linearizable mappings and the low growth criterion we examine a family of discrete second order systems which are integrable through reduction to a linear system these systems were previously identified using the singularity confinement criterion here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates we show that the linearizable mappings are characterized by a very special degree growth the ones linearizable by reduction to projective systems exhibit zero growth i e they behave like linear systems while the remaining ones derivatives of riccati and gambier mapping lead to linear growth this feature may well serve as a detector of integrability through linearization

extension of the adler bobenko suris classification of integrable lattice equations the classification of lattice equations that are integrable in the sense of higher dimensional consistency is extended by allowing directed edges we find two cases that are not transformable via the admissible transformations to the lattice equations in the existing classification

performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

a novel kurtosis driven variable step size adaptive algorithm a new variable step size lms filter is introduced the time varying step size sequence is adjusted utilizing the kurtosis of the estimation error therefore reducing performance degradation due to the existence of strong noise the convergence properties of the algorithm are analyzed and an adaptive kurtosis estimator that takes into account noise statistics and optimally adapts itself is also presented simulation results confirm the algorithm s improved performance and flexibility

on the optimum gain parameter in lms adaptation in this correspondence the optimum selection of the adaptation parameter in lms adaptive estimation is discussed for a white data covarianee matrix it is shown via analysis and numerical evaluation that the optimum selection of is a complicated function of the number of filter taps the initial weight setting the wiener weights and the number of learning samples is not chosen in general to yield the most rapid transient response for most cases of interest a smaller value of will be selected for slower adaptation and smaller misadjustment error at the end of the learning phase

adaptive lms l filters for noise suppression in images several adaptive least mean squares lms l filters both constrained and unconstrained ones are developed for noise suppression in images and compared in this paper first the location invariant lms l filter for a nonconstant signal corrupted by zero mean additive white noise is derived it is demonstrated that the location invariant lms l filter can be described in terms of the generalized linearly constrained adaptive processing structure proposed by griffiths and jim 1982 subsequently the normalized and the signed error lms l filters are studied a modified lms l filter with nonhomogeneous step sizes is also proposed in order to accelerate the rate of convergence of the adaptive l filter finally a signal dependent adaptive filter structure is developed to allow a separate treatment of the pixels that are close to the edges from the pixels that belong to homogeneous image regions

integrable systems without the painleve property we examine whether the painleve property is a necessary condition for the integrability of nonlinear ordinary differential equations we show that for a large class of linearizable systems this is not the case in the discrete domain we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non painleve continuous linearizable systems we find that while these discrete systems are themselves linearizable they possess non confined singularities

deconstructing an integrable lattice equation we show that an integrable lattice equation obtained by j hietarinta using the consistency around a cube method without the tetrahedron assumption is indeed solvable by linearization we also present its nonautonomous extension

the complex lms adaptive algorithm transient weight mean and covariance with applications to the ale the transient and steady state mean and covariance of the complex valued lms adaptive element weights are investigated when the inputs are samples from circularly normal processes it is shown that the data covariance diagonalizing matrix also diagonalizes the weight covariance matrix this result permits describing the complete transient behavior of the adaptive line enhancer ale weight covariance matrix in closed form for the case of multiple equal power narrow band statistically independent orthogonal rayleigh fading sinusoids in broad band noise based on these transient covariance results it is shown that for any stage of adaptation there exists an optimum ale gain constant that minimizes the excess mean squared prediction error this result is of particular significance when processing time limited data

integrability conditions for nonautonomous quad graph equations this paper presents a systematic investigation of the integrability conditions for nonautonomous quad graph maps using the lax pair approach the ultra local singularity confinement criterion and direct construction of conservation laws we show that the integrability conditions derived from each of the methods are the one and the same suggesting that there exists a deep connection between these techniques for partial difference equations

a new extrapolation algorithm for band limited signals using the regularization method the ill posedness of the extrapolation problem in the presence of noise is considered a stable algorithm is constructed by solving a fredholm equation based on a regularization method the algorithm appears relatively robust since the noise eta sub delta t is taken as a function in l sup 2 t t t 0 such that the error energy integral mod eta sub delta t mod sup 2 dt

generalized symmetry classification of discrete equations of a class depending on twelve parameters we carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square which belong to a twelve parametric class the direct result of this classification is a list of equations containing no new examples however as an indirect result of this work we find a number of integrable examples pretending to be new one of them has a nonstandard symmetry structure the others are analogues of the liouville equation in the sense that they are darboux integrable we also enumerate all equations of the class which are linearizable via a two point first integral and specify the nature of integrability of some known equations

efficient subspace based algorithm for adaptive bearing estimation and tracking in some practical applications of array processing the directions of the incident signals should be estimated adaptively and or the time varying directions should be tracked promptly in this paper an adaptive bearing estimation and tracking abest algorithm is investigated for estimating and tracking the uncorrelated and correlated narrow band signals impinging on a uniform linear array ula based on the subspace based method without eigendecomposition sumwe where a linear operator is obtained from the array data to form a basis for the null space by exploiting the array geometry and its shift invariance property specifically the null space is estimated using the least mean square lms or normalized lms nlms algorithm and the directions are updated using the approximate newton method the transient analyses of the lms and nlms algorithms are studied where the weight i e the linear operator is in the form of a matrix and there is a correlation between the additive noise and input data that involve the instantaneous correlations of the received array data in the updating equation and the step size stability conditions are derived explicitly in addition the analytical expressions for the mean square error mse and mean square deviation msd learning curves of the lms algorithm are clarified the effectiveness of the abest algorithm is verified and the theoretical analyses are corroborated through numerical examples simulation results show that the abest algorithm is computationally simple and has good adaptation and tracking abilities

linear quadrilateral lattice equations and multidimensional consistency it is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations members of which are compatible between one another on a higher dimensional lattice there turn out to be two such families a natural parametrization is given for each

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

on the probability density function of the lms adaptive filter weights the joint probability density function of the weight vector in least mean square lms adaptation is studied for gaussian data models an exact expression is derived for the characteristic function of the weight vector at time n 1 conditioned on the weight vector at time n the conditional characteristic function is expanded in a taylor series and averaged over the unknown weight density to yield a first order partial differential difference equation in the unconditioned characteristic function of the weight vector the equation is approximately solved for small values of the adaption parameter and the weights are shown to be jointly gaussian with time varying mean vector and covariance matrix given as the solution to well known difference equations for the weight vector mean and covariance matrix the theoretical results are applied to analyzing the use of the weights in detection and time delay estimation simulations that support the theoretical results are also presented

microstatistic lms filtering adaptive microstatistic filters are developed for applications in which the second order statistics of the thresholded signals are not known or may be nonstationary a multilevel threshold decomposition such that real valued stochastic processes can be filtered is used and the computational complexity of the algorithm can be arbitrarily specified by the designer the adaptation uses the least mean squares error approach of the least mean square lms algorithm the convergence of the adaptive algorithm is proved due to the nonhomogeneous statistical characteristic of the threshold signals a different step size adaptation parameter can be assigned to each threshold level simple design guidelines are developed for finding the set of nonhomogeneous step sizes which in practice yield better convergence characteristics

a variable step size lms algorithm a least mean square lms adaptive filter with a variable step size is introduced the step size increases or decreases as the mean square error increases or decreases allowing the adaptive filter to track changes in the system as well as produce a small steady state error the convergence and steady state behavior of the algorithm are analyzed the results reduce to well known results when specialized to the constant step size case simulation results are presented to support the analysis and to compare the performance of the algorithm with the usual lms algorithm and another variable step size algorithm they show that its performance compares favorably with these existing algorithms

stochastic gradient adaptation under general error criteria examines a family of adaptive filter algorithms of the form w sub k 1 w sub k spl mu f d sub k w sub k sup t x sub k x sub k in which f spl middot is a memoryless odd symmetric nonlinearity acting upon the error such algorithms are a generalization of the least mean square lms adaptive filtering algorithm for even symmetric error criteria for this algorithm family the authors derive general expressions for the mean and mean square convergence of the filter coefficients for both arbitrary stochastic input data and gaussian input data they then provide methods for optimizing the nonlinearity to minimize the algorithm misadjustment for a given convergence rate using the calculus of variations it is shown that the optimum nonlinearity to minimize misadjustment near convergence under slow adaptation conditions is independent of the statistics of the input data and can be expressed as p x p x where p x is the probability density function of the uncorrelated plant noise for faster adaptation under the white gaussian input and noise assumptions the nonlinearity is shown to be x 1 spl mu spl lambda x sup 2 spl sigma sub k sup 2 where spl lambda is the input signal power and spl sigma sub k sup 2 is the conditional error power thus the optimum stochastic gradient error criterion for gaussian noise is not mean square it is shown that the equations governing the convergence of the nonlinear algorithm are exactly those which describe the behavior of the optimum scalar data nonlinear adaptive algorithm for white gaussian input simulations verify the results for a host of noise interferences and indicate the improvement using non mean square error criteria

large deviation principle in one dimensional dynamics we study the dynamics of smooth interval maps with non flat critical points for every such a map that is topologically exact we establish the full level 2 large deviation principle for empirical means in particular the large deviation principle holds for every non renormalizable quadratic map this includes the maps without physical measure found by hofbauer and keller and challenges the widely shared view of the large deviation principle as a refinement of laws of large numbers

backlund transformations for integrable lattice equations we give new backlund transformations bts for some known integrable in the sense of being multidimensionally consistent quadrilateral lattice equations as opposed to the natural auto bt inherent in every such equation these bts are of two other kinds specifically it is found that some equations admit additional auto bts with backlund parameter whilst some pairs of apparently distinct equations admit a bt which connects them

a new two dimensional lattice model that is consistent around a cube for two dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions i e that it is consistent around a cube cac as a consequence of cac one can construct a lax pair for the model recently adler bobenko and suris conducted a search based on this principle and certain additional assumptions one of those assumptions was the tetrahedron property which is satisfied by most known equations we present here one lattice equation that satisfies the consistency condition but does not have the tetrahedron property its lax pair is also presented and some basic properties are discussed

backward stability for polynomial maps with locally connected julia sets we study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability then we deduce from this that a polynomial f with a locally connected julia set is backward stable outside any neighborhood of its attracting and neutral cycles for a conformal measure this easily implies that one of the following holds 1 for a e x e j f x j f 2 for a e x j f x c x for a critical point c x depending on x

convergence analysis of lms filters with uncorrelated gaussian data statistical analysis of the least mean squares lms adaptive algorithm with uncorrelated gaussian data is presented exact analytical expressions for the steady state mean square error mse and the performance degradation due to weight vector misadjustment are derived necessary and sufficient conditions for the convergence of the algorithm to the optimal wiener solution within a finite variance are derived it is found that the adaptive coefficient which controls the rate of convergence of the algorithm must be restricted to an interval significantly smaller than the domain commonly stated in the literature the outcome of this paper therefore places fundamental limitations on the mse performance and rate of convergence of the lms adaptive scheme

conservation laws for nqc type difference equations this paper presents a classification of all three point conservation laws for a large class of integrable difference equations that has been described by nijhoff quispel and capel we show that every nonlinear equation from this class has at least two nontrivial conservation laws most of the conservation laws that are found are new

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

on the conservation laws for nonlinear partial difference equations in this paper a direct method is proposed to construct conservation laws for nonautonomous nonlinear partial difference equations p es the identified p es can be classified into qrt type kdv type mkdv type and sg type the integrability of the identified partial difference equations is also discussed

linearization through symmetries for discrete equations we show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete cole hopf transformation if the equation is linearizable then the symmetry provides the linearizing transformation at the end we present a few examples of applications for equations defined on four lattice points

on the convergence behavior of the lms and the normalized lms algorithms it is shown that the normalized least mean square nlms algorithm is a potentially faster converging algorithm compared to the lms algorithm where the design of the adaptive filter is based on the usually quite limited knowledge of its input signal statistics a very simple model for the input signal vectors that greatly simplifies analysis of the convergence behavior of the lms and nlms algorithms is proposed using this model answers can be obtained to questions for which no answers are currently available using other perhaps more realistic models examples are given to illustrate that even quantitatively the answers obtained can be good approximations it is emphasized that the convergence of the nlms algorithm can be speeded up significantly by employing a time varying step size the optimal step size sequence can be specified a priori for the case of a white input signal with arbitrary distribution