on hamiltonian stationary lagrangian spheres in non einstein kahler surfaces hamiltonian stationary lagrangian spheres in kahler einstein surfaces are minimal we prove that in the family of non einstein kahler surfaces given by the product 1 2 of two complete orientable riemannian surfaces of different constant gauss curvatures there is only a non minimal hamiltonian stationary lagrangian sphere this example defined when the surfaces 1 and 2 are spheres is unstable

representation type of jordan algebras abstract the problem of classification of jordan bimodules over non semisimple finite dimensional jordan algebras with respect to their representation type is considered the notions of diagram of a jordan algebra and of jordan tensor algebra of a bimodule are introduced and a mapping qui is constructed which associates to the diagram of a jordan algebra j the quiver of its universal associative enveloping algebra s j the main results are concerned with jordan algebras of semi matrix type that is algebras whose semi simple component is a direct sum of jordan matrix algebras in this case criterion of finiteness and tameness for one sided representations are obtained in terms of diagram and mapping qui for jordan tensor algebras and for algebras with radical square equals to 0

train algebras of rank 3 with finiteness conditions abstract in this paper we prove that a train algebra of rank 3 which is finitely generated n therian or artinian is finite dimensional

metric lie algebras with maximal isotropic centre a metric lie algebra is a lie algebra equipped with an invariant non degenerate symmetric bilinear form it is called indecomposable if it is not the direct sum of two metric lie algebras we are interested in describing the isomorphism classes of indecomposable metric lie algebras in the present paper we restrict ourselves to a certain class of solvable metric lie algebras which includes all indecomposable metric lie algebras with maximal isotropic centre we will see that each metric lie algebra belonging to this class is a twofold extension associated with an orthogonal representation of an abelian lie algebra we will describe equivalence classes of such extensions by a certain cohomology set in particular we obtain a classification scheme for indecomposable metric lie algebras with maximal isotropic centre and the classification of metric lie algebras of index 2

perturbation analysis of orthogonal canonical forms abstract we present a complete perturbation analysis of orthogonal canonical forms ocfs of linear multivariable systems both local linear and nonlocal nonlinear perturbation bounds are derived in particular the conditioning of ocfs is determined

a computational algorithm for pole assignment of linear single input systems an efficient computational algorithm for pole assignment of linear multiinput systems is proposed a preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form the gain matrix elements are then found by orthogonal transformation of the closed loop system matrix into upper quasi triangular form whose diagonal blocks correspond to the desired poles the algorithm is numerically stable the computed gain matrix being exact for a system with slightly perturbed matrices it works equally well with real and complex distinct and multiple poles and is applicable to ill conditioned and high order problems

on a type of commutative algebras abstract we introduce some basic concepts for jacobi jordan algebras such as representations crossed products or frobenius metabelian co flag objects a new family of solutions for the quantum yang baxter equation is constructed arising from any 3 step nilpotent jacobi jordan algebra crossed products are used to construct the classifying object for the extension problem in its global form for a given jacobi jordan algebra a and a given vector space v of dimension c a global non abelian cohomological object g h 2 a v is constructed it classifies from the view point of the extension problem all jacobi jordan algebras that have a surjective algebra map on a with kernel of dimension c the object g h 2 a k responsible for the classification of co flag algebras is computed all 1 dim a dimensional jacobi jordan algebras that have an algebra surjective map on a are classified and the automorphism groups of these algebras is determined several examples involving special sets of matrices and symmetric bilinear forms as well as equivalence relations between them generalizing the isometry relation are provided

an arma robust system identification using a generalized l sub p norm estimation algorithm the parameter estimation of arma systems with noisy output data is considered the system output is corrupted by measurement noise and the noise distribution is assumed to be unknown a generalized l sub p norm iterative estimation algorithm 1

on the application of orthogonal transformation for the design and analysis of feedforward networks orthogonal transformation which can lead to compaction of information has been used in two ways to optimize on the size of feedforward networks 1 through the selection of optimum set of time domain inputs and the optimum set of links and nodes within a neural network nn and 2 through the orthogonalization of the data to be used in nn s in case of processes with periodicity the proposed methods are efficient and are also extremely robust numerically the singular value decomposition svd and qr with column pivoting factorization qrcp are the transformations used svd mainly serves as the null space detector qrcp coupled with svd is used for subset selection which is one of the main operations on which the design of the optimal network is based svd has also been used to devise a new approach for the assessment of the convergence of the nn s which is an alternative to the conventional output error analysis

nonlinear autoregressive and nonlinear autoregressive moving average model parameter estimation by minimizing hypersurface distance the least squares ls can be used for nonlinear autoregressive nar and nonlinear autoregressive moving average narma parameter estimation however for nonlinear cases the ls results in biased parameter estimation due to its assumption that the independent variables are noise free the total least squares tls is another method that can used for nonlinear parameter estimation to increase the accuracy of the ls because it specifically accounts for the fact that the independent variables are noise corrupted tls has its own limitations however mainly because it is difficult for the method to isolate noise from the signal components we present a new method that is based on minimization of hypersurface distance for accurate parameter estimation for nar and narma models computer simulation examples show that the new method results in far more accurate nar and narma model parameter estimates than do either the ls and tls with noise that is either white or colored and retains its high accuracy even when the signal to noise ratio snr is as low as 10 db

on the structure of symmetric self dual lie algebras a finite dimensional lie algebra is called symmetric self dual if it possesses an invariant nondegenerate symmetric bilinear form symmetric self dual lie algebras have been studied by medina and revoy who have proven a very useful theorem about their structure in this paper we prove a refinement of their theorem that has wide applicability in conformal field theory where symmetric self dual lie algebras start to play an important role due to the fact that they are precisely the lie algebras that admit a sugawara construction we also prove a few corollaries that are important in conformal field theory

parallel and large scale matrix computations in control some ideas abstract the design and analysis of time invariant linear control systems give rise to a variety of interesting linear algebra problems numerically effective methods now exist for several of these problems however algorithms for large scale computations and efficient parallel algorithms for these problems are virtually nonexistent in this paper we propose several efficient general purpose parallel algorithms for single input and multiinput eigenvalue assignment problems a desirable feature of these algorithms is that they are composed of simple linear algebraic operations such as matrix vector multiplication solution of a linear system and computations of the eigensystem and singular values of a symmetric matrix for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms the proposed algorithms thus have potential for implementations on some existing and future parallel processors we also propose a numerical method for the sylvester matrix equation arising in the construction of a luenberger observer the method does not need reduction to condensed forms and is thus suitable for large and sparse matrices the method also exhibits a certain parallelism

a cooperative recurrent neural network for solving l1 estimation problems with general linear constraints the constrained l1 estimation is an attractive alternative to both the unconstrained l1 estimation and the least square estimation in this letter we propose a cooperative recurrent neural network crnn for solving l1 estimation problems with general linear constraints the proposed crnn model combines four individual neural network models automatically and is suitable for parallel implementation as a special case the proposed crnn includes two existing neural networks for solving unconstrained and constrained l1 estimation problems respectively unlike existing neural networks with penalty parameters for solving the constrained l1 estimation problem the proposed crnn is guaranteed to converge globally to the exact optimal solution without any additional condition compared with conventional numerical algorithms the proposed crnn has a low computational complexity and can deal with the l1 estimation problem with degeneracy several applied examples show that the proposed crnn can obtain more accurate estimates than several existing algorithms

the global extension problem crossed products and co flag non commutative poisson algebras let p be a poisson algebra e a vector space and e p an epimorphism of vector spaces with v ker the global extension problem asks for the classification of all poisson algebra structures that can be defined on e such that e p becomes a morphism of poisson algebras from a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component all such poisson algebra structures on e are classified by an explicitly constructed classifying set gph2 p v which is the coproduct of all non abelian cohomological objects ph2 p v v v which are the classifying sets for all extensions of p by v v v the second classical poisson cohomology group h2 p v appears as the most elementary piece among all components of gph2 p v several examples are provided in the case of metabelian poisson algebras or co flag poisson algebras over p the latter being poisson algebras q which admit a finite chain of epimorphisms of poisson algebras pn q npn 1 p1 1p0 p such that dim ker i 1 for all i 1 n

an algorithm for pole assignment in high order multivariable systems an efficient computational method is presented for solving the problem of pole assignment by state feedback in linear multivariable systems with large dimensions a given multiinput system is first transformed to an upper block hessenberg form by means of orthogonal state coordinate transformations the state feedback problem is then reformulated in terms of the sylvester equation the transformed system matrices along with certain assumed block forms for unknown matrices enable the sylvester equation to be decomposed and solved effectively a distinct point of the proposed algorithm is that the solution procedure can be tailored to parallel implementation and is therefore fast computational aspects of the algorithm are discussed and numerical examples are provided

harmonic maps and constant mean curvature surfaces in h 2 x r we introduce a it hyperbolic gauss map into the poincar e disk for any surface in bbb h 2 times bbb r with regular vertical projection and prove that if the surface has constant mean curvature h 1 2 this hyperbolic gauss map is harmonic conversely we show that every nowhere conformal harmonic map from an open simply connected riemann surface sigma into the poincar e disk is the hyperbolic gauss map of a two parameter family of such surfaces as an application we obtain that any holomorphic quadratic differential on sigma can be realized as the abresch rosenberg holomorphic differential of some and generically infinitely many complete surfaces with h 1 2 in bbb h 2 times bbb r a similar result applies to minimal surfaces in the heisenberg group rm nil 3 finally we classify all complete minimal vertical graphs in bbb h 2 times bbb r

fast algorithms for l p deconvolution this paper discusses two algorithms for l p 1 leq p leq 2 deconvolution one iterative reweighted least squares irls is discussed with an eye toward improving its computational efficiency while the other residual steepest descent rsd is considered in an attempt to improve its convergence properties in the first case fast fourier transforms are used to reduce the number of computations in the second a new rescaling procedure which enhances rsd convergence is introduced the lack of stability of l 1 filters and its implications are discussed simple examples are used to illustrate pertinent concepts

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

detection filter design spectral theory and algorithms a new formulation of the detection filter problem is generated by assignment of the closed loop eigenstructure under certain constraints detection filters which are actually a specific class of observers fix the output error direction of the system so that it can be associated with a particular failure mode and its known design failure direction the derivation of detection filters from an eigensystem assignment approach permits a very transparent theory the detection filter gains and closed loop eigenvectors are obtained from a set of simultaneous equations necessary and sufficient conditions for the solution of these algebraic equations are determined which produce a complete theory for detection filters

the moduli space of 4 dimensional nilpotent complex associative algebras abstract in this paper we study 4 dimensional nilpotent complex associative algebras this is a continuation of the study of the moduli space of 4 dimensional algebras the nonnilpotent algebras were analyzed in an earlier paper even though there are only 15 families of nilpotent 4 dimensional algebras the complexity of their behavior warranted a separate study which we give here

reduced dimension blind space time 2 d rake receivers for ds cdma communication systems blind space time rake receivers for classical ds cdma that cancel strong multiuser access interference while optimally combining the desired user s multipath are presented the delay spread is assumed to be fraction of a symbol interval as in the is 95 cdma standard the post correlation symbol interval is segmented into that which encompasses the rake fingers and that away from the rake fingers the signal plus interference and interference alone space time correlation matrices are estimated during the former and latter respectively reduced dimension space time rake receivers are investigated for alleviation of the computational burden of computing the largest generalized eigenvector of the resulting matrix pencil theoretical analysis and supporting simulations reveal that if the space and time compressing transformations are designed judiciously reduced dimension two dimensional 2 d rake receivers offer faster convergence at the expense of only a slight loss in asymptotic signal to interference plus noise ratio sinr relative to the full dimension space time rake receiver applications to the is 95 uplink are also presented along with supporting simulations

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

on nuclear bernstein algebras abstract in 1975 p holgate proved that the core of any orthogonal bernstein algebra is a special train algebra and consequently a genetic algebra let a be a bernstein algebra then a is a nuclear algebra if and only if the core of a is a the present paper proves that the core of a bernstein algebra is a special train algebra

principal pivot transforms properties and applications the principal pivot transform ppt is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right hand side of the system the notion of the ppt is encountered in mathematical programming statistics and numerical analysis among other areas the purpose of this paper is to draw attention to the main properties and uses of ppts make some new observations and motivate further applications of ppts in matrix theory special consideration is given to ppts of matrices whose principal minors are positive 2000 elsevier science inc all rights reserved

a gametheoretic approach for non uniform pole shifting and pole homothety this paper investigates the problems of non uniform pole shifting or pole homothety in multivariable linear systems by using a gametheoretic approach and more specifically a particular nash strategy with an open loop information structure for a game including a time preference rate in the quadratic criteria such a result is possible due to properties of non symmetric algebraic riccati equations associated with an open loop nash strategy numerical examples allow to illustrate the efficiency of the proposed approach

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

matrix bidiagonal form the matrix bidiagonal form is analysed and some applications to linear system theory are shown in the first part it is shown that the reduction of a given matrix a to the mbf is immediate in the very common situation in which a vector b such that a b is controllable is known in the second part some applications are presented such as the evaluation of a function of a matrix and the solution of the lyapunov equation

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

cooperative recurrent neural networks for the constrained l_ 1 estimator the constrained least absolute deviation l1 estimator is an attractive alternative to both the unconstrained l1 estimator and the least square estimator this paper introduces a constrained l1 method and proposes two cooperative recurrent neural networks crnns for the constrained l1 estimator unlike existing cooperative neural networks the proposed two crnns have a novel weighting cooperation scheme to integrate individual neural network information automatically as a special case the proposed continuous time crnn includes the existing continuous time neural network for unconstrained l1 estimator compared with existing continuous time neural networks for the constrained l1 estimator the proposed continuous time crnn has a lower model complexity and the finite time convergence to the exact optimal solution without any additional condition furthermore compared with existing numerical algorithms for the constrained l1 estimator in addition to a low computational complexity the proposed two crnns are suitable for parallel implementation and can deal with the l1 estimation problem with degeneracy the proposed two crnns are applied to parameter estimation problems under non gaussian noise environments simulation results demonstrate that the proposed crnns are indeed effective in dealing with the l1 estimation problem with nonunique solutions and in obtaining a better solution than relevant algorithms

orthogonal canonical forms for second order systems it is shown that a linear second order system with arbitrary damping cannot be reduced to hessenberg triangular form by means of orthogonal transformations however it is also shown that such an orthogonal reduction is always possible for the modal damping commonly assumed for models of flexible structures it is shown that modally damped models can be orthogonally reduced to a new triangular second order schur form

an algorithm for the assignment of system zeros abstract this paper presents an algorithm for assigning the locations and number of zeros to a linear system through synthesis of the system input and output space couplings the approach imposes no restrictions on the state space model can assign both real and complex zeros using only real arithmetic and is computationally efficient numerical properties of the algorithm are discussed and examples are given to illustrate its performance

jacobi jordan algebras abstract we study finite dimensional commutative algebras which satisfy the jacobi identity such algebras are jordan algebras we describe some of their properties and give a classification in dimensions n 7 over algebraically closed fields of characteristic not 2 or 3

linear and numerical linear algebra in control theory some research problems abstract this paper presents an informal survey of theoretical and computational aspects of three linear algebra problems arising in control theory namely the sylvester observer matrix equation the matrix eigenvalue assignment problem and the stability and stabilization of large second order systems difficulties are highlighted recent developments are mentioned some research problems in these areas which should be of interest to both linear algebraists and control theorists are described

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

on train algebras of rank 3 abstract we classify train algebras of rank 3 having type n 1 or 3 n 2 for an arbitrary value of n then we classify train algebras of rank 3 which have an anisotropic kernel

computing the radius of controllability for state space systems abstract in this paper we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair a b in order to do so we construct a sequence of structured matrices from given system matrices a and b controllability of the pair a b is equivalent to a condition on the null space dimension of an appropriate matrix in this sequence we show that the dimension of the reachability space is also related to the above condition further it is shown that the nearest structured low rank approximation slra of this structured matrix corresponds to a nearest uncontrollable system to the pair a b

a hopf theorem for ambient spaces of dimensions higher than three we consider surfaces m 2 immersed in e c n times mathbb r where e c n is a simply connected n dimensional complete riemannian manifold with constant sectional curvature e c n and assume that the mean curvature vector of the immersion is parallel in the normal bundle we consider further a hopf type complex quadratic form q on m 2 where the complex structure of m 2 is compatible with the induced metric it is not hard to check that q is holomorphic see 3 p 289 we will use this fact to give a reasonable description of immersed surfaces in e c n times mathbb r that have parallel mean curvature vector

jordan lie super algebra and jordan lie triple system abstract we introduce notions of jordan lie super algebras and jordan lie triple systems as well as doubly graded lie super algebras they are intimately related to both lie and jordan super algebras as well as antiassociative algebra

special and exceptional mock lie algebras abstract we observe several facts and make conjectures about commutative algebras satisfying the jacobi identity the central question is which of those algebras admit a faithful representation i e in lie parlance satisfy the ado theorem or in jordan parlance are special

maximum likelihood estimation of a class of non gaussian densities with application to i sub p deconvolution the properties of the maximum likelihood estimator of the generalized p gaussian gpg probability density function from n independent identically distributed samples is investigated especially in the context of the deconvolution problem under gpg white noise specifically the properties in the estimator are first described independently of the application then the solution of the above mentioned deconvolution problem is obtained as the solution of a minimum norm problem in an l sub p normed space it is shown that such minimum norm solution is the maximum likelihood estimate of the system function parameters and that such an estimate is unbiased with the lower bound of the variance of the error equal to the cramer rao lower bound and the upper bound derived from the concept of a generalized inverse the results are illustrated by computer simulations

system parameter estimation with input output noisy data and missing measurements an investigation is undertaken to examine the parameter estimation problem of linear systems when some of the measurements are unavailable i e missing data and the probability of occurrence of missing data is unknown a priori the system input and output data are also assumed to be corrupted by measurement noise and the knowledge of the noise distribution is unknown under the unknown noise distribution and missing measurements a consistent parameter estimation algorithm which is based on an l sub p norm iterative estimation algorithm iteratively reweighted least squares irls is proposed to estimate the system parameters we show that if the probability of missing measurement is less than one half the parameter estimates via the proposed estimation algorithm will converge to the true parameters as the number of data tends to infinity finally several simulation results are presented to illustrate the performance of the proposed l sub p norm iterative estimation algorithm simulation results indicate that under input output missing data and noise environment the proposed parameter estimation algorithm is an efficient approach toward the system parameter estimation problem

representations on train algebras of rank 3 we give a characterization of the representations on train algebras of rank 3 we prove that the subalgebra of the algebra of endomorphisms of a module generated by the representation of the nil ideal of the algebra is nilpotent finally we prove that every irreducible module has dimension one over the field under consideration

stratification of moduli spaces of lie algebras similar matrices and bilinear forms abstract in this paper the authors apply a stratification of moduli spaces of complex lie algebras to analyzing the moduli spaces of n n matrices under scalar similarity and bilinear forms under the cogredient action for similar matrices we give a complete description of a stratification of the space by some very simple projective orbifolds of the form p n g where g is a subgroup of the symmetric group n 1 acting on p n by permuting the projective coordinates for bilinear forms we give a similar stratification up to dimension 4

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

surfaces with parallel mean curvature vector in complex space forms we consider a quadratic form defined on the surfaces with parallel mean curvature vector of an any dimensional complex space form and prove that its 2 0 part is holomorphic when the complex dimension of the ambient space is equal to 2 we define a second quadratic form with the same property and then determine those surfaces with parallel mean curvature vector on which the 2 0 parts of both of them vanish we also provide a reduction of codimension theorem and prove a non existence result for 2 spheres with parallel mean curvature vector

a new parameter estimate in singular perturbations abstract a new upper bound is obtained for the singular perturbation parameter of an asymptotically stable singularly perturbed system general time invariant systems with a single small parameter are considered the paper employs a riccati equation whose solution is known to facilitate the exact decoupling of fast and slow dynamics an application of the brouwer fixed point theorem to the riccati equation and of liapunov s direct method to the fast and slow subsystems results in the desired upper bound computation of the estimate requires only the solution of two liapunov matrix equations

bias reduction in parameter estimation abstract the upper bound of the bias in the estimated parameters is derived in terms of the spectral norm it is then established that the bias is related to the condition of the problem as well as the noise existing in the data conventional methods such as the generalized least squares gls the instrumental variables iv and the extended matrix em method seek to minimize bias by reducing the level of noise correlation in the data the method presented in this paper approaches the bias reduction problem via the condition number which acts as a multiply factor on the bias due to noise correlation by improving the condition of the problem it can be shown that bias in the estimated parameters is significantly reduced

on the computation of sensitivity functions of linear time invariant systems responses via diagonalization this note presents a method for computing the sensitivity functions of parametrized linear time invariant systems for the case where the system matrix a is diagonalizable the method is based on a formula derived in the note for the sensitivity of an exponential of a diagonalizable matrix

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

decomposition and stability for multiparameter singular perturbation problems time scale separation and stability of linear time varying and time invariant multiparameter singular perturbation problems are analyzed the first problem considered in the paper is that of deriving upper bounds on the small parasitic parameters ensuring the existence of an invertible bounded transformation exactly separating fast and slow dynamics this problem is most interesting for the time varying case the analysis of this problem in the time varying case requires the two time scale setting introduced by h k khalil and p v kokotovic 16 this entails that the mutual ratios of the small parameters are bounded by known positive constants the second problem considered is to derive parameter bounds ensuring that the system in question is uniformly asymptotically stable the results on decomposition are used to facilitate the derivation of these latter bounds fortunately the analysis of decomposition and stability questions for time invariant multiparameter singular perturbation problems requires no restriction on the relative magnitudes of the small parameters a concept of strong d stability is introduced and shown to greatly simplify the stability analysis of time invariant multiparameter problems

properties of numerical algorithms related to computing controllability the numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations in particular the numerical rank of a matrix the numerical stability of algorithms the sensitivity of problems and the scaling of problems are discussed a numerically stable algorithm is given for computing controllability but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful and this appears to be an open computational problem

numerical linear algebra aspects of control design computations the interplay between recent results and methodologies in numerical linear algebra and mathematical software and their application to problems arising in systems control and estimation theory is discussed the impact of finite precision finite range arithmetic including the implications of the proposed ieee floating point standard s on control design computations is illustrated with numerous examples as are pertinent remarks concerning numerical stability and conditioning basic tools from numerical linear algebra such as linear equations linear least squares eigenproblems generalized eigenproblems and singular value decomposition are then outlined a selected list of applications of the basic tools then follows including algorithms for solution of problems such as matrix exponentials frequency response system balancing and matrix riccati equations the implementation of such algorithms as robust mathematical software is then discussed a number of issues are addressed including characteristics of reliable mathematical software availability and evaluation language implications fortran ada etc and the overall role of mathematical software as a component of computer aided control system design

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

surfaces with parallel mean curvature in sasakian space forms we study the global geometry of surfaces in sasakian space forms whose mean curvature vector is parallel in the normal bundle these include the riemannian heisenberg space of dimension 2n 1 we prove a codimension reduction theorem we introduce two holomorphic quadratic differentials on anti invariant such surfaces and use them to obtain classification theorems

cmc biconservative surfaces in sn r and hn r abstract we classify non minimal biconservative surfaces with parallel mean curvature vector field in s n r and h n r when these surfaces do not lie in s n or h n and they are not vertical cylinders we find their explicit local equation we also prove a result on the compactness of biconservative surfaces with constant mean curvature in hadamard manifolds

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

structure des algebres de bernstein abstract the theory of bernstein s algebras is revisited many results concerning the structure of bernstein algebras ideals nilpotency and so on are stated for bernstein jordan algebras we give a definitive formulation of the theorem which characterizes them finally we emphasize the importance of the invariant l this ideal doesn t depend on the peirce decomposition of the algebra in the study of the structure of bernstein s algebras

the geometry of finite topology bryant surfaces we prove that in the three dimensional hyperbolic space h 3 the properly embedded constant mean curvature one surfaces with finite topology are of finite total curvature and each ends are regular this imply in particular that the hrosphere is the only properly embedded simply connected constant mean curvature one surface in h 3

a hopf theorem for non constant mean curvature and a conjecture of a d alexandrov we prove a uniqueness theorem for immersed spheres of prescribed non constant mean curvature in homogeneous three manifolds in particular this uniqueness theorem proves a conjecture by a d alexandrov about immersed spheres of prescribed weingarten curvature in mathbb r 3 for the special but important case of prescribed mean curvature as a consequence we extend the classical hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in mathbb r 3

surfaces with parallel mean curvature in ℂℙⁿ ℝ and ℂℍⁿ ℝ we consider surfaces with parallel mean curvature vector pmc surfaces in cp r and ch r and more generally in cosymplectic space forms we introduce a holomorphic quadratic differential on such surfaces this is then used in order to show that the anti invariant pmc 2 spheres of a 5 dimensional non flat cosymplectic space form of product type are actually the embedded rotational spheres s h m r of hsiang and pedrosa where m is a complete simply connected surface with constant curvature when the ambient space is a cosymplectic space form of product type and its dimension is greater than 5 we prove that an immersed non minimal non pseudo umbilical anti invariant 2sphere lies in a product space m r where m is a space form we also provide a reduction of codimension theorem for the pmc surfaces of a non flat cosymplectic space form

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft