invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

strong integrality of quantum invariants of 3 manifolds we prove that the quantum so 3 invariant of an arbitrary 3 manifold m is always an algebraic integer if the order of the quantum parameter is co prime with the order of the torsion part of h 1 m z an even stronger integrality known as cyclotomic integrality was established by habiro for integral homology 3 spheres here we also generalize habiro s result to all rational homology 3 spheres

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

adaptation and parameter estimation in systems with unstable target dynamics and nonlinear parametrization in this paper we propose a solution to the problem of adaptive control and parameter estimation in systems with unstable target dynamics models of uncertainties are allowed to be nonlinearly parameterized and required to be smooth and monotonic functions of linear functionals of the parameters the mere assumption of existence of nonlinear operator gains for the target dynamics is sufficient to guarantee that system solutions are bounded reach a neighborhood of the target set and the mismatches between the modeled uncertainties and their compensator converge to zero with respect to parameter convergence a standard persistent excitation condition suffices to ensure that it is exponential when a weaker nonlinear version of persistent excitation is satisfied asymptotic convergence is guaranteed the spectrum of possible applications ranges from tyre road slip control to asynchronous message transmission in spiking neural oscillators

estimation of states and parameters for linear systems with nonlinearly parameterized perturbations we consider systems that can be described by a linear part with a nonlinear perturbation where the perturbation is parameterized by a vector of unknown constant parameters under a set of technical assumptions about the perturbation and its relationship to the outputs we present a modular design technique for estimating the system states and the unknown parameters the design consists of a high gain observer that estimates the states of the system together with the full perturbation and a parameter estimator constructed by the designer to invert a nonlinear equation we illustrate the technique on a simulated dc motor with friction

semismooth potentials of stochastic systems with degenerate diffusions we prove that for a number of optimal control problems including finite horizon long run average and optimal stopping with one dimensional degenerate diffusion processes the potential function solution to poisson equation and hence the value function solution to hjb equation are semismooth at the degenerate points i e the left and right hand side derivatives exist but may not be equal this allows applying the ito tanaka formula in the direct comparison based optimization approach and previous results on semismooth value functions depend heavily on this property this result will facilitate further research in stochastic optimal control

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

on adaptive control of nonlinearly parameterized nonlinear systems towards a constructive procedure abstract a new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized nonlinear systems was recently proposed by the authors the key step is the construction of a monotone mapping via a suitable selection of a controller tuning function which has to satisfy some integrability conditions this translates into the need to solve a partial differential equation pde in this paper this result is extended providing some answers to the questions of characterization of monotonizable systems and solvability of the pde first adding to the design a nonlinear dynamic scaling we obviate the need to solve the pde second for the case of factorizable nonlinearities the following results are established i it is shown that the monotonicity condition is satisfied if a linear matrix inequality is feasible ii directly verifiable involutivity conditions that ensure the solution of the pde are presented iii an explicit formula for the required tuning function is given provided the regressor matrix satisfies some rank conditions hence adding a dynamic scaling this yields a constructive solution to the problem

technical communique a remark on passivity based and discontinuous control of uncertain nonlinear systems we address the problem of robust stabilisation of nonlinear systems affected by time varying uniformly bounded in time affine perturbations our approach relies on the combination of sliding mode techniques and passivity based control roughly speaking we show that under suitable conditions the sliding mode variable can be chosen as the passive output of the perturbed system then we show how to construct a controller which guarantees the global uniform convergence of the plant s outputs towards a time varying desired reference even in the presence of permanently exciting time varying disturbances we illustrate our result on the tracking control of the van der pol oscillator

the coloured jones function the invariantsjk k of a framed knotk coloured by the irreduciblesu 2 q module of dimensionk are studied as a function ofk by means of the universalr matrix it is shown that whenjk k is written as a power series inh withq eh the coefficient ofhd is an odd polynomial ink of degree at most 2d 1 this coefficient is a vassiliev invariant ofk in the second part of the paper it is shown that ask varies these invariants span ad dimensional subspace of the space of all vassiliev invariants of degreed for framed knots the analogous questions for unframed knots are also studied

adaptive observers and parameter estimation for a class of systems nonlinear in the parameters we consider the problem of asymptotic reconstruction of the state and parameter values in systems of ordinary differential equations a solution to this problem is proposed for a class of systems of which the unknowns are allowed to be nonlinearly parameterized functions of state and time reconstruction of state and parameter values is based on the concepts of weakly attracting sets and non uniform convergence and is subjected to persistency of excitation conditions in the absence of nonlinear parametrization the resulting observers reduce to standard estimation schemes in this respect the proposed method constitutes a generalization of the conventional canonical adaptive observer design

distributed cooperative adaptive identification and control for a group of continuous time systems with a cooperative pe condition via consensus in this paper we first address the uniformly exponential stability ues problem of a group of distributed cooperative adaptive systems in a general framework inspired by consensus theory distributed cooperative adaptive laws are proposed to estimate the unknown parameters of these systems it is shown that not only is the entire closed loop system stable but also both the identification tracking error and the parameter estimation error converge to zero uniformly exponentially under a cooperative persistent excitation pe condition of a regressor matrix in each system which is weaker than the traditionally defined pe condition the effects of network topology on ues of the closed loop system are also explored if the topology is time invariant it needs to be undirected and connected however when the topology is time varying it is just required that the integration of the topology over an interval with fixed length is undirected and connected the established results are then employed to identify and control several classes of linearly parameterized systems simulation examples are also provided to demonstrate the effectiveness and applications of the proposed distributed cooperative adaptive laws

a solution for two dimensional mazes with use of chaotic dynamics in a recurrent neural network model chaotic dynamics introduced into a neural network model is applied to solving two dimensional mazes which are ill posed problems a moving object moves from the position at t to t 1 by simply defined motion function calculated from firing patterns of the neural network model at each time step t we have embedded several prototype attractors that correspond to the simple motion of the object orienting toward several directions in two dimensional space in our neural network model introducing chaotic dynamics into the network gives outputs sampled from intermediate state points between embedded attractors in a state space and these dynamics enable the object to move in various directions system parameter switching between a chaotic and an attractor regime in the state space of the neural network enables the object to move to a set target in a two dimensional maze results of computer simulations show that the success rate for this method over 300 trials is higher than that of random walk to investigate why the proposed method gives better performance we calculate and discuss statistical data with respect to dynamical structure

immersion and invariance a new tool for stabilization and adaptive control of nonlinear systems a new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented the method relies upon the notions of system immersion and manifold invariance and in principle does not require the knowledge of a control lyapunov function the construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the invariant output zeroing manifold and its friend the method is well suited in situations where we know a stabilizing controller of a nominal reduced order model which we would like to robustify with respect to higher order dynamics this is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one we also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective our construction does not invoke certainty equivalence nor requires a linear parameterization furthermore viewed from a lyapunov perspective it provides a procedure to add cross terms between the parameter estimates and the plant states finally it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form yielding new stabilizing control laws we illustrate the method with several academic and practical examples including a mechanical system with flexibility modes an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

nonlinear pi control of uncertain systems an alternative to parameter adaptation a novel approach to stabilization and trajectory tracking for nonlinear systems with unknown parameters and uncertain disturbances is developed we take a drastic departure from the classical adaptive control approach consisting of a parameterized feedback law and an identifier which tries to minimize a tracking or prediction error instead we propose a simple nonlinear pi structure that generates a stable error equation with a perturbation function that exhibits at least one root trajectories are forced to converge to this root by suitably adjusting the nonlinear pi gains we consider the two basic problems of i matched uncertainties when the uncertain terms are in the image of the input matrix and ii unknown control directions when the control signal is multiplied by a gain of unknown sign we show that without knowing the system parameters and with only basic information on the uncertainties we can achieve global asymptotic stability and global tracking without injecting high gains into the loop interestingly we prove that we can take as our nonlinear pi structure an activation function reminiscent of that used in neural networks although most of the results are derived assuming full state measurement we also present an observer based solution for a chain of integrators with unknown control direction the procedure is shown to provide simple solutions to the classical problems of neural network function approximation as well as eccentricity control and friction compensation of mechanical systems

identification of linearly overparametrized nonlinear systems often a dynamical model is nonlinear in the unknown parameters but it can be transformed into an overparametrized linear regression model where the components of the overparametrization vector are nonlinear functions of the smaller number of unknown parameters an algorithm that directly identifies the unknown parameters is presented and the authors characterize the convergence domains under two different sets of assumptions on the excitation of the signals the corresponding convergence rates are computed

a formula for casson s invariant suppose h is a homology 3 sphere obtained by dehn surgery on a link l in a homology 3 sphere m if every pair of components of l has zero linking number in m then we give a formula for the casson invariant x h in terms of a m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks a few consequences of this formula are given introduction andrew casson has discovered an integral invariant of oriented homology 3 spheres which reduces mod 2 to the rochlin or n invariant 1 in this paper we give a formula for casson s invariant in the case where the homology sphere h is obtained by dehn surgery on a link l in a homology sphere m and furthermore the linking number between every pair of components of l is zero note that every homology 3 sphere can be obtained in this way with m s3 in this case the casson invariant x h can be expressed in terms of x m the surgery coefficients of l and a certain coefficient from each of the conway polynomials of l and all its sublinks to illustrate the utility of this formula we compute the casson invariant of homology spheres that arise as cyclic branched covers branched along untwisted double knots we also describe how the x invariant changes when a crossing of the framed link l between two strands of the same component is changed this provides at least in this setting an effective method for computing the invariant an interesting question is whether it is possible to show directly that the formula we give is preserved by the kirby rolfsen calculus and thus give an alternative proof of the invariance of x if l1 and l2 are two rationally framed links representing the same manifold then l1 can be transformed into l2 by a sequence of three types of moves homeomorphism twisting and trivial insertion or deletion 8 if the linking number between every pair of components of l1 is zero then the first two types of moves will preserve this property while the third move may not however if l1 can be transformed into l2 through a sequence of links all of whose linking numbers are all zero then we show that the formula for x is preserved we do not know if such a transformation is in general possible in order to state casson s theorem and the formula for x h we first establish some notation suppose l k1 kn is a framed link in a 3 manifold m with received by the editors october 22 1985 1980 mathematics subject classification 1985 revision primary 57m25 supported in part by the nsf under grant no dms 8502940 x 1986 american mathematical society 0002 9947 86 1 00 25 per page

identification of non linear system structure and parameters using regime decomposition abstract an off line algorithm for empirical modeling and identification of non linear dynamic systems is presented the minimal input to the algorithm is a sequence of empirical data and the model order using this information the algorithm searches for an optimal model structure and parameters within a rich non linear model set the model representation is based on the interpolation of a number of simple local models where each local model has a limited range of validity but the local models yield a complete global model when interpolated the method is illustrated using simulated data

parameter estimation and compensation in systems with nonlinearly parameterized perturbations we consider a class of systems influenced by perturbations that are nonlinearly parameterized by unknown constant parameters and develop a method for estimating the unknown parameters the method applies to systems where the states are available for measurement and perturbations with the property that an exponentially stable estimate of the unknown parameters can be obtained if the whole perturbation is known the main contribution is to introduce a conceptually simple modular design that gives freedom to the designer in accomplishing the main task which is to construct an update law to asymptotically invert a nonlinear equation compensation for the perturbations in the system equations is considered for a class of systems with uniformly globally bounded solutions for which the origin is uniformly globally asymptotically stable when no perturbations are present we also consider the case when the parameters can only be estimated when the controlled state is bounded away from the origin and show that we may still be able to achieve convergence of the controlled state we illustrate the method through examples and apply it to the problem of downhole pressure estimation during oil well drilling

stochastic control of multidimensional systems with relative optimization we derive the tanaka formula for multidimensional semismooth functions with the local time on the semismooth curve along the gradient direction with this formula we extend the relative optimization based approach to stochastic control to multidimensional systems optimality conditions are derived for systems with semismooth value functions and no viscosity solution is involved this approach provides new insights and motivates the research on stochastic control and stochastic calculus in particular for problems with nonsmooth features and degenerate points

a unified witten reshetikhin turaev invariant for integral homology spheres we construct an invariant j m of integral homology spheres m with values in a completion widehat mathbb z q of the polynomial ring q such that the evaluation at each root of unity gives the the su 2 witten reshetikhin turaev invariant m of m at thus j m unifies all the su 2 witten reshetikhin turaev invariants of m it also follows that m as a function on behaves like an analytic function defined on the set of roots of unity

stable adaptive control of a class of first order nonlinearly parameterized plants stable adaptive control algorithms are designed for a class of first order nonlinearly parameterized plants containing nonlinearities similar to those arising in fermentation process models when the plant nonlinearity is conveniently parameterized and when the controller parameters are adjusted adaptively the output error is shown to converge to zero stability of the overall system is proved using the lyapunov function with a suitably chosen cubic term

a parameter estimation approach to state observation of nonlinear systems abstract a novel approach to the problem of partial state estimation of nonlinear systems is proposed the main idea is to translate the state estimation problem into one of estimation of constant unknown parameters related to the systems initial conditions the class of systems for which the method is applicable is identified via two assumptions related to the transformability of the system into a suitable cascaded form and our ability to estimate the unknown parameters the first condition involves the solvability of a partial differential equation while the second one requires some persistency of excitation like conditions the proposed observer is shown to be applicable to position estimation of a class of electromechanical systems for the reconstruction of the state of power converters and for speed observation of a class of mechanical systems

on the melvin morton rozansky conjecture we prove a conjecture stated by melvin and morton and elucidated further by rozansky saying that the alexander conway polynomial of a knot can be read from some of the coefficients of the jones polynomials of cables of that knot i e coefficients of the colored jones polynomial we first reduce the problem to the level of weight systems using a general principle which may be of some independent interest and which sometimes allows to deduce equality of vassiliev invariants from the equality of their weight systems we then prove the conjecture combinatorially on the level of weight systems finally we prove a generalization of the melvin morton rozansky mmr conjecture to knot invariants coming from arbitrary semi simple lie algebras as side benefits we discuss a relation between the conway polynomial and immanants and a curious formula for the weight system of the colored jones polynomial

immersion and invariance adaptive control of nonlinearly parameterized nonlinear systems a new framework to design adaptive controllers for nonlinearly parameterized systems is proposed in this technical note the key step is the construction of a monotone mapping which explicitly depends on some of the estimator tuning parameters convexity that is related to monotonicity has been explored by several authors the property is assumed a priori and is useful for the design only in some region of state space which is completed switching the controller in our approach monotonicity can be used in the whole state space and moreover is enforced by the designer effectively becoming a synthesis tool one consequence of these features is that the controller does not rely on state dependent switching in order to dispose of degrees of freedom to render the function monotone we depart from standard adaptive control and adopt instead the recently introduced immersion and invariance approach

the universalr matrix burau representation and the melvin morton expansion of the colored jones polynomial abstract p melvin and h morton 9 studied the expansion of the colored jones polynomial of a knot in powers of q 1 and color they conjectured an upper bound on the power of color versus the power of 1 they also conjectured that the bounding line in their expansion generated the inverse alexander conway polynomial these conjectures were proved by d bar natan and s garoufalidis 1 we have conjectured 12 that other lines in the melvin morton expansion are generated by rational functions with integer coefficients whose denominators are powers of the alexander conway polynomial here we prove this conjecture by using the r matrix formula for the colored jones polynomial and presenting the universal r matrix as a perturbed burau matrix

approximation with random bases in this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a hilbert space of functions we consider and analyze published procedures both randomized and deterministic for selecting elements from these families that have been shown to ensure the rate of convergence in l2 norm of order o 1 n where n is the number of elements we show that both randomized and deterministic procedures are successful if additional information about the families of functions to be approximated is provided in the absence of such additional information one may observe exponential growth of the number of terms needed to approximate the function and or extreme sensitivity of the outcome of the approximation to parameters implications of our analysis for applications of neural networks in modeling and control are illustrated with examples

stabilization of nonlinear systems nonlinearly depending on fast time varying parameters an immersion and invariance approach the problem of stabilization of nonlinear systems which depend nonlinearly on fast time varying parameters is considered in the technical note it is assumed that if the plant parameters were known a static state feedback controller that achieves the stabilization objective with a lyapunov like function that is independent of the parameters is known a constructive procedure to update the unknown parameters of the controller based on the immersion and invariance approach is proposed the main contribution of the paper is to show that the proposed controller guarantees global convergence to zero of the systems state for arbitrary time variations of the plant parameters provided the controller parameters are bounded to ensure the latter condition an assumption that in the single parameter case is strictly weaker than the monotonicity condition invoked in previous studies is imposed stabilization is achieved via a rather unique combination of gradient like parameter estimation and the construction of a monotonic signal that counters the deleterious effect of the parameter variations several simulation examples illustrate the applicability of the suggested method

performance enhancement of parameter estimators via dynamic regressor extension and mixing a new procedure to design parameter estimators with enhanced performance is proposed in the technical note for classical linear regression forms it yields a new parameter estimator whose convergence is established without the usual requirement of regressor persistency of excitation the technique is also applied to nonlinear regressions with partially monotonic parameter dependence giving rise again to estimators with enhanced performance simulation results illustrate the advantages of the proposed procedure in both scenarios

on the series expansion approach to the identification of hammerstein systems a polynomial identification algorithm for recovering a nonlinearity in the hammerstein system is proposed the estimate employs the legendre orthogonal system with adaptively selected number of terms the global consistency along with rates of convergence are established no assumptions concerning continuity of the nonlinearity or its functional form are made a data driven method using the cross validation technique for selecting the number of terms in the estimate is presented

double construction for crossed hopf coalgebras abstract we provide an analog of the drinfeld quantum double construction in the context of crossed hopf group coalgebras introduced by turaev we prove that provided the base group is finite the double of a semisimple crossed hopf group coalgebra is both modular and unimodular

frequency domain identification of hammerstein models discusses hammerstein model identification in the frequency domain using sampled input output data by exploring the fundamental frequency and harmonics generated by the unknown nonlinearity we propose a frequency domain approach and show its convergence for both the linear and nonlinear subsystems in the presence of noise no a priori knowledge of the structure of the nonlinearity is required and the linear part can be nonparametric

adaptive control of nonlinearly parameterized systems the smooth feedback case studies global adaptive control of nonlinearly parameterized systems with uncontrollable linearization using a parameter separation technique and the tool of adding a power integrator we develop a feedback domination design approach for the explicit construction of a smooth adaptive controller that solves the problem of global state regulation in contrast to the existing results in the literature a key feature of our adaptive regulator is its minimum order property namely no matter how big the number of unknown parameters is the order of the dynamic compensator is identical to one and is therefore minimal as a consequence global state regulation of feedback linearizable systems with nonlinear parameterization is achieved by one dimensional adaptive controllers without imposing any extra e g convex concave conditions on the unknown parameters

adaptive control with nonconvex parameterization deals with the problem of nonconvex parameterization in adaptive control for a class of nonlinear dynamic systems new algorithms for stable adaptive control are introduced sufficient conditions for achieving control objectives of adaptive control with new algorithms are investigated simulation examples are presented for illustration

a categorification of the jones polynomial author s khovanov mikhail abstract we construct a bigraded cohomology theory of links whose euler characteristic is the jones polynomial

parameter and state estimation of nonlinear systems using a multi observer under the supervisory framework we present a hybrid scheme for the parameter and state estimation of nonlinear continuous time systems which is inspired by the supervisory setup used for control state observers are synthesized for some nominal parameter values and a criterion is designed to select one of these observers at any given time instant which provides state and parameter estimates assuming that a persistency of excitation condition holds the convergence of the parameter and state estimation errors to zero is ensured up to a margin which can be made as small as desired by increasing the number of observers to reduce the potential computational complexity of the scheme we explain how the sampling of the parameter set can be dynamically updated using a zoom in procedure this strategy typically requires a fewer number of observers for a given estimation error margin compared to the static sampling policy the results are shown to be applicable to linear systems and to a class of nonlinear systems we illustrate the applicability of the approach by estimating the synaptic gains and the mean membrane potentials of a neural mass model

identification of approximated hammerstein models in a worst case setting the identification of hammerstein models for nonlinear systems in considered in a worst case setting assuming unknown but bounded measurement noise a new approach is proposed in which the identification of a low complexity hammerstein model amounts to the computation of the chebichev center of a set of matrices conditioned to the manifold of rank one matrices an identification algorithm based on a relaxation technique is proposed and its consistency is proven the algorithm is computationally attractive in two cases noise bounded either in 2 or in norm the effectiveness of the proposed central algorithm and the comparison with the corresponding projection algorithm which is based on the singular value decomposition are investigated both analytically and through numerical examples in particular tight error bounds are obtained for the projection algorithm

local times on curves and uniform invariance principles sufficient conditions are given for a family of local times l t ofd dimensional brownian motion to be jointly continuous as a function oft and then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of brownian motion uniformly over a large family of measures applications included some new results for intersection local times for brownian motions on 2 and 2

parameter convergence in nonlinearly parameterized systems a large class of problems in parameter estimation concerns nonlinearly parametrized systems over the past few years a stability framework for estimation and control of such systems has been established we address the issue of parameter convergence in such systems in this paper systems with both convex concave and general parameterizations are considered in the former case sufficient conditions are derived under which parameter estimates converge to their true values using a min max algorithm in the latter case to achieve parameter convergence a hierarchical min max algorithm is proposed where the lower level consists of a min max algorithm and the higher level component updates the bounds on the parameter region within which the unknown parameter is known to lie using this hierarchical algorithm a necessary and sufficient condition is established for global parameter convergence in systems with a general nonlinear parameterization in both cases the conditions needed are shown to be stronger than linear persistent excitation conditions that guarantee parameter convergence in linearly parametrized systems explanations and examples of these conditions and simulation results are included to illustrate the nature of these conditions a general definition of nonlinear persistent excitation that leads to parameter convergence is proposed at the end of this paper

quantum koszul complexes for majid s braided lie algebras a yang baxter operator was canonically associated to any cocommutative braided lie algebra as defined by majid applying previous results we construct quantum koszul complexes for braided lie algebras with respect to this operator in the usual case the complexes reduce to the chevalley eilenberg complexes for classical lie algebras

lyapunov design for convergence rate improvement in adaptive control this note deals with the improvement of the convergence rate and robust properties in an adaptive error model a new adaptive control algorithm is presented which includes both the estimate of the prediction error to drive the parameter adaptation and the proportional integral or integral relay adjustment law to prevent unbounded growth of feedback gains a least squares gain update is used the adaptive controller is constructed using measurements of state variables only the design is based on the lyapunov function method and a new general form of the lyapunov function was found as a result exponential convergence of the lyapunov function in the presence of persistent excitation is obtained

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

adaptive classification of temporal signals in fixed weight recurrent neural networks an existence proof recurrent neural networks with fixed weights have been shown in practice to successfully classify adaptively signals that vary as a function of time in the presence of additive noise and parametric perturbations we address the question can this ability be explained theoretically we provide a mathematical proof that these networks have this ability even when parametric perturbations enter the signals nonlinearly the restrictions that we impose on the signals to be classified are that they satisfy an assumption of nondegeneracy and that noise amplitude is sufficiently small further we demonstrate that the recurrent neural networks may not only classify uncertain signals adaptively but also can recover the values of uncertain parameters of the signals up to their equivalence classes