an iterative method for the skew symmetric solution and the optimal approximate solution of the matrix equation axb c in this paper an iterative method is constructed to solve the linear matrix equation axb c over skew symmetric matrix x by the iterative method the solvability of the equation axb c over skew symmetric matrix can be determined automatically when the equation axb c is consistent over skew symmetric matrix x for any skew symmetric initial iterative matrix x 1 the skew symmetric solution can be obtained within finite iterative steps in the absence of roundoff errors the unique least norm skew symmetric iterative solution of axb c can be derived when an appropriate initial iterative matrix is chosen a sufficient and necessary condition for whether the equation axb c is inconsistent is given furthermore the optimal approximate solution of axb c for a given matrix x 0 can be derived by finding the least norm skew symmetric solution of a new corresponding matrix equation email protected email protected finally several numerical examples are given to illustrate that our iterative method is effective

the constrained inverse eigenvalue problem and its approximation for normal matrices abstract in this paper the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered the solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented and the expression of the solution for the optimal approximation problem is obtained

vessiot structure for manifolds of hyperbolic type darboux integrability and symmetry it is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is darboux integrable then its local cauchy problem may be solved by ordinary differential equations in addition such an equation has infinitely many non trivial conservation laws moreover darboux integrable equations have properties in common with infinite dimensional completely integrable systems in this paper we employ a geometric object intrinsically associated with any hyperbolic partial differential equation its hyperbolic structure to study the darboux integrability of the class e of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables it is shown that the problem of classifying the darboux integrable equations in e contains as a subproblem that of classifying the manifolds of p q hyperbolic type of rank 4 and dimension 2k 3 k 2 p 2 q 2 in turn it is shown that the problem of classifying these manifolds in the two lowest cases p q 2 2 2 3 contains as a subproblem the classification problem for lie groups this generalizes classical results of e vessiot the main result is that if an equation in e is 2 2 or 2 3 darboux integrable on the k jets k 2 then its intrinsic hyperbolic structure admits a lie group of symmetries of dimension 2k 1 or 2k 2 respectively it follows that part of the moduli space for the darboux integrable equations in e is determined by isomorphism classes of lie groups the lie group in question is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic or riemann invariants of the equation the tangential characteristic symmetries the isomorphism class of the tangential characteristic symmetries is a contact invariant of the corresponding darboux integrable partial differential equation

inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors xi i 1m in cn and a set of complex numbers i i 1m find a centrosymmetric or centroskew matrix c in rn n such that xi i 1m and i i 1m are the eigenvectors and eigenvalues of c respectively we then consider the best approximation problem for the ieps that are solvable more precisely given an arbitrary matrix b in rn n we find the matrix c which is the solution to the iep and is closest to b in the frobenius norm we show that the best approximation is unique and derive an expression for it

dead beat control of simple hammerstein models dead beat controllers for simple hammerstein systems are investigated several designs for nonminimum time state dead beat controllers are given for certain classes of simple hammerstein systems a general minimum time state dead beat controller is presented for a class of simple hammerstein systems a design for a family of minimum time control laws is provided this enables to a certain extent shaping of transient response via choosing an appropriate control law finally the authors design an output dead beat controller for a class of hammerstein systems that are not necessarily state dead beat controllable

time series modeling by a second order hammerstein system in this paper we discuss the modeling of an observed time series as the output of a second order hammerstein model driven by an independent and identically distributed i i d gaussian noise the modeling is discussed from the point of view of the hypothesis testing ht of statistical indexes derived from higher order statistics hos three points of view can be adopted in ht the more general approach is to not make any assumption on the index statistics notably on the variance and thus to use the t 2 hotelling test the second point of view involves calculating a theoretical variance in order to define normalized variables whose statistics are known the last approach is a mixed version of the others two these three approaches are applied to statistical indexes derived from linear combination of slices of spectra from the third order up the fifth order some subcases need to use the derivatives of the polyspectra since general relationships are null and cannot be used for the ht the second part of this paper is devoted to linear and quadratic kernel identification using the relationship defined for ht definition as for the ht several subcases have to be discussed using either the polyspectra or the derivatives of polyspectra

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

a class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices in this paper a class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices are essentially decomposed into the same kind subproblems for real antisymmetric matrices with smaller dimensions we present the general solution of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices in real number field in addition in corresponding solution set of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices the explicit expression of the nearest matrix to a given matrix in the frobenius norm is obtained

least square solutions for inverse problems of centrosymmetric matrices let csrnxn a aij rnxn aij an 1 j i j 1 2 n in this paper we mainly discuss solving the following problems problem io given x b rnxm find a csrn n such that ax b problem i given x b rn m find a csrn n such that ax b min problem ii given a rn n find a se such that where is the frobenius norm and se is the solution set of problem i or i0 the expressions of the general solution of problems i and 10 have been derived the sufficient and necessary conditions for the solvability of problem i0 have been provided the numerical algorithm to find the optimal approximate solution and some numerical experiments have been given

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

output dead beat control for a class of planar polynomial systems output dead beat control for a class of nonlinear discrete time systems which are described by a single input output i o polynomial difference equation is considered the class of systems considered is restricted to systems with a two dimensional state space description it is assumed that the highest degree with which the present input appears in the equation is odd necessary and sufficient conditions for the existence of output dead beat control and for the stability of the zero output constrained dynamics are presented we also design a minimum time output dead beat control algorithm feedback controller which yields stable zero dynamics whenever this is feasible a number of interesting phenomena are discussed and illustrated with examples

classification of second order ordinary differential equations admitting lie groups of fibre preserving point symmetries we use elie cartan s method of equivalence to give a complete classification in terms of differential invariants of second order ordinary differential equations admitting lie groups of fibre preserving point symmetries we then apply our results to the determination of all second order equations which are equivalent under fibre preserving transformations to the free particle equation in addition we present those equations of painleve type which admit a transitive symmetry group finally we determine the symmetry group of some equations of physical interest such as the duffing and holmes rand equations which arise as models of non linear oscillators

matrices with multiple symmetry properties applications of centrohermitian and perhermitian matrices abstract twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system we investigate matrices satisfying more than one symmetry pattern we show how a group of operators on gl n c gives rise to distinct types of matrices which satisfy sets of patterns and which give unique decompositions of matrices into components of each type these give a new characterization of normal and unitary matrices we extend symmetry patterns to vectors to study spectral properties of these matrices when a skew symmetric basis of eigenvectors exist we can infer symmetry properties of these matrices

characterization and properties of r s commutative matrices let r pdiag 0im0 1im1 k 1imk 1 p 1 cm m and s qdiag 0 in0 1 in1 k 1 ink 1 q 1 cn n where m0 m1 mk 1 m n0 n1 nk 1 n 0 1 k 1 are distinct complex numbers and zktozk 0 1 k 1 we say that a cm n is r s commutative if ra as we characterize the class of r s commutative matrices and extend results obtained previously for the case where l e2 il k and l l modk 0 l k 1 with zk our results are independent of 0 1 k 1 so long as they are distinct i e if ra as for some choice of 0 1 k 1 all distinct then ra as for arbitrary of 0 1 k 1

matrix inverse problem and its optimal approximation problem for r symmetric matrices abstract let r cn n be a nontrivial involution i e r2 i and r i a matrix a cn n is called r skew symmetric if rar a the least squares solutions of the matrix inverse problem for r skew symmetric matrices with r r are firstly derived then the solvability conditions and the solutions of the matrix inverse problem for r skew symmetric matrices with r r are given the solutions of the corresponding optimal approximation problem with r r for r skew symmetric matrices are also derived at last an algorithm for the optimal approximation problem is given it can be seen that we extend our previous results g x huang f yin matrix inverse problem and its optimal approximation problem for r symmetric matrices appl math comput 189 2007 482 489 and the results proposed by zhou et al f z zhou l zhang x y hu least square solutions for inverse problem of centrosymmetric matrices comput math appl 45 2003 1581 1589

the solutions to linear matrix equations ax b ya d with k involutory symmetries let rcmm and scnn be nontrivial k involutions if their minimal polynomials are both xk1 for some k2 i e rk1 r1i and sk1 s1i we say that acmn is r s symmetric if ras1 a and a is r s symmetric if ras a with 0 1 k1 and 0 let s be one of the subsets of all r s symmetric and r s symmetric matrices given xcnr ycsm bcmr and dcsn we characterize the matrices a in s that minimize axb2 yad2 frobenius norm under the assumption that r and s are unitary moreover among the set s x y b d s of the minimizers of axb2 yad2 min we find the optimal approximate matrix as x y b d that minimizes ag to a given unstructural matrix gcmn we also present the necessary and sufficient conditions such that ax b ya d is consistent in s if the conditions are satisfied we characterize the consistent solution set of all such a finally a numerical algorithm and some numerical examples are given to illustrate the proposed results

left and right inverse eigenvalue problem of r s symmetric matrices and its optimal approximation problem the left and right inverse eigenvalue problem which mainly arises in perturbation analysis of matrix eigenvalue and recursive matters has some practical applications in engineer and scientific computation fields in this paper we give the solvability conditions of and the general expressions to the left and right inverse eigenvalue problem for the r s symmetric and r s skew symmetric solutions the corresponding best approximation problems for the left and right inverse eigenvalue problem are also solved that is given an arbitrary complex n by n matrix a find a r s symmetric or r s skew symmetric matrix a a which is the solution to the left and right inverse eigenvalue problem such that the distance between a and a a is minimized in the frobenius norm we give an explicit solution to the best approximation problem in the r s symmetric and r s skew symmetric solution sets of the left and right inverse eigenvalue problem under the assumption that r r and s s a numerical example is given to illustrate the effectiveness of our method

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

classical invariant theory and the equivalence problem for particle lagrangians i binary forms abstract the problem of equivalence of binary forms under linear changes of variables is shown to be a special case of the problem of equivalence of particle lagrangians under the pseudogroup of transformations of both the independent and dependent variables the latter problem has a complete solution based on the equivalence method of cartan there are two particular rational covariants of any binary form which are related by a universal function the main result is that two binary forms are equivalent if and only if their universal functions are identical construction of the universal function from the syzygies of the covariants and explicit reconstruction of the form from its universal function are also discussed new results on the symmetries of forms and necessary and sufficient conditions for the equivalence of a form to a monomial or to a sum of two n th powers are consequences of this result finally we employ some syzygies due to stroh to relate our result to a theorem of clebsch on the equivalence of binary forms

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

the solvability conditions for the inverse eigenvalue problems of centro symmetric matrices this paper involves related inverse eigenvalue problems of centro symmetric matrices and their optimal approximation the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue and the general provided form of the solution furthermore the algorithm to compute the optimal approximate solution and some numerical experiments are given

constrained inverse eigenproblem and associated approximation problem for anti hermitian r symmetric matrices abstract let r c n n be a nontrivial involution i e r 2 i and r i a c n n is called anti hermitian r symmetric if a a and rar a the presentation and some properties for an arbitrary anti hermitian r symmetric matrix with r r and the relations between the eigenproblem for a and the corresponding eigenproblems for anti hermitian matrices are given then the solutions of constrained inverse eigenproblem and approximation problem are essentially decomposed into the same kind subproblems for anti hermitian matrices in complex field with smaller dimensions the explicit solutions for the later subproblems are arrived the corresponding problems which are the formulations of constrained inverse eigenproblem and approximation problem in complex field was first given then the solutions of constrained inverse eigenproblem and approximation problem with r r are derived

stabilizability and dead beat controllers for two classes of wiener hammerstein models two classes of block oriented models of the wiener hammerstein type are considered we prove that a generic condition is sufficient for a null controllable discrete time system of this form to have a stabilizing minimum time dead beat controller when the condition is violated we show how to design a nonminimum time stabilizing dynamic dead beat controller the result is used to obtain stabilizability conditions for these systems

the general solutions on the minimum residual problem and the matrix nearness problem for symmetric matrices or anti symmetric matrices this paper gives the precise solutions on the minimum residual problem and the matrix nearness problem for symmetric matrices or anti symmetric matrices firstly an equivalent linear equation is given for the minimum residual problem for symmetric matrices then by generalized singular value decomposition gsvd the expression of a general solution on the minimum residual problem for symmetric matrices is arrived furthermore for the matrix nearness problem for symmetric matrices an unique optimal approximate solution is given by first find the general symmetric solution of a corresponding equivalent matrix equation of the matrix nearness problem in addition explicit solutions on the minimum residual problem and the matrix nearness problem for anti symmetric matrices are derived

a differential geometric characterization of the first painleve transcendent on considere l equation de painleve d 2 y dx 2 6y 2 x et on montre qu elle est caracterisee uniquement parmi les equations generiques par la condition qu un certain invariant differentiel est une fonction polynomiale des trois invariants de base fournis par la methode de cartan

solution of the equivalence problem for the third painleve equation we find necessary conditions for a second order ordinary differential equation to be equivalent to the painleve iii equation under a general point transformation their sufficiency is established by reduction to known results for the equations of the form y f x y we consider separately the generic case and the case of reducibility to an autonomous equation the results are illustrated by the primary resonance equation

dead beat controllability of polynomial systems symbolic computation approaches state and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the quantifier elimination by partial cylindrical algebraic decomposition qepcad symbolic computation program the method is unified for a very large class of systems and can handle one or two sided control constraints families of minimum time state output dead beat controllers are obtained the computational complexity of the test is prohibitive for general polynomial systems but by constraining the structure of the system we may beat the curse of complexity a computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented necessary and sufficient conditions are given they are still very difficult to check therefore a number of easier to check sufficient conditions are also provided the latter are based on the grobner basis method and qepcad it is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system

on the eigenvectors of symmetric toeplitz matrices this paper presents a number of results concerning the eigenvectors of a symmetric toeplitz matrix and the location of the zeros of the filters eigenfilters whose coefficients are the elements of the eigenvectors one of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues if distinct have their zeros on the unit circle while the zeros of the other eigenfilters may or may not have their zeros on the unit circle even if the zeros of the eigenfilters of a matrix are all on the unit circle the matrix need not be toeplitz examples are given to illustrate the different properties

controllability for a class of simple wiener hammerstein systems abstract controllability for a class of simple wiener hammerstein systems is considered necessary and sufficient conditions for dead beat and complete controllability for these systems are presented the controllability tests consist of two easy to check tests for the subsystems

state deadbeat control of nonlinear systems construction via sets a geometric generalization of the discrete time linear deadbeat control problem is studied the proposed method to generate a deadbeat tracker for a given nonlinear system is constructive and makes use of sets that can be computed iteratively for demonstration derivations of the deadbeat feedback law and tracker dynamics are provided for an example system based on the method a simple algorithm that computes the deadbeat gain for a linear system with scalar input is given

the discrete linear time invariant time optimal control problem an overview a review of two decades of investigation into one of the fundamental problems of control theory the discrete linear time invariant time optimal control problem is presented two classes of multi input time optimal controller one based on system controllability and the other on eigenvalue assignment are considered in some detail also discussed are the dual state reconstruction problem the deadbeat control of linear systems with inaccessible state and other extensions the exposition is a unifying one and involves only that matrix algebra commonly encountered in control theory

a note on dead beat controllability of generalised hammerstein systems necessary and sufficient conditions for dead beat and complete controllability for a class of generalised hammerstein systems are presented since the system s structure is very close to linear only linear algebra is used for the controllability test the test is very simple and easy to use a closed loop minimum time dead beat controller can be designed for a subclass of generalised hammerstein systems using the grobner basis method

the local equivalence problem for d 2 y dx 2 f x y dy dx and the painleve transcendents on resout le probleme d equivalence locale pour d 2 y dx 2 f x y dy dx sous le groupe naturel des changements de coordonnees x x y x y il y a trois invariants de base qui s annulent si et seulement si l equation est equivalente a d 2 y dx 2 0 on montre que deux des invariants s annulent pour les 6 transcendants de painleve et le 3eme peut etre utilise pour produire un ensemble complet d invariants

characterisation of the algebraic properties of first integrals of scalar ordinary differential equations of maximal symmetry abstract we undertake a study of the first integrals of linear n th order scalar ordinary differential equations with maximal symmetry we establish patterns for the first integrals associated with these equations it is shown that second and third order equations are the pathological cases in the study of higher order differential equations the equivalence of contact symmetries for third order equations to non cartan symmetries of second order equations is highlighted

structure identification of nonlinear dynamic systems a survey on input output approaches abstract in the past several methods have been elaborated for the identification of nonlinear dynamic systems most of the methods assume that the structure of the system is given a priori therefore they are in reality parameter estimation algorithms and structure identification is thus usually performed by repeated parameter estimation however in nonlinear system theory several methods are known to determine the structure of a system in this paper structure identification of block oriented especially cascade models of semi linear dynamic models with signal dependent parameters and of nonlinear dynamic models being linear in the parameters will be considered different structure selection methods are summarized based on step and impulse tests frequency response measurements correlation analysis repeated reproducible tests and normal operating data

exceptional properties of second and third order ordinary differential equations of maximal symmetry the riccati transformation is used in the reduction of order of second and third order ordinary differential equations of maximal symmetry the sl 2 r subalgebra is preserved under this transformation the riccati transformation is itself associated with the symmetry that is annihilated in the reduction of order the solution symmetries and the intrinsically contact symmetries become nonlocal symmetries under the riccati transformation we investigate the fate and origins of the contact symmetries arising from the riccati transformation the exceptional properties of the second and third order equations of maximal symmetry are indicated in the context of generalised symmetries we express the solution symmetries contact symmetries and the sl 2 r elements in terms of a jacobian we show that a basis for the solution set of equations of maximal symmetry is given in terms of the solution set of a second order ordinary differential equation

further comments on on the eigenvectors of symmetric toeplitz matrices attention is drawn to some previous work on the matrices discussed in the comments of datta and morgera on the above paper