a flow model based on polylinking system we introduce polylinking networks which is a flow model based on polylinking systems that generalizes the classical flow model of ford and fulkerson on acyclic networks and has applications in the context of wireless networks more precisely a flow model recently introduced by avestimehr et al proceedings of the allerton conference on communication control and computing 2007 used in the context of wireless information networks is a special case of the presented model we define a notion of source destination cut and derive a max flow min cut theorem additionally we present various properties of polylinking networks that can be seen as generalizations of properties for classical flows using submodular function minimization and submodular flow algorithms one can efficiently determine a maximum flow a minimum source destination cut as well as a minimum cost flow these algorithms lead to new efficient algorithms for the information flow model

multiple subtraction using statistically estimated inverse wavelets surface related multiple elimination srme typically consists of two steps the first step is prediction and the second step is subtraction in subtraction it is important to effectively attenuate multiple events and preserve primary events when multiples cross with or overlap on primaries least square subtraction usually cannot subtract multiples effectively and may also damage the primaries when multiples overlap with primaries least square subtraction cannot always subtract multiples accurately and often damages the primaries to remedy this problem we propose to statistically estimate the inverse source wavelet correct for errors in the estimate of the inverse wavelet and then use the corrected inverse wavelets for multiple subtraction synthetic tests and real data examples show that the proposed method can effectively attenuate multiples while they also preserve the continuity of reflection events and successfully avoid amplitude distortion the proposed method is characterized by low computational costs and ease of implementation

multidimensional reflected brownian motions having exponential stationary distributions on considere la distribution stationnaire d un mouvement brownien reflechi dans un domaine d dimensionnel g on etudie deux classes de processus de ce type

lie algebras graded by finite root systems and intersection matrix algebras this paper classifies the lie algebras graded by doubly laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized cartan matrices of types b c f and g this completes the determination of the lie algebras graded by finite root systems initiated by berman and moody who studied the simply laced finite root systems of rank 2

least controls for a class of constrained linear stochastic systems this paper introduces a class of constrained linear stochastic systems the main objective is to investigate whether there exists a least control in a sense that it exerts least effort in controlling the system to within a predetermined region our approach of finding a least control is a to characterize a class of functional polyhedral sets of functions which have least elements and b to construct a least control process through these least elements existence of a least control is established and the connection between a least control and a solution of the dynamic complementarity problem dcp is also discussed

a note on computing the inverse of a triangular toeplitz matrix abstract using trigonometric polynomial interpolation a fast and effective numerical algorithm for computing the inverse of a triangular toeplitz matrix with real numbers has been recently proposed lin et al 2004 7 the complexity of the algorithm is two fast fourier transforms ffts and one fast cosine transform dct of 2 n vectors in this paper we present an algorithm with two fast fourier transforms ffts of 2 n vectors for calculating the inverse of a triangular toeplitz matrix with real and or complex numbers a theoretical accuracy and error analysis is also considered numerical examples are given to illustrate the effectiveness of our method

fixed order stabilising controller design by a mixed randomised deterministic method a novel mixed randomised deterministic method for fixed order controller design is proposed for discrete time single input single output plants it is based on the random generation of schur stable polynomials using reflection coefficients and reflection segments of polynomials stable reflection segments are then projected onto the affine set of closed loop characteristic polynomials and the stable line segments in the controller parameter space are determined a novel approach is proposed for global and local optimisation over reflection segment bunches on the basis of the weighted sum of absolute values of reflection coefficients

error sources and error propagation in the levinson durbin algorithm it is proved that there are two types of numerical error due to finite precision in the levinson durbin algorithms an erratic and a systematic one the erratic one depends on the value the input autocorrelation accidentally takes at an iteration and essentially it affects only the results obtained at this particular recursion on the contrary the systematic numerical error increases with the information the system carries and propagates essentially throughout the algorithm it is shown that for both types of error as well as the overall one there are specific intermediate quantities calculated in the evolution of the algorithm which may serve as precise indicators of the exact number of erroneous digits with which the various quantities are computed including the parcor coefficients and the filter coefficients therefore the generated numerical error can be accurately traced

trigonometric transform splitting methods for real symmetric toeplitz systems abstract in this paper we study efficient iterative methods for real symmetric toeplitz systems based on the trigonometric transformation splitting tts of the real symmetric toeplitz matrix a theoretical analyses show that if the generating function f of the n n toeplitz matrix a is a real positive even function then the tts iterative methods converge to the unique solution of the linear system of equations for sufficient large n moreover we derive an upper bound of the contraction factor of the tts iteration which is dependent solely on the spectra of the two tts matrices involved different from the cscs iterative method in ng 2003 in which all operations counts concern complex operations when the dfts are employed even if the toeplitz matrix a is real and symmetric our method only involves real arithmetics when the dcts and dsts are used the numerical experiments show that our method works better than cscs iterative method and much better than the positive definite and skew symmetric splitting pss iterative method in bai et al 2005 and the symmetric gauss seidel sgs iterative method

fast algorithms for generalized displacement structures and lossless systems abstract we derive an efficient recursive procedure for the triangular factorization of strongly regular matrices with generalized displacement structure that includes as special cases a variety of previously studied classes such as toeplitz like and hankel like matrices the derivation is based on combining a simple gaussian elimination procedure with displacement structure and leads to a transmission like interpretation in terms of two cascades of first order sections we further derive state space realizations for each section and for the entire cascades and show that these realizations satisfy a generalized embedding result and a generalized notion of j losslessness the cascades turn out to have intrinsic blocking properties which can be shown to be equivalent to interpolation constrains

the quantum group structure associated with non linearly extended virasoro algebras recently an infinite family of chiral virasoro vertex operators whose exchange algebra is given by the universalr matrix ofsl 2 q has been constructed in the present paper the case of non linearly w extended virasoro symmetries related to the algebrasan n 1 is considered along the same line contrary to the previous case a1 the standardr matrix ofsl n 1 q does not come out and a different solution of yang and baxter s equations is derived the associated quantum group structure is displayed

fast inversion of triangular toeplitz matrices in this paper we present an approximate inversion method for triangular toeplitz matrices based on trigonometric polynomial interpolation to obtain an approximate inverse of high accuracy for a triangular toeplitz matrix of size n our algorithm requires two fast fourier transforms ffts and one fast cosine transform of 2n vectors we then revise the approximate method proposed by bini siam j comput 13 1984 268 the complexity of the revised bini algorithm is two ffts of 2n vectors

closed form analytical results for the rejection of narrow band interference in pn spread spectrum systems part ii linear interpolation filters the performance of pn spread spectrum communication systems in the presence of narrow band interference is studied when linear interpolation filters are employed for the estimation and subsequent suppression of the interference closed form analytical expressions for the system s performance are established for a broad class of interference processes the advantages of linear interpolation filters over predictive filters with identical number of taps are examined analytically and some unexpected results are obtained the analytical results are illustrated by examples

a constructive proof of the gohberg semencul formula abstract gohberg and semencul gave some elegant formulas for the inverse of a toeplitz matrix as a difference of products of lower and upper triangular toeplitz matrices there are several algebraic and analytic proof of these formulas here we give a constructive proof for two of the gohberg semencul formulas under the assumption that the matrices are strongly nonsingular i e all leading minors are nonzero this assumption is stronger than necessary but it enables fast o n 2 constructions for the entries in the gohberg semencul formulas our method also gives a new proof of the relation between the reflection coefficients of a toeplitz matrix and its inverse

efficient inversion of toeplitz block toeplitz matrix an iterative algorithm for the inversion of a toeplitz block toeplitz matrix consisting of m m blocks of size p p is described the algorithm presented exploits the structure of the toeplitz block toeplitz matrix and outperforms akaike s algorithm by a factor of max 2 p m 2 the use of this algorithm for an iterative solution of a toeplitz block toeplitz set of linear equations is also presented

toeplitz properties of the block matrices encountered in the processing of spatiotemporal signals various signal processing applications such as system identification fir finite impulse response wiener filtering linear prediction and spectral estimation are known to lead to the solution of toeplitz systems for the case of one dimensional signals and to toeplitz block toeplitz systems for the case of two dimensional signals the authors investigate such systems for the case of l dimensional signals with l 2 and emphasize their block and persymmetrical nature such important characteristics allow the use of fast algorithms for signals of any dimension the case of three dimensional signals is treated in detail not only because it is of special interest for video applications but also because the reasoning used for this case can be straightforwardly applied to higher dimension signals

fast conversion algorithms for orthogonal polynomials we discuss efficient conversion algorithms for orthogonal polynomials we describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis and deduce a new algorithm of the same complexity for the converse operation

levinson type algorithms for polynomial fitting and for cholesky and q factors of hankel and vandermonde matrices this paper presents levinson 1947 type algorithms for i polynomial fitting ii obtaining a q decomposition of vandermonde matrices and a cholesky factorization of hankel matrices iii obtaining the inverse of hankel matrices the algorithm for the least squares solution of hankel systems of equations requires 3n sup 2 9n 3 multiply and divide operation mdo the algorithm for obtaining an orthogonal representation of an m spl times n vandermonde matrix x and computing the cholesky factors f of hankel matrices requires 5mn n sup 2 2n 3m mdo and the algorithm for generating the inverse of hankel matrices requires 3 n sup 2 n 2 2 mdo our algorithms have been tested by means of fitting of polynomials of various orders and fortran versions of all subroutines are provided in the appendix

the deconvolution problem fast algorithms including the preconditioned conjugate gradient to compute a map estimator different aspects of the deconvolution problem are discussed and a solution by the map estimator using the a priori knowledge of smoothness and nonnegativity of the unknown signal is proposed several fast algorithms are examined and the conjugate gradient cg method is show to have several advantages it allows computation of a constrained solution and when used with a preconditioning technique it can be faster than other algorithms for general near to toeplitz systems in particular for toeplitz block toeplitz systems

a fast algorithm for solving a toeplitz system of equations a fast algorithm for the solution of a toeplitz system of equations is presented the algorithm requires order n log n 2 computations where n is the number of equations for banded toeplitz matrices the order of computations is reduced to only n log n m log m 2 where 2m is the maximum number of nonzero principal subdiagonals of the toeplitz matrix the algorithm is in essence a fast implementation of the trench algorithm in reverse thus the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast hd half divisor algorithm to compute the first row of the inverse matrix the desired solution is then obtained directly from the first row by applying fast fourier transform techniques in order n log n computations finally the extension of the algorithm to block toeplitz matrices is also presented

fast algorithms for close to toeplitz plus hankel systems and two sided linear prediction the low displacement rank definition of close to toeplitz ct matrices is extended to close to toeplitz plus hankel ctph matrices fast algorithms for solving ctph systems of equations are presented a matrix is defined as ctph if the sum of a ct matrix and a second ct matrix postmultiplied by an exchange matrix an equivalent definition in terms of uv rank is also given this definition is motivated by the application of the algorithms to two sided linear prediction tsp autocorrelation and covariance forms of tsp analogous to those for one sided linear prediction osp are defined the covariance form of tsp is solved using the ctph fast algorithms just as the covariance form of osp is solved using ct fast algorithms numerical examples show that tsp produces smaller residuals than osp and resolves sharp spectral peaks better than osp and that covariance tsp produces smaller residuals than autocorrelation tsp

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

a generalization of the levinson algorithm for hermitian toeplitz matrices with any rank profile the paper describes a recursive algorithm for solving hermitian toeplitz systems of linear equations without any restriction on the ranks of their nested toeplitz subsystems such a general algorithm is needed e g to obtain the eigenfilters for signal processing applications or to compute the inverse of a nondefinite toeplitz matrix the regular portion of the algorithm is made of the classical levinson recursion the singular portion requires solving some well defined systems of linear equations with gradient structure the dimension of each of these sytems equals the amplitude of the corresponding singularity

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

classification of quantum groups and lie bialgebra structures on sl n f relations with brauer group given an arbitrary field f of characteristic 0 we study lie bialgebra structures on sl n f based on the description of the corresponding classical double for any lie bialgebra structure 5 the classical double d sl n f delta is isomorphic to sl n f circle times f a where a is either f epsilon with epsilon 2 0 or f circle plus f or a quadratic field extension of f in the first case the classification leads to quasi frobenius lie subalgebras of sl n f in the second and third cases a belavin drinfeld cohomology can be introduced which enables one to classify lie bialgebras on sl n f up to gauge equivalence the belavin drinfeld untwisted and twisted cohomology sets associated to an r matrix are computed for the cremmer gervais r matrix in sl 3 we also construct a natural map of sets between the total belavin drinfeld twisted cohomology set and the brauer group of the field f

boundary element solution for periodic acoustic problems abstract this work shows when using the boundary element method to solve 3d acoustic scattering problems from periodic structures the coefficient matrix can be represented as a block toeplitz matrix by exploiting the toeplitz structure the computational time and storage requirements to construct the coefficient matrix are significantly reduced to solve the linear system of equations the original matrix is embedded into a larger and more structured matrix called the block circulant matrix discrete fourier transform is then employed in an iterative algorithm to solve the block toeplitz system to demonstrate the effectiveness of the formulation for periodic acoustic problems two exterior acoustic case studies are considered the first case study examines a continuous structure to predict the noise generated by a sharp edged flat plate under quadrupole excitation directivity plots obtained using the periodic boundary element method technique are compared with numerical results obtained using a conventional boundary element model the second case study examines a discrete periodic structure to predict the acoustic performance of a sonic crystal noise barrier results for the barrier insertion loss are compared with both finite element results and available data in the literature

adaptive tracking of multiple sinusoids whose power levels are widely separated the behavior of the gradient transversal filter lms the gradient lattice gl and the least squares lattice lsl when used to track multiple sinusoidal or narrow band components whose power levels are widely separated is investigated these approaches to the realization of the pth order one step linear predictor of the time series are recursive in time the lags of the instantaneous frequency estimates from their actual underlying values are of particular interest the frequency tracking characteristics of the lms gl and lsl algorithms are illustrated in several situations included are simulations of dual sinusoids undergoing a variety of frequency versus time dynamics and a formant tracking example of real speech

szego polynomials applied to frequency analysis abstract this paper is concerned with the problem of determining unknown frequencies 1 i using an observed discrete time signal xn xn m arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well defined frequencies j periods p j 2 n j n z j 1 i we investigate the wiener levinson method formulated here in terms of szeg polynomials n n z orthogonal on the unit circle with respect to a distribution function n defined by the n observed values of the signal we prove that if n0 denotes the number of critical points eiwj then for every n n0 and n 1 the zeros z j n n of n n z can be arranged so that limn z j n n eiwj for each of the frequencies wj this result confirms one of the main parts of the conjecture given by jones njastad and saff this journal 1990 on the convergence of zeros of the szeg polynomials a related result on the convergence of corresponding two point pade approximants is also given

a combinatorial graph based solution method for a class of continuous time optimal control problems the paper addresses a class of continuous time optimal control problems whose solutions are typically characterized by both bang bang and singular control regimes analytical study and numerical computation of such solutions are very difficult and far from complete when only techniques from control theory are used this paper solves optimal control problems by reducing them to the combinatorial search for the shortest path in a specially constructed graph since the nodes of the graph are weighted in a sequence dependent manner we extend the classical shortest path algorithm to our case the proposed solution method is currently limited to single state problems with multiple control functions a production planning problem and a train operation problem are optimally solved to illustrate the method

fir filters and recursive forms for continuous time invariant state space models an fir finite impulse response filter and an fir smoother are introduced for continuous time invariant state space models it is shown in this note that finite impulse responses of the fir filter and smoother can be easily determined by solving a simple riccati type matrix differential equation on a finite interval especially for systems with stationary processes finite impulse responses of the fir filter and smoother become time invariant and can be computed from simpler equations for fast computational purposes recursive forms of the fir filter and smoother are derived by using adjoint variables in this case all gains for recursive forms are shown to be constant

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

nonparametric input estimation in physiological systems problems methods and case studies input estimation from output data is an important problem in the analysis of physiological systems because many signals of interest are not directly accessible to measurement when the system is time invariant this problem is often referred to as deconvolution three representative physiological problems regarding hormone secretion insulin dynamics and hepatic glucose production are used to illustrate the major challenges ill conditioning confidence intervals assessment infrequent and nonuniform sampling nonnegativity constraints and computational efficiency the paper provides a critical overview of the existing techniques focusing on regularization theory and bayesian estimation in order to overcome some inadequacies of the existing methods some new results are derived in particular the connection between the maximum likelihood estimate of the regularization parameter and the notion of equivalent degree of freedom is studied moreover a fast svd based numerical algorithm is developed that includes the optimization of the regularization parameter and the computation of confidence intervals the proposed techniques are validated on a benchmark problem and are shown to provide effective solutions to the three physiological case studies

reflected brownian motion with skew symmetric data in a polyhedral domain this paper is concerned with the characterization and invariant measures of certain reflected brownian motions rbm s in polyhedral domains the kind of rbm studied here behaves like d dimensional brownian motion with constant drift in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in harrison williams 9 it is shown that such an rbm can be characterized in terms of a family of submartingales and that it reaches non smooth parts of the boundary with probability zero in 9 a purely analytic problem associated with such an rbm was solved here the exponential form solution obtained in 9 is shown to be the density of an invariant measure for the rbm furthermore if the density is integrable over the polyhedral state space then it yields the unique stationary distribution for the rbm in the proofs of these results a key role is played by a dual process for the rbm and by results in 9 for reflected brownian motions on smooth approximating domains

on the family of ml spectral estimates for mixed spectrum identification a recently developed point spectrum identification procedure based on a family of ar and ml spectral estimates is exploited to arrive at a mixed spectrum identification procedure to this end a variety of properties of the ar and ml estimates as a function of model order are described these properties relate to amplitude convergence resolution and a characterization of the ar spectral artifact which is used to arrive at improved continuous spectral estimates a variety of examples are presented

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

euclid algorithm orthogonal polynomials and generalized routh hurwitz algorithm this paper is concerned with the relations between the euclid algorithm the theory of orthogonal polynomials and the problem of locating the zeros of a complex polynomial with respect to the imaginary axis in particular a simple generalized routh hurwitz algorithm is proposed which allows one to determine in any situation the numbers of zeros of an arbitrary complex polynomial in the right half plane on the imaginary axis and hence in the left half plane moreover it turns out that this algorithm yields as a side result a well defined factorization of the considered polynomial furthermore as a straightforward consequence of the adopted approach two presumably original algorithms are put into light which involve linear arithmetic operations only a polynomial nonnegativity test on the real axis and a characterization of positive real functions

order selection for and design of iir filters the design of infinite impulse response digital filters can be viewed as a spectral factorization problem when one knows what the numerator and denominator orders are this design problem is greatly simplified selection of the appropriate orders has long been a significant problem in recursive filter design recent work by gray et al suggests a general method by which the orders may be selected prior to design in this paper infinite impulse response digital filters are designed from finite impulse response prototypes using the gray technique to first select the order once the order has been determined the recursive filter coefficients are determined directly from the finite impulse filter because of the wealth of knowledge available concerning order lengths for nonrecursive filters the procedures used here permit direct comparison between the two filter methods

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

superfast tikhonov regularization of toeplitz systems toeplitz structured linear systems arise often in practical engineering problems correspondingly a number of algorithms have been developed that exploit toeplitz structure to gain computational efficiency when solving these systems the earliest fast algorithms for toeplitz systems required o n 2 operations while more recent superfast algorithms reduce the cost to o n log n 2 or below in this work we present a superfast algorithm for tikhonov regularization of toeplitz systems using an extension and transformation technique our algorithm translates a tikhonov regularized toeplitz system into a type of specialized polynomial problem known as tangential interpolation under this formulation we can compute the solution in only o n log n 2 operations we use numerical simulations to demonstrate our algorithm s complexity and verify that it returns stable solutions

a hermitian toeplitz matrix is unitarily similar to a real toeplitz plus hankel matrix two unitary matrices are presented that transform a hermitian toeplitz matrix into a real toeplitz plus hankel matrix and vice versa additional properties and consequences of these unitary transformations are also presented for a certain class of n dimensional toeplitz plus hankel systems of equations an efficient method of solution is presented an example is given to illustrate the similarity transforms toeplitz plus hankel systems arise in a wide variety of applications such as linear filtering theory discrete inverse scattering and discretization of certain integral equations arising in mathematical physics

the density of occupied nearest neighbor pairs on a rectangular 2 n lattice space utilizing a 15 term recursion that describes exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space 11 the expectation of the normalized number of occupied nearest neighbor pairs is calculated as a function of coverage and the reduced interaction potential

a fast direct method for block triangular toeplitz like with tri diagonal block systems from time fractional partial differential equations in this paper we study the block lower triangular toeplitz like with tri diagonal blocks system which arises from the time fractional partial differential equation existing fast numerical solver e g fast approximate inversion method cannot handle such linear system as the main diagonal blocks are different the main contribution of this paper is to propose a fast direct method for solving this linear system and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system our idea is based on the divide and conquer strategy and together with the fast fourier transforms for calculating toeplitz matrix vector multiplication the complexity needs o m n log 2 m arithmetic operations where m is the number of blocks the number of time steps in the system and n is the size number of spatial grid points of each block numerical examples from the finite difference discretization of time fractional partial differential equations are also given to demonstrate the efficiency of the proposed method

transverse nearest neighbor degeneracies on a 2 n lattice an exact 15 term recursion relation and the associated generating function are derived for the number of arrangements of q particles on a 2 n lattice such that s occupied nearest neighbor pairs v of which are transverse and t unoccupied nearest neighbor pairs are formed

classification of quantum groups via galois cohomology the first example of a quantum group was introduced by p kulish and n reshetikhin in the paper kulish et al j soviet math 23 2435 2441 1983 they found a new algebra which was later called uq sl 2 their example was developed independently by v drinfeld and m jimbo which resulted in the general notion of quantum group later a complimentary approach to quantum groups was developed by l faddeev n reshetikhin and l takhtajan in faddeev et al leningr math j 1 193 225 1990 recently the so called belavin drinfeld cohomology twisted and non twisted have been introduced in the literature to study and classify certain families of quantum groups and lie bialgebras later the last two authors interpreted non twisted belavin drinfeld cohomology in terms of non abelian galois cohomology h1 f h for a suitable algebraic f group h here f is an arbitrary field of zero characteristic the non twisted case is thus fully understood in terms of galois cohomology the twisted case has only been studied using galois cohomology for the so called standard drinfeld jimbo structure the aim of the present paper is to extend these results to all twisted belavin drinfeld cohomology and thus to present classification of quantum groups in terms of galois cohomology and the so called orders low dimensional cases sl 2 and sl 3 are considered in more details using a theory of cubic rings developed by b n delone and d k faddeev in delone and faddeev the theory of irrationalities of the third degree translations of mathematical monographs vol 10 ams providence 1964 our results show that there exist yet unknown quantum groups for lie algebras of the types an d2n 1 e6 not mentioned in etingof et al j am math soc 13 595 609 2000

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

fast inversion of triangular toeplitz matrices an algorithm requiring 0 n log n operations is proposed for the inversion of triangular toeplitz matrices this algorithm is faster than existing techniques even for relatively small values of n and can easily be written in fortran

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

best stable and invertible approximations for arma systems a method is proposed for finding the best stable and invertible approximations for an autoregressive moving average arma system relative to a general quadratic metric in the coefficient space mathematically the problem is equivalent to projecting the regression and moving average vectors of the system onto the set s of coefficients of monic schur polynomials the geometry of s is too complex to allow the problem to be approached directly in the arma coefficient space a solution is obtained by constrained steepest descent in the hypercube of reflection coefficients which is homomorphic to s

the weak and strong stability of algorithms in numerical linear algebra the stability of algorithms in numerical linear algebra is discussed the concept of stability is extended to notions of weak stability and strong stability justifications are given for these extensions and the implications of error analyses in terms of these definitions are discussed the concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for toeplitz systems

computing the polynomial remainder sequence via bezout matrices in this paper we address the task of computing the polynomial remainder sequence appearing in the euclidean algorithm applied to two polynomials u x and v x of degree n and m respectively m n via renewed approach based on the approximate block diagonalization of a b ezout matrix b u v more specifically this algorithm consists on using the e cient schur complementation method given in terms of a bmatrix b u v associated with the input polynomials we also compare the proposed approach to the approximate block diagonalization of a hankel matrix h u v in which the successive transformation matrices are upper triangular toeplitz matrices all algorithms have been implemented in matlab numerical experiments performed with a wide variety of test problems show the e ectiveness of this algorithm in terms of e cience stability and robustness be two polynomials with deg u x n deg v x m and m n the classical euclidean algorithm applied to u x and v x returns a sequence of quotients qk x and remainders rk x such that r 1 x u x r0 x v x rk 2 x rk 1 x qk x rk x k 1 k 2 where rk x is the polynomial remainder of the division of rk 2 x and rk 1 x and rk x is the greatest common divisor gcd of u x and v x the question of computing the quotients and the remainders is beginning to be studied by collins and brown see 11 9 10 via e cient euclidean based methods naturally the classical algorithm turns out to be unstable when applied to approximate polynomials one might think of designing stabilized numerical versions of the euclidean algorithm to the approximate case see 25 23 several papers have been devoted to describe the natural relation between the euclidean algorithm and the block lu factorization of the hankel and b

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

comparing ls fir filtering and l step ahead linear prediction this comparative study of the l step ahead linear prediction and least squares finite impulse response ls fir filtering problems emphasizes the numerical behavior of the resulting toeplitz systems it is shown that although these systems are similar the restraints on the autocorrelation coefficients fundamentally differentiate them in the process of doing so a new algorithmic scheme for the computation of the lagged lattice coefficients is developed which exhibits fundamentally improved numerical behavior moreover explicit formulas for the supremums of the absolute values of both the lagged lattice and filter coefficients are found theoretically and are experimentally confirmed by using the proposed algorithm finally the bounds of the ls fir filter coefficients are treated in comparison with the supremums of the lagged quantities

a new algorithm for computing orthogonal polynomials the aim of this paper is to present an algorithm for computing orthogonal polynomials the fast fourier transform completes the connection given by draux between the three term recurrence relationship and the euclidian algorithm applications to hankel systems and numerical examples illustrate our purpose

maximally robust 2 d channel estimation for ofdm systems two dimensional minimum mean square error mmse channel estimation for orthogonal frequency division multiplexing ofdm systems is known to perform better than the least squares as well as the conventional 1 d mmse estimation owing to its ability of exploiting both the time and the frequency correlations among the channel frequency response cfr coefficients its superior performance comes however at the price of increased requirements the knowledge of observation noise power and that of the channel frequency as well as time correlation functions in practical transmission scenarios where channel correlation functions are not known or cannot be easily estimated it is desirable to have an estimator that is robust to mismatches between the assumed and the actual channel correlation function while such a robust estimator for the case of an infinite number of observations is well known for various uncertainty classes not much attention has been paid to the practical case of a finite number of observations we derive in this paper the maximally robust mr 2 d channel estimator for the case of a finite number of pilot observations under some realistic constraints on the uncertainty class to which the 2 d channel correlation sequence belongs we demonstrate that the correlation sequence associated with the mr estimator can be obtained by a simple semi definite optimization procedure and is interestingly different from the well known heuristic proposals simulation results establish the superiority of the proposed mr estimator over commonly employed heuristic robust estimator by as much as 3 db in terms of the worst case estimation mse and around 1 db in terms of the average ber performance under different practical transmission scenarios of interest

lie bialgebras fields of cohomological dimension at most 2 and hilbert s seventeenth problem we investigate lie bialgebra structures on simple lie algebras of non split type a it turns out that there are several classes of such lie bialgebra structures and it is possible to classify some of them the classification is obtained using belavin drinfeld cohomology sets which are introduced in the paper our description is particularly detailed over fields of cohomological dimension at most two and is related to quaternion algebras and the brauer group we then extend the results to certain rational function fields over real closed fields via pfister s theory of quadratic forms and his solution to hilbert s seventeenth problem

efficient construction of canonical ladder forms for vector autoregressive processes the paper treats the problem of constructing canonical ladder realizations for vector autoregressive ar processes specified by their characteristic matrix polynomials the difficulty of this problem is rooted in the fact that the backward matrix polynomial corresponding to a given vector ar process is a nontrivial function of the forward matrix polynomial the construction calls for solving a discrete lyapunov equation in block controller form two efficient procedures for solving this equation are presented both requiring a number of operations that is proportional to at most the square of the model order applications of the new procedures to stability tests simulation of ar processes and model reduction are described

discrete flow networks bottleneck analysis and fluid approximations we conduct bottleneck analysis of a deterministic dynamic discrete flow network the analysis presupposes only the existence of long run averages and is based on a continuous fluid approximation to the network in terms of these averages the results provide functional strong laws of large numbers for stochastic jackson queueing networks since they apply to their sample paths with probability one

generalized chandrasekhar recursions from the generalized schur algorithm presents a new approach to the chandrasekhar recursions and some generalizations thereof the derivation uses the generalized schur recursions which are o n sup 2 recursions for the triangular factorization of n spl times n matrices having a certain toeplitz like displacement structure it is shown that when the extra structure provided by an underlying state space model is properly incorporated into the generalized schur algorithm it reduces to the chandrasekhar recursions which are o nn sup 2 recursions for estimating the n dimensional state of a time invariant or constant parameter system from n measured outputs it is further noted that the generalized schur algorithm factors more general structured matrices and this fact is readily used to extend the chandrasekhar recursions to a class of time variant state space models special cases of which often arise in adaptive filtering

superfast solution of linear convolutional volterra equations using qtt approximation we address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular toeplitz matrices and the recently proposed qtt format the inverses of such matrices can be computed by the divide and conquer and modified bini s algorithms for which we present the versions with the qtt approximation we also present an efficient formula for the shift of vectors given in qtt format which is used in the divide and conquer algorithm as a result we reduce the complexity of inversion from the fast fourier level o n log n to the speed of superfast fourier transform i e o log 2 n the results of the paper are illustrated by numerical examples

complements to cost equivalent realizations abstract skelton 1981 and skelton and yousuff 1983 use component cost analysis to obtain a reduced order model that is coat equivalent to a full order stochastic system this paper presents a more direct and numerically efficient method to obtain the same reduced order model which is shown to be a first order autoregressive model

direct multichannel predictive deconvolution the levinson principle generally can be used to compute recursively the solution of linear equations it can also be used to update the error terms directly this is used to do single channel deconvolution directly on seismic data without computing or applying a digital filter multichannel predictive deconvolution is used for seismic multiple attenuation in a standard procedure the prediction error filter matrices are computed with a levinson recursive algorithm using a covariance matrix of the input data the filtered output is the prediction errors or the nonpredictable part of the data starting with the classical levinson recursion wehave derived new algorithms for direct recursive calculationof the prediction errors without computing the data covariance matrix or computing the prediction error filters one algorithm generates recursively the one step forward and backward predic tion errors and the l step forward prediction error computing only the filter matrices with the highest index a numeri

on low complexity approximation of matrices the operation multiplication of a vector by a matrix can be represented by a computational scheme or model that acts on the entries of the vector sequentially the number of intermediate quantities states that are needed in the computations is a measure of the complexity of the model if that complexity is low then not only multiplication but also other operations such as inversion can be carried out efficiently using the model rather than the original matrix in the introductory sections we describe an algorithm to derive a computational model of minimal complexity that gives an exact representation of an arbitrary upper triangular matrix the main result of the paper is an algorithm for computing an approximating matrix with a model of much lower complexity than the original as low as possible for a given tolerance on the approximation error as measure for the tolerance we will use a strong norm which we will call the hankel norm it is a generalization of the hankel norm which is used in the classical model approximation theory for complex analytical functions

maximum entropy and spectral estimation a review the method of maximum entropy is reviewed with emphasis on its relationship to entropy rate wiener filters autoregressive processes extrapolation the levinson algorithm lattice all pole and all pass filters and stability

olriv a new fast adaptive algorithm for rectangular block toeplitz systems in this paper we propose a new algorithm for solving overdetermined systems when they are rectangular block toeplitz the blocks can have more rows than columns it is based on the expression of the matrix of the system to solve as the intercorrelation between two vectorial processes namely the original and the instrumental processes the instrumental process has generally more components than the original one therefore it takes the overdetermined character of the system into consideration the proposed algorithm known as the overdetermined lattice recursive instrumental variable olriv belongs to the fast rls family and lies on a double lattice structure where one lattice performs the prediction of the original process and the other one the prediction of the instrumental process the geometric complete derivation of the proposed algorithm is given in the paper afterwards we show how olriv can be applied to perform blind adaptive identification of ar channels using high order statistics where the systems to solve are often overdetermined to ensure identifiability both the scalar and the vectorial cases are investigated simulation results are finally given to show the performances of olriv

a heavy traffic limit theorem for networks of queues with multiple customer types the central result of this paper is a heavy traffic limit theorem for the vector of total station workloads in an open network of queues with multiple customer types under first come first served and priority disciplines the limit process is a regulated brownian motion on the nonnegative orthant with parameters specified from the first two moments of the interarrival and service time distributions and a matrix of reduced routing information through the phenomenon of state space collapse associated limit results for queue length workload and sojourn time processes by customer type are obtained jointly as simple transformations of the total workload limit process diffusion approximations based on the theorem are discussed

a fast but unstable orthogonal triangularization technique for toeplitz matrices abstract d sweet s clever qr decomposition algorithm for toeplitz matrices is considered it requires only o n 2 flops to factor an n n matrix we analyze the algorithm to point out its deficiencies and compare it against two other toeplitz solvers

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

numerical solution of the eigenvalue problem for efficiently structured hermitian matrices abstract an iterative procedure is proposed for computing the eigenvalues and eigenvectors of a class of specially structured hermitian toeplitz matrices which includes hermitian toeplitz and toeplitz plus hankel matrices the computational cost per eigenvalue eigenvector for a matrix of order n is o n 2 in serial mode results of numerical experiments on kac murdock szego matrices and randomly generated real symmetric toeplitz matrices and toeplitz plus hankel matrices of orders as high as 2000 are included

causal compensation for erasures in frame representations in a variety of signal processing and communications contexts erasures occur inadvertently or can be intentionally introduced as part of a data reduction strategy this paper discusses causal compensation for erasures in frame representations of signals the approach described assumes linear synthesis of the signal using a prespecified frame but no specific generation mechanism for the coefficients under this assumption it is demonstrated that erasures can be compensated for using low complexity causal systems if the transmitter is aware of the occurrence of the erasure an optimal compensation is to project the erasure error to the remaining coefficients it is demonstrated that the same compensation can be executed using a transmitter receiver combination in which the transmitter is not aware of the erasure occurrence the transmitter precompensates using projections as if assuming erasures will occur the receiver undoes the compensation for the coefficients that have not been erased thus maintaining the compensation only of the erased coefficients the stability of the resulting systems is explored and stability conditions are derived it is shown that stability for any erasure pattern can be enforced by optimizing a constrained quadratic program at the system design stage the paper concludes with examples and simulations that verify the theoretical results and illustrate key issues in the algorithms

classification of quantum groups and belavin drinfeld cohomologies for orthogonal and symplectic lie algebras in this paper we continue to study belavin drinfeld cohomology introduced in kadets et al commun math phys 344 1 1 24 2016 and related to the classification of quantum groups whose quasi classical limit is a given simple complex lie algebra g here we compute belavin drinfeld cohomology for all non skewsymmetric r matrices on the belavin drinfeld list for simple lie algebras of type b c and d

arma spectral estimation of narrow band processes via model reduction the problem of estimating autoregressive moving average arma models for narrowband processes is considered the following approach is proposed estimate a high order autoregressive ar approximation of the process by model reduction based on a truncated internally balanced realization or optimal hankel norm model reduction reduce the order of this high order ar estimate to find a lower order arma model this algorithm gives arma spectral estimates with excellent resolution properties without using iterative numerical minimization methods as for the maximum likelihood method how to take the narrowband assumption into account in the model reduction step is discussed in detail

the statistics of interacting dumbbells on a 2 n lattice space a set theoretic argument is utilizied to determine a 31 term recursion that describes exactly the composite nearest neighbor degeneracy for indistinguishable dumbbell particles distributed on a rectangular 2 n lattice space the associated generating functions the expectation of the lattice coverage and the density of occupied nearest neighbor pairs are also determined

fast algorithms for linear prediction and system identification filters with linear phase a general finite impulse response fir filter can be used as a linear prediction filter if given only an input sample sequence or as a system identification model if given the input and output sequences from an unknown system with known correlation the coefficients of the fir filter that minimize the mean square error in both applications are found by solution of a set of normal equations with toeplitz structure using only data samples the coefficients that yield the least squared error in both applications are found by solution of a set of normal equations with near to toeplitz structure computationally efficient fast algorithms have been published to solve for the coefficients from both types of normal equation structures if the fir filter is constrained to have a linear phase then the impulse response must be symmetric this then leads to normal equations with toeplitz plus hankel or near to toeplitz plus hankel structure fast algorithms for solving these normal equations for the filter coefficients are developed in this paper they have computational complexity proportional to m2and parameter storage proportional to m where m is the filter order an application of one of these algorithms for spectral estimation concludes the paper

a fast accurate approximation method with multigrid solver for two dimensional fractional sub diffusion equation a fast accurate approximation method with multigrid solver is proposed to solve a two dimensional fractional sub diffusion equation using the finite difference discretization of fractional time derivative a block lower triangular toeplitz matrix is obtained where each main diagonal block contains a two dimensional matrix for the laplacian operator our idea is to make use of the block circulant approximation via fast fourier transforms so that the resulting task is to solve a block diagonal system where each diagonal block matrix is the sum of a complex scalar times the identity matrix and a laplacian matrix we show that the accuracy of the approximation scheme is of o because of the special diagonal block structure we employ the multigrid method to solve the resulting linear systems the convergence of the multigrid method is studied numerical examples are presented to illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver an approximation method is proposed to approximate the original block lower triangular toeplitz linear system sufficient condition is given to guarantee a high accuracy of the approximation method multigrid solver is employed to solve the resulting block diagonal linear system multigrid solver is proven to be linear convergence numerical examples illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver

wiener hopf and spectral factorization of real polynomials by newton s method abstract we present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle respectively the approach is based on solving a nonlinear system by newton s method the jacobi matrix is a polynomial of a companion matrix and thus a bezoutian times the inverse of a triangular toeplitz matrix after getting deeper understanding of the displacement structure of the bezoutian each newton step can be reduced to a few applications of discrete fourier or cosine transforms and the lu decomposition of a conceived linear system with a cauchy like matrix these structural features are employed to design a fast algorithm which shows excellent numerical behavior for example in the case of the extremely ill conditioned test polynomial 2 z n k 0 n 1 z k 2 k 1 n z k of the degree 2 n 20000 we obtain the factorization after 22 newton steps with an error of 3 4 10 8 in the euclidean norm the execution time on a laptop being a couple of minutes the local quadratic convergence of our newton method is proved and explicit convergence balls are given on the basis of estimates for the lipschitz constant which occurs in kantorovich s theorem we also design and test two superfast versions of our method which for polynomials of degree 20000 typically need about half a minute to produce an initial coefficient vector and then perform the newton iteration within 1 s

fast subsampled updating stabilized fast transversal filter fsu sftf rls algorithm for adaptive filtering we present a new doubly fast algorithm for recursive least squares rls adaptive filtering that uses displacement structure and subsampled updating the fast subsampled updating stabilized fast transversal filter fsu sftf algorithm is mathematically equivalent to the classical fast transversal filter ftf algorithm the ftf algorithm exploits the shift invariance that is present in the rls adaptation of an fir filter the ftf algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the levinson 1947 algorithm in the subsampled updating approach we accumulate the rotation matrices over some time interval before applying them to the filters it turns out that the successive rotation matrices themselves can be obtained from a schur type algorithm that once properly initialized does not require inner products the various convolutions that appear in the algorithm are done using the fast fourier transform fft the resulting algorithm is doubly fast since it exploits ftf and ffts the roundoff error propagation in the fsu sftf algorithm is identical to that in the sftf algorithm a numerically stabilized version of the classical ftf algorithm the roundoff error generation on the other hand seems somewhat smaller for relatively long filters the computational complexity of the new algorithm is smaller than that of the well known lms algorithm rendering it especially suitable for applications such as acoustic echo cancellation

a levinson like algorithm for symmetric strongly nonsingular higher order semiseparable plus band matrices in this paper we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix the solver we will derive is based on the levinson algorithm which is used for solving strongly nonsingular toeplitz systems in the first part an o p2n solver for a semiseparable matrix of semiseparability rank p is derived and in a second part we derive an o l2n solver for a band matrix with bandwidth 2l 1 both solvers are constructed in a similar way firstly a yule walker like equation needs to be solved and secondly this solution is used for solving a linear equation with an arbitrary right hand side finally a combination of the above methods is presented to solve linear systems with semiseparable plus band coefficient matrices the overall complexity of this solver is 6 l p 2n plus lower order terms in the final section numerical experiments are performed attention is paid to the timing and the accuracy of the described methods

human ankle joint stiffness over the full range of muscle activation levels abstract system identification techniques have been used to track changes in dynamic stiffness of the human ankle joint over a wide range of muscle contraction levels subjects lay supine on an experimental table with their left foot encased in a rigid low inertia cast which was fixed to an electro hydraulic actuator operating as a position servo subjects generated tonic plantarflexor or dorsiflexor torques of different magnitudes ranging from rest to maximum voluntary contractions mvc during repeated presentations of a stochastic ankle angular position perturbation compliance impulse response functions irf were determined from every 2 5 s perturbation sequence the gain g natural freqency n and damping parameters of the second order model providing the best fit to each irf were determined and used to compute the corresponding inertial i viscous b and elastic k stiffness parameters the behaviour of these parameters with mean torque was found to follow two simple rules first the elastic parameter k increased in proportion to mean ankle torque as it was varied from rest to mvc these changes were considerable involving increases of more than an order of magnitude second the damping parameter remained almost invariant over the entire range of contractions despite the dramatic changes in k