modeling accuracy may result in a deterioration of the asymptotic performance that is yielding a larger steady state mse linear system identification is studied as a special case

analysis of adaptive step size sa algorithms for parameter tracking we present proofs and data for adaptive step size algorithms for tracking time varying parameters when recursive stochastic approximation type algorithms are used a classical problem in adaptive control and communication theory concerns the tracking of the best fit of a given form when the statistics or the parameters change slowly a major and yet unresolved problem has been the choice of the step sizes in the tracking algorithm an algorithm for adapting the step size using the same system measurements which are used for the tracking was suggested by benveniste and various examples worked out by brossier the numerical results were very encouraging but proofs were lacking these proofs are supplied here together with supporting numerical data the proofs are based on recent results in stochastic approximation the adaptive step size technique works very well indeed much supporting analysis is presented particularly concerning the interpretation of certain stationary processes as stationary pathwise derivatives finite difference forms are also treated these are mathematically simpler and can be applied to a wide variety of systems even when the system is not well modeled the data shows that they work well

period index bounds for arithmetic threefolds the standard period index conjecture for brauer groups of p adic surfaces s predicts that mathrm ind alpha mathrm per alpha 3 for every alpha in mathrm br mathbf q p s using gabber s theory of prime to ell alterations and the deformation theory of twisted sheaves we prove that mathrm ind alpha mathrm per alpha 4 for alpha of period prime to 6p giving the first uniform period index bounds over such fields

c a b pairs in infinite dimensions abstract we introduce the notion of c a b pairs for infinite dimensional systems and relate this to the concept of invariance under an extended operator as an application we solve the disturbance decoupling problem with measurement feedback

performance analysis of general tracking algorithms a general family of tracking algorithms for linear regression models is studied it includes the familiar least mean square gradient approach recursive least squares and kalman filter based estimators the exact expressions for the quality of the obtained estimates are complicated approximate and easy to use expressions for the covariance matrix of the parameter tracking error are developed these are applicable over the whole time interval including the transient and the approximation error can be explicitly calculated

surjectivity of the total clifford invariant and brauer dimension abstract a celebrated theorem of merkurjev that the 2 torsion of the brauer group is represented by clifford algebras of quadratic forms is in general false when the base is no longer a field the first counterexamples when the base is among certain arithmetically subtle hyperelliptic curves over local fields were constructed by parimala scharlau and sridharan we prove that considering clifford algebras of all line bundle valued quadratic forms such counterexamples disappear and we recover merkurjev s theorem in these cases for any smooth curve over a local field or any smooth surface over a finite field the 2 torsion of the brauer group is always represented by clifford algebras of line bundle valued quadratic forms

stochastic gradient identification of polynomial wiener systems analysis and application this paper presents analytical numerical and experimental results for a stochastic gradient adaptive scheme that identifies a polynomial type nonlinear system with memory for noisy output observations the analysis includes the computation of the stationary points the mean square error surface and the stability regions of the algorithm for gaussian data convergence of the mean is studied using l sub 2 and euclidian norms monte carlo simulations confirm the theoretical predictions that show a small sensitivity to the observation noise an application is presented for the identification of a nonlinear time delayed feedback system