a fast covariance type algorithm for sequential least squares filtering and prediction fast implementation of sequential least squares ls algorithms is of great importance in various applications of signal processing estimation system identification and control the purpose of this note is the introduction of an efficient sequential ls algorithm for multichannel unwindowed signals covariance case the new scheme in the single channel case requires 10 m madpr multiplications and divisions per recursion m being the number of estimated parameters this offers a saving of 5 m madpr compared to other existing algorithms

an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

a unified derivation for fast estimation algorithms by the conjugate direction method abstract fast estimation algorithms inspired by the classical method of levinson have been developed in the areas of time series analysis system identification and signal processing this paper provides a unified derivation for the levinson durbin whittle wiggins robinson fast recursive least squares rls ladder or lattice and fast cholesky algorithms as special cases of the conjugate direction method cdm this gives a novel derivation and interpretation for all these methods

the computational complexity of knot genus and spanning area we investigate the computational complexity of some problems in three dimensional topology and geometry we show that the problem of determining a bound on the genus of a knot in a 3 manifold is np complete using similar ideas we show that deciding whether a curve in a metrized pl 3 manifold bounds a surface of area less than a given constant c is np hard

fast adaptive algorithms for multichannel filtering and system identification fast transversal and lattice least squares algorithms for adaptive multichannel filtering and system identification are developed models with different orders for input and output channels are allowed four topics are considered multichannel fir filtering rational iir filtering arx multichannel system identification and general linear system identification possessing a certain shift invariance structure the resulting algorithms can be viewed as fast realizations of the recursive prediction error algorithm computational complexity is then reduced by an order of magnitude as compared to standard recursive least squares and stochastic gauss newton methods the proposed transversal and lattice algorithms rely on suitable order step up step down updating procedures for the computation of the kalman gain stabilizing feedback for the control of numerical errors together with long run simulations are included

least squares adaptive lattice and transversal filters a unified geometric theory a unified theory is presented to characterize least squares adaptive filters in either lattice or transversal filter form for nonstationary processes the derivations are based upon a geometric formulation of least squares estimation and on the concept of displacement rank a few basic geometric relations are shown to underlie the various algorithms insights into the fundamental concepts that unify lattice and transversal filter approaches to least squares adaptive filters are also given the general results are illustrated by applications to the so called pre windowed and growing memory covariance formulations of the deterministic least squares problem

qr decomposition based least squares lattice interpolators qr decomposition based least squares lattice qrd lsl predictors have been extensively applied in the design of order recursive adaptive filters this work presents qrd lsl interpolators that use both past and future data samples to estimate the present data sample based on a novel decoupling property we show that the order recursive qrd lsl interpolators display better numerical properties and achieve a higher level of computational efficiency than the conventional lsl interpolators except for an overall delay needed for physical realization qrd lsl interpolators can substantially outperform qrd lsl predictors while retaining all the desirable features of the latter

a self orthogonalizing efficient block adaptive filter this paper deals with the development of a unique self orthogonalizing block adaptive filter sobaf algorithm that yields efficient finite impulse response fir adaptive filter structures computationally the sobaf is shown to be superior to the least mean squares lms algorithm the consistent convergence performance which it provides lies between that of the lms and the recursive least squares rls algorithm but unlike the lms is virtually independent of input statistics the block nature of the sobaf exploits the use of efficient circular convolution algorithms such as the fft the rectangular transform rt the fermat number transform fnt and the fast polynomial transform fpt in performance the sobaf achieves the mean squared error mse convergence of a self orthogonalizing structure that is the adaptive filter converges under any input conditions at the same rate as an lms algorithm would under white input conditions furthermore the selection of the step size for the sobaf is straightforward as the range and the optimum value of the step size are independent of the input statistics

sliding window adaptive fast qr and qr lattice algorithms sliding window formulations of the fast qr and fast qr lattice algorithms are presented the derivations are based on the partial triangularization of raw data matrices three methods for window downdating are discussed the method of plane hyperbolic rotations the chambers method and the linpack algorithm a numerically ill conditioned stationary signal and a speech signal are used in finite wordlength simulations of the full qr nonfast fast qr and qr lattice algorithms all algorithms are observed to be numerically stable over billions of iterations for double precision mantissas 53 bits but as the number of bits is decreased in the mantissa the algorithms exhibit divergent behavior hence practically the algorithms can de regarded as numerically stable for long wordlengths

the block processing ftf adaptive algorithm a new block type adaptive filtering algorithm is presented this new block adaptive filter differs from the frequency domain block adaptive filters of ferrara 1980 and of clark mitra and parker 1980 in that the new method applies a deterministic time domain least squares criteria within each of the data blocks information is carried from block to block via a weighted initial condition this new block fast transversal filters bftf algorithm is a numerically stable algorithm and can also be used to perform efficient least squares system identification on any one data block in which case it shows a moderate computational advantage over the previous most efficient single data block algorithms of morf et al 1977 of marple 1981 and of kalouptsidis manolakis and carayannis 1984 1985 mechanisms for tracking and varying block length from block to block are also presented and evaluated finally we indicate how the new algorithm could be pipelined for maximum throughput with delay proportional to the number of parameters after computation of the sample correlation lags

adaptive multiregression in reproducing kernel hilbert spaces the multiaccess mimo channel case this paper introduces a wide framework for online i e time adaptive supervised multiregression tasks the problem is formulated in a general infinite dimensional reproducing kernel hilbert space rkhs in this context a fairly large number of nonlinear multiregression models fall as special cases including the linear case any convex continuous and not necessarily differentiable function can be used as a loss function in order to quantify the disagreement between the output of the system and the desired response the only requirement is the subgradient of the adopted loss function to be available in an analytic form to this end we demonstrate a way to calculate the subgradients of robust loss functions suitable for the multiregression task as it is by now well documented when dealing with online schemes in rkhs the memory keeps increasing with each iteration step to attack this problem a simple sparsification strategy is utilized which leads to an algorithmic scheme of linear complexity with respect to the number of unknown parameters a convergence analysis of the technique based on arguments of convex analysis is also provided to demonstrate the capacity of the proposed method the multiregressor is applied to the multiaccess multiple input multiple output channel equalization task for a setting with poor resources and nonavailable channel information numerical results verify the potential of the method when its performance is compared with those of the state of the art linear techniques which in contrast use space time coding more antenna elements as well as full channel information

fast orthogonal least squares algorithm for efficient subset model selection an efficient implementation of the orthogonal least squares algorithm for subset model selection is derived computational complexity of the algorithm is examined and the result shows that this new fast orthogonal least squares algorithm significantly reduces computational requirements

enhancement of sinusoids in colored noise and the whitening performance of exact least squares predictors the extraction of sinusoids in white noise using least squares predictors has attracted a lot of attention in the past mainly in the context of the adaptive line enhancer ale however very few results exist for the practical colored noise case or for the whitening performance of the predictors we use a matrix formulation to derive the optimal least squares coefficients and frequency response of the d step predictor or ale for sinusoids real or complex in additive colored noise several cases are considered and in particular new formulas for the amplitude gain are obtained in low pass background noise the amplitude gain of the sinusoids becomes essentially a monotonically increasing function of their frequency and a decreasing function for high pass noise for the whitening application signal to noise ratio snr bounds of the output are derived when the input is a white signal plus a sinusoidal interference we also give a state space model and stochastic interpretations of our analysis of the d step predictor providing connections to other related areas to enable filtering of nonstationary complex inputs as well as multichannel and multiexperiment data a complex vector version of the ladder algorithm is presented that can be used to implement the ale noise cancelling and noise inversion for narrow band interference rejection

an unwindowed rls adaptive lattice algorithm a covariance rls lattice adaptive filtering algorithm is presented the algorithm permits nonzero initial conditions and need not warn n 1 iterations in advance of the first iteration to preserve its low computational requirements these are both improvements over previous lattice algorithms the algorithm also has a slight computational advantage over previous solutions rendering it more applicable to adaptive filtering applications such as fast starting adaptive equalizers and echo cancellers where the initial data in the adaptive filter are not and cannot be assumed to be zero

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

efficient block implementation of exact sequential least squares problems an efficient blockwise algorithm namely the block sequential least squares bsls algorithm is presented for sequentially solving ls problems in realtime the information is carried from block to block by iterating some correlation vectors in the case of successive data blocks the exactness of the bsls algorithm is achieved at approximately the same computational requirement as characterizes the nonexact bftf block fast transversal filter algorithm which is significantly less than sample by sample rls recursive least squares algorithms however the bsls cannot accommodate the case of discontinuous blocks of data which can be accommodated at the expense of a nonexact solution by the bftf it is shown that the bsls algorithm allows efficient use of the fft fast fourier transform technique to make remarkable gains in computational complexity savings additionally the bsls algorithm can provide an improved numerical stability over the existing fast rls algorithms the numerical performance is illustrated by applications to adaptive equalization and online parameter identification

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

instrumental variable methods for arma spectral estimation the modified yule walker technique of arma spectral estimation is shown to be a special case of the instrumental variable method of system identification several recursive instrumental variable algorithms are proposed for adaptive spectral estimation including a new algorithm for the overdetermined case an efficient lattice algorithm is presented for solving the modified yule walker equations given the sample correlation coefficients the performance of some of the algorithms is illustrated by simulation results

echo cancellation of voiceband data signals using recursive least squares and stochastic gradient algorithms the convergence properties of adaptive least squares ls and stochastic gradient sg algorithms are studied in the context of echo cancellation of voiceband data signals the algorithms considered are the sg transversal sg lattice ls transversal fast kalman and ls lattice it is shown that for the channel estimation problem considered here ls algorithms converge in approximately 2n iterations where n is the order of the filter in contrast both sg algorithms display inferior convergence properties due to their reliance upon statistical averages simulations are presented to verify this result and indicate that the fast kalman algorithm frequently displays numerical instability which can be circumvented by using the lattice structure finally the equivalence between an ls algorithm and a fast converging modified sg algorithm which uses a maximum length input data sequence is shown

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

a fast recursive least squares adaptive second order volterra filter and its performance analysis a fast recursive least squares rls adaptive nonlinear filter modeled using a second order volterra series expansion is presented the structure uses the ideas of fast rls multichannel filters and has a computational complexity of o n sup 3 multiplications where n 1 represents the memory span in number of samples of the nonlinear system model a theoretical performance analysis of its steady state behaviour in both stationary and nonstationary environments is presented the analysis shows that when the input is zero mean and gaussian distributed and the adaptive filter is operating in a stationary environment the steady state excess mean squared error due to the coefficient noise vector is independent of the statistics of the input signal the results of several simulation experiments show that the filter performs well in a variety of situations the steady state behaviour predicted by the analysis is in very good agreement with the experimental results

kernel based approximation in sobolev spaces with radial basis functions in this paper we study several radial basis function approximation schemes in sobolev spaces we obtain an optional error estimate by using a class of smoothing operators we also discussed sufficient conditions for the smoothing operators to attain the desired approximation order we then construct the smoothing operators by some compactly supported radial kernels and use them to approximate sobolev space functions with optimal convergence order these kernels can be simply constructed and readily applied to practical problems the results show that the approximation power depends on the precision of the sampling instrument and the density of the available data

a topological proof of the equivariant dehn lemma an elementary topological proof is given for a completely general version of the equivariant dehn lemma in the spirit of the original proof of the nonequivariant version due to c d papakyriakopolous in 1957

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

least squares arma modeling of linear time varying systems lattice filter structures and fast rls algorithms the problem of modeling linear time varying systems is one of considerable importance in many varied and diverse applications this paper looks at ar ma and arma modeling of linear time varying systems with lattice filters there are two parts to this paper the first part develops the lattice filter structures in particular a new arma n m lattice filter structure is presented n ar order m ma order which is fully consistent with the geometric characteristics of the ar and ma lattice filter structures in that it evaluates all optimal arma i j filters of order lower than n m and each such filter is realized in terms of a fully orthogonal set of realization vectors for uncorrelated input this arma lattice filter simplifies greatly resulting in the white input arma lattice filter the second part of the paper develops new fast rls algorithms for the evaluation of the lattice filter coefficients there are two classes of algorithms presented one class is based on projection error sample averaging while the other uses input output sample averaging and some orthogonality relations this latter approach is equivalent to the exact rls formulation the algorithms of the former class are the most efficient modeling algorithms known to the authors and the algorithms of the latter class are of comparable or better efficiency than other algorithms in the same class

a method for interpolating on manifolds structural dynamics reduced order models a rigorous method for interpolating a set of parameterized linear structural dynamics reduced order models roms is presented by design this method does not operate on the underlying set of parameterized full order models hence it is amenable to an online real time implementation it is based on mapping appropriately the rom data onto a tangent space to the manifold of symmetric positive definite matrices interpolating the mapped data in this space and mapping back the result to the aforementioned manifold algorithms for computing the forward and backward mappings are offered for the case where the roms are derived from a general galerkin projection method and the case where they are constructed from modal reduction the proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input in all cases good accuracy is demonstrated at real time processing speeds copyright 2009 john wiley sons ltd

spherical trigonometry yule s parcor identity and frls fully normalized lattice yule s parcor identity is recognized as the cosine law of spherical trigonometry the six parcors propagated by the fully normalized frls lattice filter are the cosines of the six elements of a spherical triangle and this lattice algorithm is one solution to a spherical triangle problem that arises naturally in navigation and astronomy exploiting this new geometric interpretation yields unnoticed and potentially useful recursions among frls quantities

tracking improvements in fast rls algorithms using a variable forgetting factor the concept of a variable forgetting factor vff is incorporated into fast recursive least squares frls algorithms compromises in the data matrix that are needed to do this are examined both prewindowed and growing memory covariance algorithms are presented in transversal and lattice structures forgetting factor adaptation schemes which improve tracking performance over conventional frls algorithms are suggested finally the bias introduced by the use of the vff is analyzed

ar and arma identification algorithms of levinson type an innovations approach fast recursive in time identification procedures for both ar and arma processes e g chandrasekhar square root algorithms have been available for a few years these algorithms realize the desired transfer functions in the classical polynomial or rational form on the other hand the synthesis of polynomial and rational transfer functions in lattice and ladder form has fostered great interest in network theory by virtue of its pleasant properties this type of synthesis is strongly related to the theory of orthogonal polynomials on the unit circle an identification procedure with the realization of the desired whitening filter in lattice form was available for ar processes we give here a simple approach for obtaining such algorithms investigating furthermore the connections between the so obtained algorithms and the classical ones least squares procedure in the same way we obtain identification procedures with realization of the desired filter in lattice and ladder form for arma processes together with the connection with the classical extended least squares procedure the method is based upon a fairly general levinson orthogonalization lemma in hilbert space involving innovation techniques we extend the method to various other estimation problems the algorithms we obtain are fast even somewhat faster than the previous fast ones and seem to be well conditioned

a new unwindowed lattice filter for rls estimation adaptive lattice algorithms are derived for the solution of unwindowed least squares estimation problems for ar and fir models the basic approach is to embed the unwindowed problem in a larger prewindowed problem and then eliminate superfluous terms in the lattice initializations are given to allow the lattice to use no initial parameter estimates or to include initial parameter estimates with desired weightings in the quadratic criterion for parameter estimation a numerical example is given

a new recursive pseudo least squares algorithm for arma filtering and modeling ii this study is based on the observation that if the bootstrapping is combined with different parameterizations of the arma autoregressive moving average process then different linearized problems are obtained for the underlying nonlinear arma modeling problem in this part a specific parameterization termed the predictor space representation for an arma process which decouples the estimation for the ar and the ma parameters is used a vector space formalism for the given data case is then defined and the least squares arma filtering problem is interpreted in terms of projection operations on some linear spaces a new projection operator update formula which is particularly suited for the underlying problem is then used in conjunction with the vector space formalism to develop a computationally efficient pseudo least squares algorithm for arma filtering it is noted that these recursions can be put in the form of a filter structure

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

surrogate modeling of multiscale models using kernel methods this work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods the key element is to learn the interface between the different scales using a fast surrogate for the microscale model which is given by multivariate kernel expansions the expansions are computed using statistically representative samples of in and output of the microscale model as learning methods we apply both support vector machines and a vectorial kernel greedy algorithm we demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines first we consider a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and second a model for simulation of saturation overshoots in porous media involving nonclassical shock waves

inherent matrix identities on arma lattice filter realization algorithm and their application presents the matrix identities that are inherent in the solution of the normal equations for an arma lattice filter this derivation also makes clear the relationship between the recursive least squares rls method and the arma lattice filter realization algorithm further as an application of the matrix identities a new method for model identification with frequency weighting mifw is presented

on the identification of systems from data measurements using arma lattice models methods for the autoregressive moving average arma modeling of digital systems using a two channel autoregressive ar lattice are presented one method has the advantage that one half of the lattice parameters are zero when the system s excitation signal is white noise

crosscap numbers and the jones polynomial abstract we give sharp two sided linear bounds of the crosscap number non orientable genus of alternating links in terms of their jones polynomial our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings we also discuss generalizations of our results for classes of non alternating links

an unwindowed multichannel lattice filter with orthogonal channels an unwindowed or covariance multichannel lattice filter for recursive least squares estimation is derived the channels are orthogonalized to eliminate the need for matrix inversion the channel orthogonalization process leads to forward propagating and backward propagating blocks in both the lattice filter and the model parameter construction algorithm these blocks are particularly suitable for array processing as illustrated by arrays presented in the paper

pure time updates for least squares lattice algorithms this note presents some new time update formulas for certain types of lattice algorithms used in autoregressive modeling of stationary time series the new formulas enable the computation of the autoregressive coefficients in a number of operations per time step proportional to the model order

efficient recursive in order least squares fir filtering and prediction this paper is concerned with the efficient determination of the optimum in the least squares sense fir filter on the basis of data samples of the input and desired response signals by procedures recursive in the filter order this situation typically arises when no a priori statistics are available and the system order is not known the general multiinput multioutput multichannel case is considered here and a fast algorithm is presented requiring for single channel signals approximately 2s 15m multiplications mps per order m s being the number of samples in the special case of linear prediction it calls for about s 12m mps hence it offers a computational reduction of 5m and 2m mps in comparison to the methods of marple 1 and morf et al 2 respectively additionally the proposed scheme is inherently symmetric and is suited very well to initialization of fast sequential algorithms as well as algorithms searching for the optimum lag filter

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

recursive least squares ladder estimation algorithms recursive least squares ladder estimation algorithms have attracted much attention recently because of their excellent convergence behavior and fast parameter tracking capability compared to gradient based algorithms we present some recently developed square root normalized exact least squares ladder form algorithms that have fewer storage requirements and lower computational requirements than the unnormalized ones a hilbert space approach to the derivations of magnitude normalized signal and gain recursions is presented the normalized forms are expected to have even better numerical properties than the unnormalized versions other normalized forms such as joint process estimators e g adaptive line enhancer and arma pole zero models will also be presented applications of these algorithms to fast or zero startup equalizers adaptive noise and echo cancellers non gaussian event detectors and inverse models for control problems are also mentioned

square root covariance ladder algorithms square root normalized ladder algorithms provide an efficient recursive solution to the problem of multichannel autoregressive model fitting a simplified derivation of the general update formulas for such ladder forms is presented and is used to develop the growing memory and sliding memory covariance ladder algorithms new ladder form realizations for the identified models are presented leading to convenient methods for computing the model parameters from estimated reflection coefficients a complete solution to the problem of possible singularity in the ladder update equations is also presented

the number of reidemeister moves needed for unknotting there is a positive constant c1 such that for any diagram d representing the unknot there is a sequence of at most 2 c1n reidemeister moves that will convert it to a trivial knot diagram where n is the number of crossings in d a similar result holds for elementary moves on a polygonal knot k embedded in the 1 skeleton of the interior of a compact orientable triangulated p l 3 manifold m there is a positive constant c2 such that for each t 1 if m consists of t tetrahedra and k is unknotted then there is a sequence of at most 2 c 2t elementary moves in m which transforms k to a triangle contained inside one tetrahedron of m we obtain explicit values for c1 and c2

the global modal parameterization for non linear model order reduction in flexible multibody dynamics in flexible multibody dynamics advanced modelling methods lead to high order non linear differential algebraic equations daes the development of model reduction techniques is motivated by control design problems for which compact ordinary differential equations odes in closed form are desirable in a linear framework reduction techniques classically rely on a projection of the dynamics onto a linear subspace in flexible multibody dynamics we propose to project the dynamics onto a submanifold of the configuration space which allows to eliminate the non linear holonomic constraints and to preserve the lagrangian structure the construction of this submanifold follows from the definition of a global modal parameterization gmp the motion of the assembled mechanism is described in terms of rigid and flexible modes which are configuration dependent the numerical reduction procedure is presented and an approximation strategy is also implemented in order to build a closed form expression of the reduced model in the configuration space numerical and experimental results illustrate the relevance of this approach

bases for kernel based spaces since it is well known de marchi and schaback 2001 4 that standard bases of kernel translates are badly conditioned while the interpolation itself is not unstable in function space this paper surveys the choices of other bases all data dependent bases turn out to be defined via a factorization of the kernel matrix defined by these data and a discussion of various matrix factorizations e g cholesky qr svd provides a variety of different bases with different properties special attention is given to duality stability orthogonality adaptivity and computational efficiency the newton basis arising from a pivoted cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the native hilbert space of the kernel efficient adaptive algorithms for calculating the newton basis along the lines of orthogonal matching pursuit conclude the paper

the reduced model multiscale method r3m for the non linear homogenization of hyperelastic media at finite strains this paper presents a new multi scale method for the homogenization analysis of hyperelastic solids undergoing finite strains the key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method in order to alleviate the complexity of multiscale procedures which usually involve a large number of nonlinear nested problems to be solved the problem associated with the representative volume element rve is solved via a model reduction method proper orthogonal decomposition the reduced basis is obtained through pre computations on the rve the technique coined as reduced model multiscale method r3m allows reducing significantly the computation times as no large matrix needs to be inverted and as the convergence of both macro and micro problems is enhanced furthermore the r3m drastically reduces the size of the data base describing the history of the micro problems in order to validate the technique in the context of porous elastomers at finite strains a comparison between a full and a reduced multiscale analysis is performed through numerical examples involving different micro and macro structures as well as different nonlinear models neo hookean mooney rivlin it is shown that the r3m gives good agreement with the full simulations at lower computational and data storage requirements

an adaptive arma four line lattice filter for spectral estimation with frequency weighting a method for designing an adaptive four line lattice filter which can perform frequency weighting spectral estimation which provides more accurate spectral estimation for some frequency bands than for others is proposed using a suitable frequency weighting function denoted as an arma autoregressive moving average model an estimated spectrum is obtained by arbitrarily weighing some frequency bands more heavily than others if the frequency weighting function has the property of a low pass filter the spectrum of the reference model can be estimated accurately with a reduced arma order in the low frequency band spectra of time varying models can be estimated with an exponentially weighted sliding window and the input signal of the reference model can be estimated by assumption the order update and the time update recursive formulas and the frequency weighting method for the filter are described the algorithm is verified by experimental results

sliding window order recursive least squares algorithms order recursive least squares orls algorithms employing a sliding window sw are presented the authors demonstrate that standard architectures that are well known for growing memory orls estimation e g triangular array lattice and multichannel lattice also apply to sliding window orls estimation a specific sw orls algorithm is the combination of two independent attributes its global architecture and its local cell implementation various forms of local cell implementation based on efficient time recursions of time varying coefficients are discussed in particular the authors show that time and order updates of any order recursive sliding window least squares algorithm can be realized solely in terms of 3 spl times 3 hyperbolic householder transformations hht finally the authors present two hht based algorithms the hht triangular array algorithm and the hht lattice algorithm

the reduced basis element method application to a thermal fin problem the reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization the basic idea is to decompose the domain of computation into a series of subdomains the elements that are similar to a few reference domains these reference domains are actually filled with reduced basis functional spaces that are mapped to each subdomain together with the geometry the discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces finally a mortar type method is applied to glue the various local functions in this paper we focus on the definition of the reference shapes and together with theoretical and numerical justifications of the method we provide a posteriori error analysis tools that allow us to certify the computational results

an inverse micro mechanical analysis toward the stochastic homogenization of nonlinear random composites abstract an inverse mean field homogenization mfh process is developed to improve the computational efficiency of non linear stochastic multiscale analyzes by relying on a micro mechanics model first full field simulations of composite stochastic volume element sve realizations are performed to characterize the homogenized stochastic behavior the uncertainties observed in the non linear homogenized response which result from the uncertainties of their micro structures are then translated to an incremental secant mfh formulation by defining the mfh input parameters as random effective properties these effective input parameters which correspond to the micro structure geometrical information and to the material phases model parameters are identified by conducting an inverse analysis from the full field homogenized responses compared to the direct finite element analyzes on sves the resulting stochastic mfh process reduces not only the computational cost but also the order of uncertain parameters in the composite micro structures leading to a stochastic mean field reduced order model mf rom a data driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic mf rom embedded in a stochastic finite element method sfem the two cases of elastic unidirectional ud fibers embedded in an elasto plastic matrix and of elastic ud fibers embedded in a damage enhanced elasto plastic matrix are successively considered in order to illustrate the capabilities of the method the stochastic response of a ply is studied under transverse loading condition

fast algorithm of chandrasekhar type for arma model identification abstract this paper presents a new fast identification algorithm based on chandrasekhar type equations by appropriately defining extended state vectors and corresponding matrices a state space model is obtained from the arma representation so that the kalman filter can be used as a parameter estimator because the resulting system is time invariant it is possible to apply chandrasekhar factorization techniques to calculate the filter gain and thus produce a fast algorithm the sparse nature of the state space matrices greatly facilitates the initial factorization required to launch the recursive algorithm the new algorithm is very simple and efficient its computational complexity per iteration requires 14n multiplications n number of arma parameters consequently a substantial gain in computing time is obtained compared to most other algorithms partaicularly those of lattice type

a robust recursive least squares algorithm a new algorithm is developed which guarantees the normalized bias in the weight vector due to persistent and bounded data perturbations to be bounded robustness analysis for this algorithm has been presented an approximate recursive implementation is also proposed it is termed as the robust recursive least squares rrls algorithm since it resembles the rls algorithm in its structure and is robust with respect to persistent bounded data perturbation simulation results are presented to illustrate the efficacy of the rrls algorithm

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

a class of mixed transversal and ladder adaptive filters with pure order updates a class of exact recursive least squares rls adaptive filter structures with pure order updates called mixed transversal and ladder mital filters is presented in which the propagation of numerical errors along in time is limited to at most m sampling periods so the long term numerical stability can be guaranteed the adaptive structures can be treated either as transversal forms or as ladder forms to meet the requirements of different applications the outstanding numerical performances of the adaptive filters are illustrated by computer simulations

a fast arma transversal rls filter algorithm a fast pole zero arma transversal rls recursive least squares algorithm is derived using a geometric formulation and the concept of projection onto a vector subspace to derive a recursive solution the algorithm estimates a parameter vector that contains both numerator and denominator coefficients of an unknown system transfer function i e models an arma pole zero process the algorithm has a transversal filter structure but is distinguished from previous multichannel transversal algorithms wherein each input channel is constrained to have the same order here the pole and zero orders can be independently and arbitrarily specified the derivation of the algorithm uses permutation matrices similar to those in the arma fast kalman algorithm but achieves a significant reduction in computations when compared to that algorithm it is shown that when the pole and zero orders of the arma process are correctly specified the algorithm generates an extremely good estimate furthermore if the poles and zeros are overspecified it is shown that a spectral match is still achieved by mutual cancellation of superfluous poles and zeros

decreasing the temporal complexity for nonlinear implicit reduced order models by forecasting abstract implicit numerical integration of nonlinear odes requires solving a system of nonlinear algebraic equations at each time step each of these systems is often solved by a newton like method which incurs a sequence of linear system solves most model reduction techniques for nonlinear odes exploit knowledge of a system s spatial behavior to reduce the computational complexity of each linear system solve however the number of linear system solves for the reduced order simulation often remains roughly the same as that for the full order simulation we propose exploiting knowledge of the model s temporal behavior to 1 forecast the unknown variable of the reduced order system of nonlinear equations at future time steps and 2 use this forecast as an initial guess for the newton like solver during the reduced order model simulation to compute the forecast we propose using the gappy pod technique the goal is to generate an accurate initial guess so that the newton solver requires many fewer iterations to converge thereby decreasing the number of linear system solves in the reduced order model simulation

positive definite toeplitz embedding based on the cyclic extension of a data matrix embedding in a block toeplitz matrix is one of the techniques for efficient inversion of arbitrary covariance matrices arising in least squares estimation delosme and morf have shown how the concept of toeplitz distance leads to a positive definite embedding via a mixed form representation of the given matrix in this paper a new method is proposed for the positive definite block toeplitz embedding which has the particular feature of taking the signal itself as a starting point it consists of a suitable cyclic extension and embedding of the given signal in a multichannel stationary signal the main advantage of the method is that given the signal the embedding can be performed by inspection

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

one fixed point actions and homology 3 spheres one of the best known three dimensional manifolds is the poincare homology three sphere 1 2 3 5 this closed 3 manifold has the same homology groups as the standard sphere s3 but is not simply connected see pnc p 106 or br2 section 1 8 numerous investigations during the past century have shown that this manifold has many remarkable properties several equivalent descriptions of 1 2 3 5 are summarized in kis from the viewpoint of transformation groups one noteworthy property is that 1 2 3 5 is the only nonsimply connected homology sphere admitting a transitive action of a compact lie group brl the manifold 1 2 3 5 also figures importantly in regularity questions for group actions on spherelike manifolds linear or orthogonal actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions one distinguishing feature of linear actions is that their fixed point sets if nonempty are also spheres the well known results of p a smith cf br2 show that continuous p group actions on homology spheres all resemble linear actions in this respect i e the fixed point sets are always zp homology cohomology spheres on the other hand outside the family of p groups there are many exotic group actions perhaps the most elementary example of this type involves 1 2 3 5 if a is the subgroup of the rotation group so3 given by the symmetries of a regular icosahedron so that a is isomorphic to the alternating group a5 then the quotient space s03 a is the poincare homology 3 sphere and the induced action of a5 a has exactly one fixed point this example suggests that general actions of finite groups on homology spheres can be quite different from linear actions and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action

on the least squares tap adjustment algorithm in adaptive digital echo cancellers fast recursive algorithms for updating coefficients in digital echo cancellers can be derived from the well known method of least squares unless zero initial conditions are assumed the exact initialization of these algorithms is yet unknown for random data of more than twice the order of the filter the existence of a unique least squares solution is proven a constructive recursive procedure in time and order for computing the pseudoinverse solution for the initial steps is derived since the data matrix is composed of integers this technique facilitates the implementation of stable tap update algorithms

the computational complexity of knot and link problems we consider the problem of deciding whether a polygonal knot in 3 dimensional euclidean space is unknotted ie capable of being continuously deformed without self intersection so that it lies in a plane we show that this problem unknotting problem is in np we also consider the problem splitting problem of determining whether two or more such polygons can be split or continuously deformed without self intersection so that they occupy both sides of a plane without intersecting it we show that it also is in np finally we show that the problem of determining the genus of a polygonal knot a generalization of the problem of determining whether it is unknotted is in pspace we also give exponential worst case running time bounds for deterministic algorithms to solve each of these problems these algorithms are based on the use of normal surfaces and decision procedures due to w haken with recent extensions by w jaco and j l tollefson

normalized lattice algorithms for least squares fir system identification recently developed algorithms for least squares identification of autoregressive models are extended in this paper so as to facilitate least squares identification of finite impulse response models the algorithms belong to the class of square root normalized lattice algorithms hence they share the computational efficiency and good numerical behavior of the latter two versions are presented one for identifying time invariant models and the other for tracking time varying parameters new lattice form realizations of the identified fir models are given the general framework is then specialized to the important cases of prediction and smoothing

order recursive factorization of the pseudoinverse of a covariance matrix a numerically reliable algorithm is developed to recursively update the square root of a pseudoinverse matrix using the square root of a lower dimension pseudoinverse matrix the numerical computations are based on a generalized singular value decomposition which is used to do a canonical correlation analysis an operation count is given for sequential and parallel implementation of a partitioned order recursive algorithm these methods are useful for covariance analysis to determine the contributions due to various modeling errors in the design of a kalman filter

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

underdetermined order recursive least squares adaptive filtering the concept and algorithms underdetermined recursive least squares urls adaptive filtering is introduced in particular the urls algorithm is derived and shown to be a direct consequence of the principle of minimal disturbance by exploiting the hankel structure of the filter input matrix the fast transversal filter ftf version of the urls algorithm urls ftf is derived including sliding window and growing window types the computational complexity is reduced to o n o m where n is the adaptive filter length and m is the order of the urls algorithm in addition the efficient urls eurls algorithm which does not compute the filter coefficients explicitly thereby significantly reducing the computational load is presented some earlier adaptive algorithms such as the averaged lms filtered x lms and fast conjugate gradient are shown to be suboptimal approximations of the urls algorithm instrumental variable approximations are also discussed the urls algorithm has a whitening effect on the input signal which provides immunity to the eigenvalue spread of the input signal correlation matrix although the algorithm is sensitive to observation noise it has good tracking characteristics and tradeoffs can be found by tuning the step size the utility of the urls algorithms in its basic form and ftf realization depends heavily on the practical applicability of the mth order sliding window estimate of the covariance matrix and mth order ptf relations the feasibility of the urls family in practical applications is demonstrated in channel equalization and acoustic echo cancellation