the q version of a theorem of bochner askey and wilson 1985 found a family of orthogonal polynomials in the variable s k 1 2 k 1 k that satisfy a q difference equation of the form a k p n s qk p n s k b k p n s k q p n s k theta n p n s k n 0 1 we show here that this property characterizes the askey wilson polynomials the proof is based on an operator identity of independent interest this identity can be adapted to prove other characterization results indeed it was used in grunbaum and haine 1996 to give a new derivation of the result of bochner alluded to in the title of this paper we give the appropriate identity for the case of difference equations leading to the wilson polynomials but pursue the consequences only in the case of q difference equations leading to the askey wilson and big q jacobi polynomials this approach also works in the discrete case and should yield the results in leonard 1982

some multivariable orthogonal polynomials of the askey tableau continuous families a multivariable generalization is presented for all the continuous families of the askey tableaux this significantly extends some previously known multivariable jacobi polynomials the latter are recovered as a limit case from a family of multivariable continuous hahn polynomials which in turn are obtained as a limit of some multivariable wilson polynomials a continuous analog of the single variable charlier polynomials is also introduced

on uncontrollable discrete time bilinear systems which are nearly controllable in this note for a class of uncontrollable discrete time bilinear systems it is shown that the controllable region nearly covers the whole space while the uncontrollable region is only a hypersurface as a result for almost any initial state and any terminal state of the system the former can be transferred to the latter in addition the two dimensional controllability counterexamples in are generalized to arbitrary finite dimensional cases

on near controllability and stabilizability of a class of discrete time bilinear systems in this paper the near controllability and stabilizability of a class of discrete time bilinear systems are studied a necessary and sufficient condition for the near controllability of the systems is first derived then on the basis of the corresponding near controllability criterion the stabilizability problem of the systems is solved in turn

on a class of bivariate second order linear partial difference equations and their monic orthogonal polynomial solutions abstract in this paper we classify the bivariate second order linear partial difference equations which are admissible potentially self adjoint and of hypergeometric type using vector matrix notation explicit expressions for the coefficients of the three term recurrence relations satisfied by monic orthogonal polynomial solutions are obtained in terms of the coefficients of the partial difference equation finally we make a compilation of the examples existing in the literature belonging to the class analyzed in this paper namely bivariate charlier meixner kravchuk and hahn orthogonal polynomials

stochastic optimization on continuous domains with finite time guarantees by markov chain monte carlo methods we introduce bounds on the finite time performance of markov chain monte carlo mcmc algorithms in solving global stochastic optimization problems defined over continuous domains it is shown that mcmc algorithms with finite time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm this work is inspired by the concept of finite time learning with known accuracy and confidence developed in statistical learning theory

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

on controllability and near controllability of discrete time inhomogeneous bilinear systems without drift in this paper controllability and near controllability of discrete time inhomogeneous bilinear systems without drift are studied necessary and sufficient conditions for controllability and near controllability of the systems are respectively presented in particular algorithms for computing the control inputs to achieve the transition of a given pair of states for the controllable systems and nearly controllable systems are also provided examples are shown to demonstrate the conceptions and results of the paper

bivariate bannai ito polynomials a two variable extension of bannai ito polynomials is presented they are obtained via q 1 limits of the bivariate q racah and askey wilson orthogonal polynomials introduced by gasper and rahman their orthogonality relation is obtained these new polynomials are also shown to be multispectral two dunkl shift operators are seen to be diagonalized by the bivariate bannai ito polynomials and 3 and 9 term recurrence relations are provided a two variable extension of bannai ito polynomials is presented they are obtained via q 1 limits of the bivariate q racah and askey wilson orthogonal polynomials introduced by gasper and rahman their orthogonality relation is obtained these new polynomials are also shown to be multispectral two dunkl shift operators are seen to be diagonalized by the bivariate bannai ito polynomials and 3 and 9 term recurrence relations are provided

the q onsager algebra and multivariable q special functions two sets of mutually commuting q difference operators x i and y j i j 1 n such that x i and y i generate a homomorphic image of the q onsager algebra for each i are introduced the common polynomial eigenfunctions of each set are found to be entangled product of elementary pochhammer functions in n variables and n 3 parameters under certain conditions on the parameters they form two dual bases of polynomials in n variables the action of each operator with respect to its dual basis is block tridiagonal the overlap coefficients between the two dual bases are expressed as entangled products of q racah polynomials and satisfy an orthogonality relation the overlap coefficients between either one of these bases and the multivariable monomial basis are also considered one obtains in this case entangled products of dual q krawtchouk polynomials finally the split basis in which the two families of operators act as block bidiagonal matrices is also provided

discrete orthogonal polynomials and difference equations of several variables the goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions under some mild assumptions we give a complete solution of the problem

analysis still matters a surprising instance of failure of runge kutta felberg ode solvers this paper provides a nice example to illustrate that without supporting analysis a numerical simulation may lead to incorrect conclusions we explore a pedagogical example of failure of runge kutta felberg rkf algorithms for a simple dynamical system that models the coupling of two oscillators although the system appears to be well behaved the explicit rkf solvers provide erratic numerical solutions the mode of failure is based in a period doubling route to chaos due to the existence of stable linear solutions in the problem

the multivariate hahn polynomials and the singular oscillator karlin and mcgregor s d variable hahn polynomials are shown to arise in the dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in cartesian and hyperspherical coordinates these polynomials in d discrete variables depend on real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution the focus is put on the d 2 case for which the connection with the three dimensional singular oscillator is used to derive the main properties of the polynomials forward backward shift operators orthogonality relation generating function recurrence relations bispectrality difference equations and explicit expression in terms of the univariate hahn polynomials the extension of these results to an arbitrary number of variables is presented at the end of the paper

hard frequency domain model error bounds from least squares like identification techniques the problem of deriving so called hard error bounds for estimated transfer functions is addressed a hard bound is one that is sure to be satisfied i e the true system nyquist plot will be confined with certainty to a given region provided that the underlying assumptions are satisfied by blending a priori knowledge and information obtained from measured data it is shown how the uncertainty of transfer function estimates can be quantified the emphasis is on errors due to model mismatch the effects of unmodeled dynamics can be considered as bounded disturbances hence techniques from set membership identification can be applied to this problem the approach taken corresponds to weighted least squares estimation and provides hard frequency domain transfer function error bounds the main assumptions used in the current contribution are that the measurement errors are bounded that the true system is indeed linear with a certain degree of stability and that there is some knowledge about the shape of the true frequency response

cyclic tridiagonal pairs higher order onsager algebras and orthogonal polynomials abstract the concept of cyclic tridiagonal pairs is introduced and explicit examples are given for a fairly general class of cyclic tridiagonal pairs with cyclicity n we associate a pair of divided polynomials the properties of this pair generalize the ones of tridiagonal pairs of racah type the algebra generated by the pair of divided polynomials is identified as a higher order generalization of the onsager algebra it can be viewed as a subalgebra of the q onsager algebra for a proper specialization at q the primitive 2 n th root of unity orthogonal polynomials beyond the leonard duality are revisited in light of this framework in particular certain second order dunkl shift operators provide a realization of the divided polynomials at n 2 or q i

structured matrix norms for robust stability and performance with block structured uncertainty in this paper we introduce new lower and upper robust stability bounds for structured uncertainty involving arbitrary spatial norms specifically we consider a norm bounded block structured uncertainty characterization wherein the defining spatial norm is not necessarily the maximum singular value this new uncertainty characterization leads to the notion of structured matrix norms for characterizing the allowable size of the nominal transfer function for robust stability the lower and upper bounds are specialized to specific matrix norms including holder unitarily invariant and induced norms to provide conditions for robust stability with several different characterizations of plant uncertainty one of the key advantages of the proposed approach over the structured singular value is the reduction is computational complexity gained by directly addressing a given uncertainty characterization without having to transform it to a spectral norm type characterization finally we introduce a performance blo

on small controllability and controllability of a class of nonlinear systems in this paper small controllability and controllability of a class of nonlinear systems are studied both the discrete time case and continuous time case are considered by proposing an implicit function approach sufficient conditions for the systems respectively to be small controllable and controllable are obtained examples are also provided to illustrate the results of the paper

on controllability of homogeneous and inhomogeneous discrete time multi input bilinear systems in dimension two abstractin this paper the controllability problem of two dimensional discrete time multi input bilinear systems is completely solved the homogeneous and the inhomogeneous cases are studied separately and necessary and sufficient conditions for controllability are established by using a linear algebraic method which are easy to apply moreover for the uncontrollable systems near controllability is considered and similar necessary and sufficient conditions are also obtained finally examples are provided to demonstrate the results of this paper

brief robust nyquist array analysis based on uncertainty descriptions from system identification a robust nyquist array analysis for mimo systems is proposed based on uncertainty descriptions obtained from system identification two types of statistical based uncertainty error bounds for the frequency response are obtained element bounds and column bounds gershgorin s theorem and the concepts of diagonal dominance and gershgorin bands are extended to include model uncertainty robust stability theorems are developed based on ellipsoidal uncertainty descriptions obtained from system identification an example is given to illustrate the robust nyquist array analysis

comments on a controllability counterexample and the continuation lemma a technical note in this journal vol 50 no 6 pp 840 841 june 2005 by elliott gives a bilinear example showing that the euler discretization of a noncontrollable continuous time system can be controllable the example is correct but there was a flaw in a result of the tn lemma 1 for discrete time systems local controllability implies controllability that has independent interest in this note the lemma is reformulated as a conjecture for continuous in state systems and it is also proved under additional conditions for a class of two dimensional bilinear systems the euler discretization is shown directly to be small controllable a fortiori controllable

some functions that generalize the askey wilson polynomials we determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q difference operator the solution is described in terms of an arbitrary solution of a q analogue of gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the askey wilson polynomials there is some evidence that this bispectral problem for an arbitrary order q difference operator is intimately related with some q deformation of the toda lattice hierarchy and its virasoro symmetries when tridiagonal matrices are replaced by the schroedinger operator and q 1 this statement holds with toda replaced by kdv in this context this paper determines the analogs of the bessel and airy potentials

a controllability counterexample simulation by time discretizations can be qualitatively misleading as counterexamples a class of noncontrollable single input two dimensional bilinear control systems is presented whose euler discretizations are controllable on the punctured plane

a bispectral q hypergeometric basis for a class of quantum integrable models for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain class of parameters and if the n variables belong to a discrete support in this case the bispectral property finds a natural interpretation within the framework of tridiagonal pairs in the third part eigenfunctions of the q dolan grady hierarchy are considered in the polynomial basis in particular invariant subspaces are identified for certain conditions generalizing nepomechie s relations in the fourth part the analysis is extended to the special case q 1 this framework provides a q hypergeometric formulation of quantum integrable models such as the open xxz spin chain with generic integrable boundary conditions q 1 for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain

macdonald polynomials and algebraic integrability we construct explicitly nonpolynomial eigenfunctions of the difference operators by macdonald in the case t qk k z this leads to a new more elementary proof of several macdonald conjectures proved first by cherednik we also establish the algebraic integrability of macdonald operators at t qk k z generalizing the result of etingof and styrkas our approach works uniformly for all root systems including the bcn case and related koornwinder polynomials moreover we apply it for a certain deformation of the an root system where the previously known methods do not work

orthogonality of macdonald polynomials with unitary parameters for any admissible pair of irreducible reduced crystallographic root systems we present discrete orthogonality relations for a finite dimensional system of macdonald polynomials with parameters on the unit circle subject to a truncation relation

bispectral commuting difference operators for multivariable askey wilson polynomials we construct a commutative algebra a z generated by d algebraically independent q difference operators acting on variables z 1 z 2 z d which is diagonalized by the multivariable askey wilson polynomials p n z considered by gasper and rahman 2005 iterating sears 4 3 transformation formula we show that the polynomials p n z possess a certain duality between z and n analytic continuation allows us to obtain another commutative algebra a n generated by d algebraically independent difference operators acting on the discrete variables n 1 n 2 n d which is also diagonalized by p n z this leads to a multivariable q askey scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions

the higher rank q deformed bannai ito and askey wilson algebra the q deformed bannai ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra mathfrak osp q 1 vert 2 it turned out to be isomorphic to the askey wilson algebra in the present paper these results will be extended to higher rank the rank n 2 q bannai ito algebra mathcal a n q which by the established isomorphism also yields a higher rank version of the askey wilson algebra is constructed in the n fold tensor product of mathfrak osp q 1 vert 2 an explicit realization in terms of q shift operators and reflections is proposed which will be called the mathbb z 2 n q dirac dunkl model the algebra mathcal a n q is shown to arise as the symmetry algebra of the constructed mathbb z 2 n q dirac dunkl operator and to act irreducibly on modules of its polynomial null solutions an explicit basis for these modules is obtained using a q deformed mathbf ck extension and fischer decomposition

asymptotic variance expressions for identified black box transfer function models identification of black box transfer function models is considered it is assumed that the transfer function models possess a certain shift property which is satisfied for example by all polynomial type models expressions for the variances of the transfer function estimates are derived that are asymptotic both in the number of observed data and in the model orders the result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance respectively is proportional to the inverse of the joint spectrum matrix for the input and driving noise multiplied by the spectrum of the additive output noise the factor of proportionality is the ratio of model order to number of data this result is independent of the particular model structure used the result is applied to evaluate the performance degradation due to variance for a number of typical model uses some consequences for input design are also drawn

diagonal dominance for robust stability of mimo interval systems this paper provides sufficient conditions for robust stability of a multivariable interval feedback system the result is based on the robustness of the well known diagonal dominance of the system return difference matrix which allows for a simple application of the nyquist criterion to a multi input multi output mimo system

the askey wilson algebra and its avatars the original askey wilson algebra introduced by zhedanov encodes the bispectrality properties of the eponym polynomials the name askey wilson algebra is currently used to refer to a variety of related structures that appear in a large number of contexts we review these versions sort them out and establish the relations between them we focus on two specific avatars the first is a quotient of the original zhedanov algebra it is shown to be invariant under the weyl group of type d 4 and to have a reflection algebra presentation the second is a universal analogue of the first one it is isomorphic to the kauffman bracket skein algebra kbsa of the four punctured sphere and to a subalgebra of the universal double affine hecke algebra c 1 vee c 1 this second algebra emerges from the racah problem of u q mathfrak sl 2 and is related via an injective homomorphism to the centralizer of u q mathfrak sl 2 in its threefold tensor product how the artin braid group acts on the incarnations of this second avatar through conjugation by r matrices in the racah problem or half dehn twists in the diagrammatic kbsa picture is also highlighted attempts at defining higher rank askey wilson algebras are briefly discussed and summarized in a diagrammatic fashion

controllability of nonlinear discrete time systems a lie algebraic approach this paper presents a geometric study of controllability for discrete time nonlinear systems various accessibility properties are characterized in terms of lie algebras of vector fields some of the results obtained are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear