poles and zeros at infinity of linear time varying systems the notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time varying systems this study is based on the notion of a newborn systems which is in a mathematical point of view a graded module extension over the noncommutative ring of differential operators it is proved to be a relevant generalization to the time varying case of the equivalence class for the so called restricted equivalence of rosenbrock s polynomial matrix descriptions the authors approach is intrinsic and unifies the definitions previously given in the literature in the time invariant case

q difference equation and q polynomials we establish one general q exponential operator we give one general q difference equation relationship between q exponential operator and q difference equation are given using q difference equation to set up a q polynomials hn some generating functions for hn are obtained in this paper we establish one general q exponential operator identity by solving one simple q difference equation using this q difference equation some multilinear generating functions for h n are discussed

the asymptotic variance matrices of the sample correlation matrix in elliptical and normal situations and their proportionality abstract this note presents the asymptotic variance matrices of the sample correlation matrix in elliptical and normal situations and generalizes the scalar proportionality result to the multivariate case

a sandwich type standard error estimator of sem models with multivariate time series structural equation models are increasingly used as a modeling tool for multivariate time series data in the social and behavioral sciences standard error estimators of sem models originally developed for independent data require modifications to accommodate the fact that time series data are inherently dependent in this article we extend a sandwich type standard error estimator of independent data to multivariate time series data one required element of this estimator is the asymptotic covariance matrix of concurrent and lagged correlations among manifest variables whose closed form expression has not been presented in the literature the performance of the adapted sandwich type standard error estimator is evaluated using a simulation study and further illustrated using an empirical example

uniform exact model matching for a class of linear time varying analytic systems abstract the problem of uniform exact model matching is studied via state and output feedback for a class of linear time varying analytic systems the following two major issues are resolved the nessecary and sufficient conditions for the problems to have a solution and the general analytical expressions for the controller matrices a major feature of the proposed approach is that it reduces the uniform exact model matching problem to that of solving a linear nonhomogeneous algebraic matrix equation

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

the bivariate rogers szego polynomials we present an operator approach to deriving mehler s formula and the rogers formula for the bivariate rogers szego polynomials hn x y q the proof of mehler s formula can be considered as a new approach to the nonsymmetric poisson kernel formula for the continuous big q hermite polynomials hn x a q due to askey rahman and suslov mehler s formula for hn x y q involves a 32 sum and the rogers formula involves a 21 sum the proofs of these results are based on parameter augmentation with respect to the q exponential operator and the homogeneous q shift operator in two variables by extending recent results on the rogers szego polynomials hn x q due to hou lascoux and mu we obtain another rogers type formula for hn x y q finally we give a change of base formula for hn x a q which can be used to evaluate some integrals by using the askey wilson integral

q differential operator identities and applications abstract in this paper we construct a new q exponential operator and obtain some operator identities using these operator identities we give a formal extension of jackson s 2 2 transformation formula a formal extension of bailey s 3 3 summation and an extension of the sears terminating balanced 3 4 transformation formula are also derived by our operator method in addition we also derive several interesting a formal extensions involving multiple sum about three terms of sears 2 3 transformation formula and heine s 1 2 transformation formula

decoupling through input output blending the paper presents a novel decoupling method based on blending the input and output signals of linear dynamical systems for this purpose blend vectors are introduced and calculated such that the

discussing some examples of linear system interconnections the interconnection of linear systems may exhibit disturbing phenomena like hidden modes or a lack of control variables these features which cannot be taken into account by the transfer function approach are here explained by utilizing the language of modules which is strongly related to willems behavorial approach several examples are carefully discussed

the minimal delay decoupling problem feedback implementation with stability in this paper we solve the minimal delay decoupling problem of linear multivariable systems we look for feedback implementable solutions which moreover guarantee closed loop stability we prove that dynamic state feedback decoupling with stability is achievable if the number of independent inputs is large enough to compensate the intrinsic nondecouplability of the system we introduce new feedback invariants characterizing the minimal number of infinite and unstable zeros of the decoupled system

necessary and sufficient conditions for asymptotic model matching of switching linear systems this paper investigates the problem of designing a feedback controller to force the response of a given plant to match asymptotically that of a prescribed model in case both the plant and the model are switching linear systems matching has to be achieved with stability of the feedback loop for any initial conditions of the plant the model and the controller in case it is dynamic and for any choice of the switching law solvability of the problem is characterized in terms of necessary and sufficient conditions that in part refer to the geometric structure of the difference system which compares the output of the model and of the plant stability is considered both for slow switching and for arbitrary switching proofs of the solvability conditions provide viable procedures for synthesizing the static or dynamic controllers that achieve the matching respectively when the state of the model is measurable and when it is not weaker sufficient conditions that can be practically checked by simple algorithmic procedures are provided both in the general situation and under slightly restrictive hypotheses

asymptotic biases in exploratory factor analysis and structural equation modeling formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the wishart maximum likelihood estimation for normally and nonnormally distributed variables when multivariate normality is satisfied considerable simplification is obtained for the models of unstandardized variables formulas for the models of standardized variables are also provided numerical examples with monte carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data some relationships between the asymptotic biases and other asymptotic values are discussed

operator identities involving the bivariate rogers szego polynomials and their applications to the multiple q series identities abstract in this paper we first give several operator identities involving the bivariate rogers szego polynomials by applying the technique of parameter augmentation to the multiple q binomial theorems given by milne s c milne balanced 2 3 summation theorems for u n basic hypergeometric series adv math 131 1997 93 187 we obtain several new multiple q series identities involving the bivariate rogers szego polynomials these include multiple extensions of mehler s formula and rogers s formula our u n 1 generalizations are quite natural as they are also a direct and immediate consequence of their often classical known one variable cases and milne s fundamental theorem for a n or u n 1 basic hypergeometric series in theorem 1 49 of s c milne an elementary proof of the macdonald identities for a l 1 adv math 57 1985 34 70 as rewritten in lemma 7 3 on p 163 of s c milne balanced 2 3 summation theorems for u n basic hypergeometric series adv math 131 1997 93 187 or corollary 4 4 on pp 768 769 of s c milne m schlosser a new a n extension of ramanujan s 1 1 summation with applications to multilateral a n series rocky mountain j math 32 2002 759 792

une interpretation algebrique de la transformation de laplace et des matrices de transfert abstract the tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where even in the time varying case a formal laplace transform and the transfer matrix are most naturally defined several classic problems are examined in this algebraic setting the relationship between left right coprime matrix decomposition and controllability observability the state variable canonical realization the transfer algebra with respect to parallel and series connections the input output inversion and model matching

q difference equations for askey wilson type integrals via q polynomials abstract in this paper we show how to deduce askey wilson type integrals by the method of q difference equation in addition we construct the relation between bilinear generating functions and solutions of q difference equation more over we generalize bailey s 6 6 summation from the perspective of q integral by the method of q difference equation at last we deduce u n 1 type generating functions for al salam carlitz polynomials by the method of q difference equation

poisson kernel and multilinear generating functions of some orthogonal polynomials abstract multilinear generatings functions for the polynomials r 0 n n r a q r b q n r x r are derived which on specialization yield the poisson kernel and multilinear generating functions for the orthogonal polynomials like continuous q ultraspherical al salam chihara q random walk etc

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

model matching and decoupling for continuous and discrete time linear time varying systems solutions to the exact model matching and block decoupling problems for both continuous and discrete time linear time varying systems are presented the parametrisation of the whole class of proper solutions is given for the decoupling the minimal delay problem is also considered in a time varying setting the approach is algebraic and based on the smith macmillan form at infinity of a transfer matrix of a time varying system which has been recently introduced in systems theory this avoids the difficulties related to the inversion of the transfer matrices with entries in non commutative fields over which the determinants of dieudonne or ore type are much more complicated the solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown

applications of q lagrange inversion to basic hypergeometric series a family of g lagrange inversion formulas is given special cases include quadratic and cubic transformations for basic hypergeometric series the g analogs of the so called strange evaluations are also corollaries some new rogers ramanujan identities are given a connection between the work of rogers and andrews and lagrange inversion is stated

the exact model matching problem for linear time varying systems an algebraic approach the exact model matching problem is formulated and solved for linear time varying systems the condition for the existence of a proper solution which is well known in the time invariant case is proven here to still be valid in the time varying case the properness is characterized using the smith macmillan form at infinity recently defined by the authors for the transfer matrices with time varying coefficients

some multilinear generating functions for q hermite polynomials abstract as an application of the transformation theory of basic or q hypergeometric series a number of multilinear generating functions are developed for the q hermite polynomials studied by l j rogers g szego and others relevant connections of these general multilinear generating functions with various known results for the classical or basic hermite polynomials are also indicated

generalizations of certain carlitz s trilinear and srivastava agarwal type generating functions abstract in this paper a class of trilinear generating functions for generalized hahn polynomials is generalized in addition a question of carlitz is answered and some known results are deduced as special cases moreover some expansions of srivastava agarwal type generating functions for hahn polynomials are gained finally additional proofs of certain carlitz s generating functions are given

an operator approach to the al salam carlitz polynomials we present an operator approach to rogers type formulas and mehler s formula for the al salam carlitz polynomials un x y a q by using the q exponential operator we obtain a rogers type formula which leads to a linearization formula with the aid of a bivariate augmentation operator we get a simple derivation of mehler s formula due to al salam and carlitz some orthogonal q polynomials math nachr 30 47 1965 by means of the cauchy companion augmentation operator we obtain an equivalent form of mehler s formula we also give several identities on the generating functions for products of the al salam carlitz polynomials which are extensions of the formulas for the rogers szego polynomials

notes on askey roy integral and certain generating functions for q polynomials abstract in this paper generalizations of askey roy integral are given by the technique of transformation in addition a recurring formula for askey roy integral and an integral involving cauchy polynomial are deduced by askey roy integral besides a new transformation identity is gained by askey roy integral moreover a certain generating function for generalized al salam carlitz polynomials is obtained by askey roy integral at last two generalizations of u n 1 type generating function for generalized al salam carlitz polynomials are derived by askey roy integral

infinite structure and exact model matching problem a geometric approach we first generalize to a b c d quadruples the geometric characterization of the infinite zero structure we then give a new geometric proof for a solvability condition of the exact model matching problem expressed in terms of this concept and previously established in the polynomial context by morse verghese and pernebo this emphasizes the approach independent character of the infinite zero structure

bivariate generating functions for rogers szego polynomials abstract in the present paper we utilize the general q exponential operators to derive several carlitz type bivariate generating functions for rogers szego polynomials moreover we give an equivalent expansion formula of a certain bivariate generating functions for rogers szego polynomials and propose an open problem

weighted chi squared tests for partial common principal component subspaces summary we consider tests of the null hypothesis that g covariance matrices have a partial common principal component subspace of dimension s our approach uses a dimensionality matrix which has its rank equal to s when the hypothesis holds the test can then be based on a statistic computed from the eigenvalues of an estimate of this dimensionality matrix the asymptotic distribution of this statistic is that of a linear combination of independent one degree of freedom chi squared random variables simulation results indicate that this test yields significance levels that come closer to the nominal level than do those of a previously proposed method the procedure is also extended to a test that g correlation matrices have a partial common principal component subspace

asymptotic properties of a correlation matrix under a two step monotone incomplete sample abstract this paper derives an asymptotic distribution for a correlation matrix in the context of a two step monotone incomplete sample drawn from n p q a multivariate normal population with mean and covariance matrix based on the result we perform hypothesis testing for the correlation matrix and investigate its accuracy using numerical simulations

brief paper output feedback pole placement for linear time varying systems with application to the control of nonlinear systems the output feedback pole placement problem is solved in an input output algebraic formalism for linear time varying ltv systems the recent extensions of the notions of transfer matrices and poles of the system to the case of ltv systems are exploited here to provide constructive solutions based as in the linear time invariant lti case on the solutions of diophantine equations also differences with the results known in the lti case are pointed out especially concerning the possibilities to assign specific dynamics to the closed loop system and the conditions for tracking and disturbance rejection this approach is applied to the control of nonlinear systems by linearization around a given trajectory several examples are treated in detail to show the computation and implementation issues

proper rational matrix diophantine equations and the exact model matching problem relying on the theory behind the smith macmillan form of a rational matrix at s infty a necessary and sufficient condition is derived from the solvability of matrix diophantine equations of the form a s x s b s y s m s where a s b s and m s are given proper rational matrices and x s and y s are unknown proper rational matrices it is shown that the above result can be used in order to resolve in a new illuminating way the exact model matching problem emmp if a solution to emmp exists then the family of all solutions is parametrized

principal component models for correlation matrices summary distributional theory regarding principal components is less well developed for correlation matrices than it is for covariance matrices the intent of this paper is to reduce this disparity methods are proposed that enable investigators to fit and to make inferences about flexible principal components models for correlation matrices the models allow arbitrary eigenvalue multiplicities and allow the distinct eigenvalues to be modelled parametrically or nonparametrically local parameterisations and implicit functions are used to construct full rank unconstrained parameterisations first order asymptotic distributions are obtained directly from the theory of estimating functions second order accurate distributions for making inferences under normality are obtained directly from likelihood theory simulation studies show that the bartlett correction is effective in controlling the size of the tests and that first order approximations to nonnull distributions are reasonably accurate the methods are illustrated on a dataset

expansions in the askey wilson polynomials abstract we give a general expansion formula of functions in the askey wilson polynomials and using askey wilson orthogonality we evaluate several integrals moreover we give a general expansion formula of functions in polynomials of askey wilson type which are not necessarily orthogonal limiting cases give expansions in little and big q jacobi type polynomials we also give a new generating function for askey wilson polynomials and a new evaluation for specialized askey wilson polynomials

vibrational feedback control zeros placement capabilities it is shown that in addition to closed loop pole placement vibrational feedback controllers lead in the sense specified below to a possibility of open loop zeros assignability on this basis the superior performance characteristics of continuous time periodic controllers discovered in 3 6 are explained

a transfer function approach to linear time varying discrete time systems in the first part of the paper a transfer function approach is developed for the class of linear time varying discrete time systems the theory is specified in terms of skew noncommutative rings of polynomials and formal power series both with coefficients in a ring of time functions the transfer function matrix is defined to be a matrix whose entries belong to a skew ring of formal power series it is shown that various system properties such as asymptotic stability can be characterized in terms of the skew ring framework in the last part of the paper the transfer function framework is applied to the study of feedback control new results are obtained on assignability of system dynamics by using dynamic output feedback and dynamic state feedback the results are applied to the control of an armature controlled do motor with a variable loading

on carlitz s trilinear generating functions abstract in this paper carlitz s trilinear generating functions for rogers szego and hahn polynomials are obtained by using carlitz s q operator in addition an equivalent proof of srivastava jain s mixed trilinear generating function is gained moreover a carlitz s trilinear generating function is deduced at last an application to the multiple carlitz s generating functions for hahn polynomials is given

some q generating functions of the carlitz and srivastava agarwal types associated with the generalized hahn polynomials and the generalized rogers szego polynomials motivated essentially by the fact that generating functions play an important role in the investigation of many potentially useful properties of the sequences which they generate various families of q generating functions of the carlitz and srivastava agarwal types have been investigated in the literature for the continuous q hermite polynomials the hahn polynomials the rogers szego polynomials and indeed also for many other q polynomial systems the main object of this paper is to apply a certain q exponential decomposition technique with a view to deriving several bilinear and trilinear q generating functions associated with the generalized hahn polynomials as a simple consequence of one of the main results derived in this paper we are led naturally to the corrected version of a bilinear q generating function which appeared in one of carlitz s papers

set of open loop block diagonalizers of transfer matrices abstract the set of open loop right block diagonalizers of a given p m transfer matrix z with specified block sizes is studied a characterization of this set is obtained in terms of a maximal open loop block diagonalizer and a minimal proper basis of the kernel of z the relevance of the results to some non interacting control problems is pointed out in particular the results obtained yield a minimum delay solution to the block decoupling problem using dynamic state feedback

on the minimum delay problem in this note we solve the following problem given a full column rank p m transfer function matrix find the minimum non negative integers li i 1 m such that the equation g z f z diagz lstaggeredi z lstaggeredm has a proper solution g z

a note on moment integrals and some applications abstract in this paper two moment integrals are given by the method of transformation in addition generalizations of sears s transformation are obtained by moment integrals moreover certain q mehler formulas for rogers szego polynomials are gained by moment integrals besides an open problem of trilinear generating function is deduced by moment integrals at last generalizations of u n 1 type kalnins miller transformation are achieved by moment integrals

two new q exponential operator identities and their applications abstract in this paper we use the q chu vandermonde formula to prove two new operator identities which are the extensions of liu s results these two q exponential operator identities are used to derive some q summation formulas and q integrals

new proofs of generating functions for rogers szego polynomials abstract in this paper we show how to use the q exponential operator to prove four generating functions for rogers szego polynomials of l carlitz generating functions for certain q orthogonal polynomials collectanea math 23 1972 91 104 in addition four auxiliary ones under the replacement q q 1 are derived by this method