the q laplace operator and q harmonic polynomials on the quantum vector space the aim of this paper is to study q harmonic polynomials on the quantum vector space generated by q commuting elements x1 x2 xn they are defined as solutions of the equation qp 0 where p is a polynomial in x1 x2 xn and the q laplace operator q is determined in terms of q derivatives the projector hm am hm is constructed where am and hm are the spaces of homogeneous of degree m polynomials and q harmonic polynomials respectively by using these projectors a q analog of classical associated spherical harmonics is constructed they constitute an orthonormal basis of hm a q analog of separation of variables is given representations of the nonstandard q deformed algebra uq son which plays the role of the rotation group so n in the case of classical harmonic polynomials on the spaces hm are explicitly constructed

kernel weighted gmm estimators for linear time series models abstract this paper analyzes the higher order asymptotic properties of generalized method of moments gmm estimators for linear time series models using many lags as instruments a data dependent moment selection method based on minimizing the approximate mean squared error is developed in addition a new version of the gmm estimator based on kernel weighted moment conditions is proposed it is shown that kernel weighted gmm estimators can reduce the asymptotic bias compared to standard gmm estimators kernel weighting also helps to simplify the problem of selecting the optimal number of instruments a feasible procedure similar to optimal bandwidth selection is proposed for the kernel weighted gmm estimator

asymmetric simple exclusion process with open boundaries and askey wilson polynomials we study the one dimensional asymmetric simple exclusion process asep with open boundary conditions particles are injected and ejected at both boundaries it is clarified that the steady state of the model is intimately related to the askey wilson polynomials the partition function and the n point functions are obtained in the integral form with four boundary parameters the thermodynamic current is evaluated to confirm the conjectured phase diagram

comparative lagrangian formulations for localized buckling an energy functional for a strut on a nonlinear softening foundation is worked into two different lagrangian forms in fast and slow space respectively the developments originate independently of the underlying differential equation and carry some quite general features in each case the kinetic energy is an indefinite quadratic form in fast space this leads to an escape phenomenon with fractal properties in slow space kinetic energy is added to a potential contribution that is familiar from modal formulations together and in conjunction with a recent set of numerical experiments they illustrate the extra complexities of localized as opposed to distributed periodic buckling

on the existence of solutions for a class of fourth order differential equations abstract we study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations which involve particular cases of the extended fisher kolmogorov and swift hohenberg equations we address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution the main result follows from an iterative method using the sequence of solutions obtained through the mountain pass technique in order to illustrate this method some numerical examples are shown

bootstrap inference for instrumental variable models with many weak instruments this study s main contribution is to theoretically analyze the application of bootstrap methods to instrumental variable models when the available instruments may be weak and the number of instruments goes to infinity with the sample size we demonstrate that a standard residual based bootstrap procedure cannot consistently estimate the distribution of the limited information maximum likelihood estimator or fuller 1977 estimator under many many weak instrument sequence the primary reason is that the standard procedure fails to capture the instrument strength in the sample adequately in addition we consider the restricted efficient re bootstrap of davidson and mackinnon 2008 2010 2014 that generates bootstrap data under the null restricted and uses an efficient estimator of the coefficient of the reduced form equation efficient we find that the re bootstrap is also invalid however it effectively mimics more key features in the limiting distributions of interest and thus is less distorted in finite samples than the standard bootstrap procedure finally we propose modified bootstrap procedures that provide a valid distributional approximation for the two estimators with many many weak instruments a monte carlo experiment shows that hypothesis testing based on the asymptotic normal approximation can have severe size distortions in finite samples instead our modified bootstrap procedures greatly reduce these distortions

the classification of leonard triples of qracah type abstract let k denote an algebraically closed field let v denote a vector space over k with finite positive dimension by a leonard triple on v we mean an ordered triple of linear transformations in end v such that for each of these transformations there exists a basis of v with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal there is a family of leonard triples said to have qracah type this is the most general type of leonard triple we classify the leonard triples of qracah type up to isomorphism we show that any leonard triple of qracah type satisfies the z 3 symmetric askey wilson relations

an invariant of regular isotopy this paper studies a two variable laurent polynomial invariant of regular isotopy for classical unoriented knots and links this invariant is denoted lk for a link k and it satisfies the axioms 1 regularly isotopic links receive the same polynomial 2 lo0 1 3 l al l a l 4 l small diagrams indicate otherwise identical parts of larger diagrams regular isotopy is the equivalence relation generated by the reidemeister moves of type ii and type iii invariants of ambient isotopy are obtained from l by writhe normalization

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

center of quantum algebra uq so 3 we analyze the structure of the center of the quantum algebra uq so 3 this structure as expected depends substantially on the deformation parameter q when q is not a root of unity there exists only one casimir element which is the deformation of ordinary casimir element known from so3 the center has a structure of a ring of polynomials of one variable when qn 1 there are three more casimir elements of the form of polynomials in algebra generators uq so 3 can be seen as a finite dimensional module over the commutative ring of polynomials in these three casimir elements knowing three parametrical family of n dimensional irreducible representations we prove that the dimension of this module is n3 all four casimir elements are no longer algebraically independent we present the explicit description of the central variety that is polynomial dependence between these four casimir elements this dependence differs for n 2m 1 n 2 2m 1 and n 4m

least squares model averaging this paper considers the problem of selection of weights for averaging across least squares estimates obtained from a set of models existing model average methods are based on exponential akaike information criterion aic and bayesian information criterion bic weights in distinction this paper proposes selecting the weights by minimizing a mallows criterion the latter an estimate of the average squared error from the model average fit we show that our new mallows model average mma estimator is asymptotically optimal in the sense of achieving the lowest possible squared error in a class of discrete model average estimators in a simulation experiment we show that the mma estimator compares favorably with those based on aic and bic weights the proof of the main result is an application of the work of li 1987

degenerations of sklyanin algebra and askey wilson polynomials a new trigonometric degeneration of the sklyanin algebra is found and the functional realization of its representations in space of polynomials in one variable is studied a further contraction gives the standard quantum algebra uq sl 2 it is shown that the degenerate sklyanin algebra contains a subalgebra isomorphic to algebra of functions on the quantum sphere su 2 so 2 q12 the diagonalization of general quadratic form in its generators leads in the functional realization to the difference equation for askey wilson polynomials

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

quantum deformations of su 2 explicit representations of two deformations of su 2 one in the cartan basis and one in the cartesian basis are constructed the cartan deformation subsumes previously known cases while the cartesian deformation appears to be new in the latter case explicit reversible mappings in terms of representations of su 2 and a casimir for the algebra are constructed

on the classification of irreducible finite dimensional representations of uq so3 algebra in an earlier work m havlicek et al j math phys 40 2135 1999 we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation uq so3 of the lie algebra so3 where q is not a root of unity for each dimension only one of them called classical admits limit q 1 in the first part of this paper we show that any finite dimensional irreducible representation is equivalent to some of these representations in the case qn 1 we derive new casimir elements of uq so3 and show that a dimension of any irreducible representation is not higher than n these elements are casimir elements of uq som for all m and even of uq isom 1 due to inonu wigner contraction according to the spectrum of one of the generators the representations are found to belong to two main disjoint sets we give full classification and explicit formulas for all representations from the first set we call them nonsingular representations if n is odd we have full classificatio

a higher rank racah algebra and the mathbb z _2 n laplace dunkl operator a higher rank generalization of the rank one racah algebra is obtained as the symmetry algebra of the laplace dunkl operator associated to the mathbb z 2 n root system this algebra is also the invariance algebra of the generic superintegrable model on the n sphere bases of dunkl harmonics are constructed explicitly using a cauchy kovalevskaia theorem these bases consist of joint eigenfunctions of labelling abelian subalgebras of the higher rank racah algebra a method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented

representations of the cyclically symmetric q deformed algebra soq 3 an algebra homomorphism from the nonstandard q deformed cyclically symmetric algebra uq so3 to the extension uq sl2 of the hopf algebra uq sl2 is constructed not all irreducible representations of uq sl2 can be extended to representations of uq sl2 composing the homomorphism with irreducible representations of uq sl2 we obtain representations of uq so3 not all of these representations of uq so3 are irreducible reducible representations of uq so3 are decomposed into irreducible components in this way we obtain all irreducible representations of uq so3 when q is not a root of unity a part of these representations turns into irreducible representations of the lie algebra so3 when q 1 representations of the other part have no classical analog using the homomorphism it is shown how to construct tensor products of finite dimensional representations of uq so3 irreducible representations of uq so3 when q is a root of unity are constructed some of them are obtained from irreducible re

on the periodic orbits of the fourth order differential equation u qu u ef u u u u we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu u ef u u u u where q and e are real parameters e is small and f is a nonlinear function

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

quadratic harnesses q commutations and orthogonal martingale polynomials we introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a commutation relation this implies that quadratic harnesses are essentially determined uniquely by five numerical constants explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest

on the limit cycles for a class of fourth order differential equations we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation where p is a rational number different from 0 e is small and f is a nonlinear function

twisted clebsch gordan coefficients for suq 2 a new approach to the clebsch gordan problem for suq 2 is proposed the twisted clebsch gordan coefficients tcgc are defined to be the overlaps between twisted connected and unconnected bases being the q analogues of rotated bases in standard su 2 the explicit expression for tcgc is found in terms of the basic hypergeometric function 4 phi 3 racah q polynomials

invariants of 2 1 gravity in 1 2 we established and dicussed the algebra of observables for 2 1 gravity at both the classical and quantum level here our treatment broadens and extends previous results to any genusg with a systematic discussion of the centre of the algebra the reduction of the number of independent observables to 6g 6 g 1 is treated in detail with a precise classification forg 1 andg 2

normalized leonard pairs and askey wilson relations abstract let v denote a vector space with finite positive dimension and let a a denote a leonard pair on v as is known the linear transformations a a satisfy the askey wilson relations a 2 a aa a a a 2 aa a a a a 2 a i a 2 a a aa aa 2 a a aa a a 2 a i for some scalars the scalar sequence is unique if the dimension of v is at least 4 if c c t t are scalars and t t are not zero then ta c t a c is a leonard pair on v as well these affine transformations can be used to bring the leonard pair or its askey wilson relations into a convenient form this paper presents convenient normalizations of leonard pairs by the affine transformations and exhibits explicit askey wilson relations satisfied by them

model averaging by jackknife criterion in models with dependent data abstract the past decade witnessed a literature on model averaging by frequentist methods for the most part the asymptotic optimality of various existing frequentist model averaging estimators has been established under i i d errors recently hansen and racine hansen b e racine j 2012 jackknife model averaging journal of econometrics 167 38 46 developed a jackknife model averaging jma estimator which has an important advantage over its competitors in that it achieves the lowest possible asymptotic squared error under heteroscedastic errors in this paper we broaden hansen and racine s scope of analysis to encompass models with i a non diagonal error covariance structure and ii lagged dependent variables thus allowing for dependent data we show that under these set ups the jma estimator is asymptotically optimal by a criterion equivalent to that used by hansen and racine a monte carlo study demonstrates the finite sample performance of the jma estimator in a variety of model settings

structural localization phenomena and the dynamical phase space analogy the localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints a from a modal perspective b from a formulation which allows the amplitude to modulate in an asymptotically defined slow space and c from a dynamical analogy in phase space suggested by the form of the underlying differential equation a simple strut on an asymmetric nonlinear foundation provides a typical illustrative example the three approaches emphasize different features of the localization phenomenon the modal view illustrates the distinctive effects of boundary conditions the modulated approach generates a convenient second order differential equation in the amplitude function and the dynamical phase space analogy suggests a useful interpretation of localization as a homoclinic connection comparisons are also made with nonlinear numerical solutions as the strut length approaches infinity it is shown that the fully localized solution represents the unstable post buckled state with the lowest energy allowing evaluation of the minimum energy barrier relevant to dynamical impact studies attention is drawn to the possibility of spatial chaos becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection

simultaneous selection and weighting of moments in gmm using a trapezoidal kernel abstract this paper proposes a novel procedure to estimate linear models when the number of instruments is large at the heart of such models is the need to balance the trade off between attaining asymptotic efficiency which requires more instruments and minimizing bias which is adversely affected by the addition of instruments two questions are of central concern 1 what is the optimal number of instruments to use 2 should the instruments receive different weights this paper contains the following contributions toward resolving these issues first i propose a kernel weighted generalized method of moments gmm estimator that uses a trapezoidal kernel this kernel turns out to be attractive to select and weight the number of moments second i derive the higher order mean squared error of the kernel weighted gmm estimator and show that the trapezoidal kernel generates a lower asymptotic variance than regular kernels finally monte carlo simulations show that in finite samples the kernel weighted gmm estimator performs on par with other estimators that choose optimal instruments and improves upon a gmm estimator that uses all instruments

higher rank classical analogs of the askey wilson algebra from the sln onsager algebra the sl n onsager algebra has been introduced by uglov and ivanov in 1995 in this letter a frt presentation of the sl n onsager algebra is given its current algebra and commutative subalgebra are constructed certain quotients of the sl n onsager algebra are then considered which produce classical analogs of higher rank extensions of the askey wilson algebra as examples the cases n 3 and n 4 are described in details

a finite oscillator model with equidistant position spectrum based on an extension of mathfrak su 2 we consider an extension of the real lie algebra su 2 by introducing a parity operator p and a parameter c this extended algebra is isomorphic to the bannai ito algebra with two parameters equal to zero for this algebra we classify all unitary finite dimensional representations and show their relation with known representations of su 2 moreover we present a model for a one dimensional finite oscillator based on the odd dimensional representations of this algebra for this model the spectrum of the position operator is equidistant and coincides with the spectrum of the known su 2 oscillator in particular the spectrum is independent of the parameter c while the discrete position wavefunctions which are given in terms of certain dual hahn polynomials do depend on this parameter

nonsymmetric koornwinder polynomials and duality in the fundamental work of lusztig l on affine hecke algebras a special role is played by the root system of type cn the affine hecke algebra is a deformation of the group algebra of an affine weyl group which usually depends on as many parameters as there are distinct root lengths i e one or two for an irreducible root system however in the cn case the hecke algebra h has three parameters corresponding to the fact that there is a simple coroot which is divisible by 2 recently cherednik c1 c3 has introduced the notion of a double affine hecke algebra and has used it to prove several conjectures on macdonald polynomials these polynomials and cherednik s double affine hecke algebra involve two or three parameters i e one more than the number of root lengths

instrumental variable estimation in the presence of many moment conditions abstract this paper develops shrinkage methods for addressing the many instruments problem in the context of instrumental variable estimation it has been observed that instrumental variable estimators may behave poorly if the number of instruments is large this problem can be addressed by shrinking the influence of a subset of instrumental variables the procedure can be understood as a two step process of shrinking some of the ols coefficient estimates from the regression of the endogenous variables on the instruments then using the predicted values of the endogenous variables based on the shrunk coefficient estimates as the instruments the shrinkage parameter is chosen to minimize the asymptotic mean square error the optimal shrinkage parameter has a closed form which makes it easy to implement a monte carlo study shows that the shrinkage method works well and performs better in many situations than do existing instrument selection procedures

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

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

periodic solutions for extended fisher kolmogorov and swift hohenberg equations by truncature techniques abstract combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

modular leonard triples abstract let k denote a field and let v denote a vector space over k of finite positive dimension an ordered triple a a a of linear operators on v is said to be a leonard triple whenever for each b a a a there exists a basis of v with respect to which the matrix representing b is diagonal and the matrices representing the other two operators are irreducible tridiagonal a leonard triple a a a is said to be modular whenever for each b a a a there exists an antiautomorphism of end v which fixes b and swaps the other two operators we classify the modular leonard triples up to isomorphism

sl n onsager s algebra and integrability we define ansl n analog of onsager s algebra through a finite set of relations that generalize the dolan grady defining relations for the original onsager s algebra this infinite dimensional lie algebra is shown to be isomorphic to a fixed point subalgebra ofsl n loop algebra with respect to a certain involution as the consequence of the generalized dolan grady relations a hamiltonian linear in the generators ofsl n onsager s algebra is shown to posses an infinite number of mutually commuting integrals of motion

transition maps between the 24 bases for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below 1 i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal 2 ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in an earlier paper we described 24 special bases for v one feature of these bases is that with respect to each of them the matrices that represent a and a are i diagonal and irreducible tridiagonal or ii irreducible tridiagonal and diagonal or iii lower bidiagonal and upper bidiagonal or iv upper bidiagonal and lower bidiagonal for each ordered pair of bases among the 24 there exists a unique linear transformation from v to v that sends the first basis to the second basis we call this the transition map in this paper we find each transition map explicitly as a polynomial in a a

the unitary irreducible representations of the quantum heisenberg algebra the goal of this work is to describe the irreducible representations of the quantum heisenberg algebra and the unitary irreducible representation of one of its real forms the solution of this problem is obtained through the investigation of theleft spectrum of the quantum heisenberg algebra using the result about spectra of generic algebras of skew differential operators cf r

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

affine cubic surfaces and character varieties of knots it is known that the fundamental group homomorphism 1 t 2 1 s 3 k 1 t 2 1 s 3 k induced by the inclusion of the boundary torus into the complement of a knot k in s 3 s 3 is a complete knot invariant many classical invariants of knots arise from the natural restriction map induced by the above homomorphism on the sl 2 sl 2 character varieties of the corresponding fundamental groups in our earlier work 3 we proposed a conjecture that the classical restriction map admits a canonical deformation into a two parameter family of affine cubic surfaces in c 3 c 3 in this paper we show that modulo some mild technical conditions our conjecture follows from a known conjecture of brumfiel and hilden 1 on the algebraic structure of the peripheral system of a knot we then confirm the brumfiel hilden conjecture for an infinite class of knots including all torus knots 2 bridge knots and certain pretzel knots we also show the class of knots for which the brumfiel hilden conjecture holds is closed under taking connect sums and knot coverings

linear covariance algebra for slq 2 the askey wilson algebra aw 3 is shown to serve as a hidden covariance algebra for quantum algebra slq 2 the generators of aw 3 are chosen to be linear combinations of slq 2 generators with operator valued coefficients

jackknife model averaging for quantile regressions abstract in this paper we consider model averaging for quantile regressions qr when all models under investigation are potentially misspecified and the number of parameters is diverging with the sample size to allow for the dependence between the error terms and regressors in the qr models we propose a jackknife model averaging jma estimator which selects the weights by minimizing a leave one out cross validation criterion function and demonstrate its asymptotic optimality in terms of minimizing the out of sample final prediction error we conduct simulations to demonstrate the finite sample performance of our estimator and compare it with other model selection and averaging methods we apply our jma method to forecast quantiles of excess stock returns and wages

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

hybrid generalized empirical likelihood estimators instrument selection with adaptive lasso in this paper we use the adaptive lasso estimator to choose the relevant instruments and eliminate the irrelevant instruments the limit theory of zou 2006 is extended from univariate iid case to heteroskedastic and non gaussian data then we use the selected instruments in generalized empirical likelihood estimators gel in this sense these are called hybrid gel it is also shown that the lasso estimators are not model selection consistent whereas the adaptive lasso can select the correct model with fixed number of instruments in simulations we show that hybrid gel estimators have smaller bias and mean squared error than the other estimators in certain cases

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

the quantum algebra uq sl2 and its equitable presentation abstract we show that the quantum algebra u q sl 2 has a presentation with generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1 we call this the equitable presentation we show that y respectively z is not invertible in u q sl 2 by displaying an infinite dimensional u q sl 2 module that contains a nonzero null vector for y respectively z we consider finite dimensional u q sl 2 modules under the assumption that q is not a root of 1 and char k 2 where k is the underlying field we show that y and z are invertible on each finite dimensional u q sl 2 module we display a linear operator that acts on finite dimensional u q sl 2 modules and satisfies 1 x y 1 y z 1 z x on these modules we define using the q exponential function

leonard triples of q racah type and their pseudo intertwiners abstract let f denote a field and let v denote a vector space over f with finite positive dimension pick a nonzero q f such that q 4 1 and let a b c denote a leonard triple on v that has q racah type we show that there exist invertible w w w in end v such that i a commutes with w and w 1 b w c ii b commutes with w and w 1 c w a iii c commutes with w and w 1 a w b moreover each of w w w is unique up to multiplication by a nonzero scalar in f we show that the three elements w w w w w w mutually commute and their product is a scalar multiple of the identity a number of related results are obtained we call w w w the pseudo intertwiners for a b c

revisiting the askey wilson algebra with the universal r matrix of uq su 2 a description of the embedding of a centrally extended askey wilson algebra aw 3 in uq sl2 3 is given in terms of the universal r matrix of uq sl2 the generators of the centralizer of uq sl2 in its three fold tensor product are naturally expressed through conjugations of casimir elements with r they are seen as the images of the generators of aw 3 under the embedding map by showing that they obey the aw 3 relations this is achieved by introducing a natural coaction also constructed with the help of the r matrix

choosing the number of instruments properties of instrumental variable estimators are sensitive to the choice of valid instruments even in large cross section applications in this paper we address this problem by deriving simple mean square error criteria that can be minimized to choose the instrument set we develop these criteria for two stage least squares 2sls limited information maximum likelihood liml and a bias adjusted version of 2sls b2sls we give a theoretical derivation of the mean square error and show optimality in monte carlo experiments we find that the instrument choice generally yields an improvement in performance also in the angrist and krueger 1991 returns to education application when the instrument set is chosen in the way we consider it turns out that both 2sls and liml give similar large returns to education

periodic orbits of the fourth order non autonomous differential equation u qu pu ef t u u u u abstract we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu pu e f t u u u u where q p and e are real parameters e is small and f is a nonlinear non autonomous periodic function with respect to t moreover we provide some applications

unified theory of exactly and quasiexactly solvable discrete quantum mechanics i formalism we present a simple recipe to construct exactly and quasiexactly solvable hamiltonians in one dimensional discrete quantum mechanics in which the schrodinger equation is a difference equation it reproduces all the known ones whose eigenfunctions consist of the askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable the recipe also predicts several new ones an essential role is played by the sinusoidal coordinate which generates the closure relation and the askey wilson algebra together with the hamiltonian the relationship between the closure relation and the askey wilson algebra is clarified

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

from quantum affine symmetry to the boundary askey wilson algebra and the reflection equation within the quantum affine algebra representation theory we construct linear covariant operators that generate the askey wilson algebra it has the property of a coideal subalgebra which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk the generators of the askey wilson algebra are implemented to construct an operator valued k matrix a solution of a spectral dependent reflection equation we consider the open driven diffusive system where the askey wilson algebra arises as a boundary symmetry and can be used for an exact solution of the model in the stationary state we discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the askey wilson algebra to the reflection equation

the z 2 n dirac dunkl operator and a higher rank bannai ito algebra abstract the kernel of the z 2 n dirac dunkl operator is examined the symmetry algebra a n of the associated dirac dunkl equation on s n 1 is determined and is seen to correspond to a higher rank generalization of the bannai ito algebra a basis for the polynomial null solutions of the dirac dunkl operator is constructed the basis elements are joint eigenfunctions of a maximal commutative subalgebra of a n and are given explicitly in terms of jacobi polynomials the symmetry algebra is shown to act irreducibly on this basis via raising lowering operators a scalar realization of a n is proposed

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

an algebra behind the clebsch gordan coefficients of uq sl2 abstract let f denote a field fix a nonzero q f with q 4 1 let h q denote a unital associative f algebra generated by a b c and the relations assert that each of q b c q 1 c b q 2 q 2 a q c a q 1 a c q a b q 1 b a q 2 q 2 c commutes with a b c we call h q the universal q hahn algebra motivated by the clebsch gordan coefficients of u q sl 2 we find a homomorphism h q u q sl 2 u q sl 2 we show that the kernel of is an ideal of h q generated by a central element of h q the decomposition formulae for the tensor products of irreducible verma u q sl 2 modules and of finite dimensional irreducible u q sl 2 modules into the direct sums of finite dimensional irreducible h q modules are also given in the paper

constructing optimal instruments by first stage prediction averaging this paper considers model averaging as a way to construct optimal instruments for the two stage least squares 2sls limited information maximum likelihood liml and fuller estimators in the presence of many instruments we propose averaging across least squares predictions of the endogenous variables obtained from many different choices of instruments and then use the average predicted value of the endogenous variables in the estimation stage the weights for averaging are chosen to minimize the asymptotic mean squared error of the model averaging version of the 2sls liml or fuller estimator this can be done by solving a standard quadratic programming problem

asymptotic optimality for c_p c_l cross validation and generalized cross validation discrete index set c p c l cross validation and generalized cross validation are useful data driven techniques for selecting a good estimate from a proposed class of linear estimates the asymptotic behaviors of these procedures are studied some easily interpretable conditions are derived to demonstrate the asymptotic optimality it is argued that cross validation and generalized cross validation can be viewed as some special ways of applying c l applications in nearest neighbor nonparametric regression and in model selection are discussed in detail

dunkl shift operators and bannai ito polynomials abstract we consider the most general dunkl shift operator l with the following properties i l is of first order in the shift operator and involves reflections ii l preserves the space of polynomials of a given degree iii l is potentially self adjoint we show that under these conditions the operator l has eigenfunctions which coincide with the bannai ito polynomials we construct a polynomial basis which is lower triangular and two diagonal with respect to the action of the operator l this allows to express the bi polynomials explicitly we also present an anti commutator aw 3 algebra corresponding to this operator from the representations of this algebra we derive the structure and recurrence relations of the bi polynomials we introduce new orthogonal polynomials referred to as the complementary bi polynomials as an alternative q 1 limit of the askey wilson polynomials these complementary bi polynomials lead to a new explicit expression for the bi polynomials in terms of the ordinary wilson polynomials

the dual pair uq su 1 1 oq1 2 2n q oscillators and askey wilson algebras the universal askey wilson algebra aw 3 can be obtained as the commutant of uq su 1 1 in uq su 1 1 3 we analyze the commutant of oq1 2 2 oq1 2 2 oq1 2 2 in q oscillator representations of oq1 2 6 and show that it also realizes aw 3 these two pictures of aw 3 are shown to be dual in the sense of howe this is made clear by highlighting the role of the intermediate casimir elements of each member of the dual pair uq su 1 1 oq1 2 6 we also generalize these results a higher rank extension of the askey wilson algebra denoted aw n can be defined as the commutant of uq su 1 1 in uq su 1 1 n and a dual description of aw n as the commutant of oq1 2 2 n in q oscillator representations of oq1 2 2n is offered by calling upon the dual pair uq su 1 1 oq1 2 2n