efficient estimation for the proportional hazards model with interval censoring the maximum likelihood estimator mle for the proportional hazards model with case 1 interval censored data is studied it is shown that the mle for the regression parameter is asymptotically normal with n convergence rate and achieves the information bound even though the mle for the baseline cumulative hazard function only converges at n 1 3 rate estimation of the asymptotic variance matrix for the mle of the regression parameter is also considered to prove our main results we also establish a general theorem showing that the mle of the finite dimensional parameter in a class of semiparametric models is asymptotically efficient even though the mle of the infinite dimensional parameter converges at a rate slower than n the results are iliustrated by applying them to a data set from a tumorigenicity study

stability of the second order matrix polynomial this note summarizes some existing stability and instability conditions of the second order matrix polynomial which arises in the formulation of classical mechanics aerodynamics and robotic systems also some sufficient conditions for stability or instability of the second order matrix polynomial are newly developed via the relevant linear matrix equation obtained from the lyapunov theory

multiparametric quantum gl 2 lie bialgebras quantum r matrices and non relativistic limits multiparametric quantum deformations of gl 2 are studied through a complete classification of gl 2 lie bialgebra structures from these the non relativistic limit leading to harmonic oscillator lie bialgebras is implemented by means of a contraction procedure new quantum deformations of gl 2 together with their associated quantum r matrices are obtained and other known quantizations are recovered and classified several connections with integrable models are outlined

ladder operators normal ordering problem for quantum deformed systems and the q p generalization of the stirling and bell numbers we resolve the ladder operators normal ordering problem for strings in the form for q p deformed supersymmetric and shape invariant potential systems where n is a positive integer we provide exact and explicit expressions for their normal form where in all are at the right hand side and find that the solution involves expansion coefficients sequence which corresponds to the generalization for any q p deformed shape invariant potential system of the classical stirling and bell numbers we show that these numbers are given for expressions depending on parameters related to the forms of supersymmetric partner potentials and the system quantum deformation we apply the general formalism to poschl teller morse and scarf potential systems

duality and representations for new exotic bialgebras we find the exotic matrix bialgebras which correspond to the two nontriangular nonsingular 4 4 r matrices in the classification of hietarinta namely rs0 3 and rs1 4 we find two new exotic bialgebras s03 and s14 which are not deformations of the classical algebras of functions on gl 2 or gl 1 1 with this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements we also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail we also discuss in detail a special case of rs1 4 in which the corresponding algebra turns out to be a special case of the two parameter quantum group deformation glp q 2

duals of colored quantum universal enveloping algebras and colored universal t matrices we extend the notion of dually conjugate hopf super algebras to the colored hopf super algebras hc that we recently introduced we show that if the standard hopf super algebras hq that are the building blocks of hc have hopf duals hq then the latter may be used to construct coloured hopf duals hc endowed with colored algebra and antipode maps but with a standard coalgebraic structure next we review the case where the hq s are quantum universal enveloping algebras of lie super algebras uq g so that the corresponding hq s are quantum super groups gq we extend the fronsdal and galindo universal t matrix formalism to the colored pairs uc g gc by defining colored universal t matrices we then show that together with the colored universal r matrices previously introduced the latter provide an algebraic formulation of the colored rtt relations proposed by basu mallick this establishes a link between the colored extensions of drinfeld jimbo and faddeev reshetikhin takhtajan pictures of quant

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

a sufficient condition for the stability of interval matrix polynomials the root location of sets of scalar polynomials whose coefficients are confined to intervals and the associated extension to eigenvalues of sets of constant matrices whose coefficients are contained in intervals are reviewed a central result for complex scalar interval polynomials is a theorem developed by v l kharatonov 1978 which states that each member of a set of such polynomials is stable if and only if eight special polynomials from the set are stable the case of interval matrix polynomials is examined and a kharitonov like result for their strong stability is provided this in turn yields a sufficient condition for stability of a set of interval matrix polynomials

a necessary and sufficient condition for the stability of nonnegative interval discrete systems the stability of nonnegative discrete systems with interval uncertainty is discussed for this class of system a necessary and sufficient condition for the stability is provided when all elements of the system matrix are assumed to be specified by intervals an illustrative example is used as a guideline to demonstrate the effectiveness of the result simple computational tests are suggested and their usefulness in this framework is discussed the possible extension of this result to more general classes of systems is addressed

characterization of robust controllers abstract this paper presents a procedure for designing a robust controller which is able to stabilize a multivariable system with interval uncertainties in the system parameters the uncertain parameters are assumed to lie within known intervals the fundamental idea behind the procedure is to characterize the set of robust controllers as a convex set by the new matrix inequalities result a suitable controller can then be chosen by means of an optimization algorithm the controller thus chosen will be able to ensure that the closed loop matrix lies within desired intervals for all the prescribed interval uncertainties in the system parameters

noncommutative phase and unitarization of glp q 2 in this article imposing hermitian conjugate relations on the two parameter deformed quantum group glp q 2 is studied this results in a noncommutative phase associated with the unitarization of the quantum group after the achievement of the quantum group up q 2 with pq real via a noncommutative phase the representation of the algebra is built by means of the action of the operators constituting the up q 2 matrix on states

universal t matrix representations of ospq 1 2 and little q jacobi polynomials we obtain a closed form expression of the universal t matrix encapsulating the duality between the quantum superalgebra uq osp 1 2 and the corresponding supergroup ospq 1 2 the classical q 1 limit of this universal t matrix yields the group element of the undeformed ospq 1 2 supergroup the finite dimensional representations of the quantum supergroup ospq 1 2 are readily constructed employing the above mentioned universal t matrix and the known finite dimensional representations of the dually related deformed uq osp 1 2 superalgebra proceeding further we derive the product law the recurrence relations and the orthogonality of the representations of the quantum supergroup ospq 1 2 it is shown that the entries of these representation matrices are expressed in terms of the little q jacobi polynomials with q q two mutually complementary singular maps of the universal t matrix on the universal r matrix are also presented

representations of two parameter quantum algebras and p q special functions abstract in this paper the connection between p q analogue of special functions and representations of certain two parameter quantum algebras has been developed the realizations of the representations of these quantum algebras are in terms of p q derivative operators the matrix elements of these representations are computed in terms of the generalized p q hypergeometric series this leads to the developments in the theory of p q special functions

duality for exotic bialgebras in the classification of hietarinta three triangular 4 4 r matrices lead via the frt formalism to matrix bialgebras which are not deformations of the trivial one in this paper we find the bialgebras which are in duality with these three exotic matrix bialgebras we note that the l t duality of frt is not sufficient for the construction of the bialgebras in duality we find also the quantum planes corresponding to these bialgebras both by the wess zumino r matrix method and by manin s method

models of q algebra representations matrix elements of the q oscillator algebra this article continues a study of function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions here a q analog of the oscillator algebra not a quantum algebra is considered it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed this local approach applies to more general families of special functions e g with complex arguments and parameters than does the quantum group approach it is shown that the matrix elements themselves transform irreducibly under the action of the algebra q analogs of a formula are found for the product of two hypergeometric functions 1f1 and the product of a 1f1 and a bessel function they are interpreted here as expansions of the matrix elements of a group operator

z3 graded differential geometry of the quantum plane in this work the z3 graded differential geometry of the quantum plane is constructed the corresponding quantum lie algebra and its hopf algebra structure are obtained the dual algebra i e the universal enveloping algebra of the quantum plane is explicitly constructed

variance estimation of a survival function for interval censored survival data interval censored survival data often occur in medical studies especially in clinical trials in this case many authors have considered estimation of a survival function there is however relatively little discussion on estimating the variance of estimated survival functions for right censored data a special case of interval censored data the most commonly used method for variance estimation is to use the greenwood formula in this paper we propose a generalization of the greenwood formula for variance estimation of a survival function based on interval censored data also a simple bootstrap approach is presented the two methods are evaluated and compared using simulation studies and a real data set the simulation results suggest that the methods work well

duality for the matrix quantum group glp q 2 c it is shown that the algebra up q dual to glp q 2 c is isomorphic to u pq 1 2 sl 2 c z as a commutation algebra where z is a subalgebra central in up q the subalgebra z is a hopf subalgebra of up q while the commutation subalgebra u pq 1 2 sl 2 c is not a hopf subalgebra

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

a new lmi condition for robust stability of polynomial matrix polytopes a sufficient condition for checking the robust stability of a polytope of polynomial matrices is proposed in this note a simple feasibility test performed in a convex set of linear matrix inequalities defined at the vertices of the polytope yields sufficient conditions for the robust stability of the entire domain both continuous time left half plane and discrete time stability unit disk are investigated numerical comparisons with quadratic stability results and with another method recently appeared in the literature show that the conditions proposed provide in general less conservative results

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

stability analysis of interval matrices improved bounds new sufficient conditions for the stability of interval matrices are presented based on the lyapunov stability approach as well as on the frequency domain approach the conditions help to overcome the conservatism of the existing criteria for the stability of interval matrices based on a matrix approach several numerical examples are given to demonstrate the new conditions and to compare them with results previously reported

stability of a polytope of matrices counterexamples while there have been significant breakthroughs for the stability of a polytope of polynomials since v l kharitonov s 1978 seminal result on interval polynomials for a polytope of matrices the stability problem is considered far from completely resolved counterexamples are provided for three conjectures that are directly motivated by the results in the polynomial case these counterexamples illustrate the fundamental differences between polynomial stability and matrix stability problems and indicate that some obvious lines of attack on the matrix polytope stability problem will fail

sufficient condition for asymptotic stability of discrete interval systems abstract a counter example is given to a recent result on the stability analysis of interval matrices by jiang 1988 a sufficient condition is then presented for the asymptotic stability of the discrete time interval system x k 1 a i x k by testing the norms of extreme matrices of the interval matrix a i such that they are all less than one

simplified sufficient conditions for the asymptotic stability of interval matrices it will be shown that the sufficient conditions for the asymptotic stability of interval matrices proved in mansour 1988 can be further simplified by considering only a part of the extreme matrices

maximum likelihood for interval censored data consistency and computation summary standard convex optimization techniques are applied to the analysis of interval censored data these methods provide easily verifiable conditions for the self consistent estimator proposed by turnbull 1976 to be a maximum likelihood estimator and for checking whether the maximum likelihood estimate is unique a sufficient condition is given for the almost sure convergence of the maximum likelihood estimator to the true underlying distribution function

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

colored quantum universal enveloping algebras we define some new algebraic structures termed colored hopf algebras by combining the coalgebra structures and antipodes of a standard hopf algebra set h corresponding to some parameter set q with the transformations of an algebra isomorphism group g herein called color group such transformations are labeled by some color parameters taking values in a color set c we show that various classes of hopf algebras such as almost cocommutative coboundary quasitriangular and triangular ones can be extended into corresponding colored algebraic structures and that colored quasitriangular hopf algebras in particular are characterized by the existence of a colored universal r matrix satisfying the colored yang baxter equation the present definitions extend those previously introduced by ohtsuki which correspond to some substructures in those cases where the color group is abelian we apply the new concepts to construct colored quantum universal enveloping algebras of both semisimple and nonsemisimple lie algebras considering several examples with fixed or varying parameters as a by product some of the matrix representations of colored universal r matrices derived in the present paper provide new solutions of the colored yang baxter equation which might be of interest in the context of integrable models

tuning of complex coefficient pi pd pid controllers for a universal plant structure controllers are tuned either analytically or through optimisation techniques for integer fractional and fractional complex order plants with dead time by considering only the positive frequency in

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

a necessary and sufficient condition for the stability of interval matrices consider the linear dynamical system described by the equation xdot t ax t where a is a constant real matrix with dimensions n n and the elements of matrix a are not known precisely we know only the upper and lower bounds of the intervals in which the elements of matrix a are confined in this paper there is proved a necessary and sufficient condition for the stability of solutions of eqn 1 with incomplete information about the elements of matrix a

the exceptional lie algebra e7 25 multiplets and invariant differential operators in the present paper we continue the project of systematic construction of invariant differential operators on the example of the non compact exceptional algebra e7 25 our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras which we call conformal lie algebras which have very similar properties to the conformal algebras of n dimensional minkowski spacetime this class of algebras is identified and summarized in a table another motivation is related to the ads cft correspondence we give the multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators

stability analysis of interval matrices continuous and discrete systems this paper considers the stability properties of dynamical interval systems two new methods of determining an upper bound on interval perturbations are presented one of the methods is applicable to both continuous and discrete systems

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

quantum group structure on the quantum plane we show that the quantum plane introduced by manin is a quantum group according to the faddeev reshetikhin takhtajan approach we give its matrix representation r matrix and its hopf structure some quantum properties of the quantum matrix whose elements generate the quantum plane are also discussed

invariant differential operators and characters of the ads 4 algebra the aim of this paper is to apply systematically to ads4 some modern tools in the representation theory of lie algebras which are easily generalized to the supersymmetric and quantum group settings and necessary for applications to string theory and integrable models here we introduce the necessary representations of the ads4 algebra and group we give explicitly all singular null vectors of the reducible ads4 verma modules these are used to obtain the ads4 invariant differential operators using this we display a new structure a diagram involving four partially equivalent reducible representations one of which contains all finite dimensional irreps of the ads4 algebra we study in more detail the cases involving uirs in particular the di and the rac singletons and the massless uirs in the massless case we discover the structure of sets of 2s0 1 conserved currents for each spin s0 uir all massless cases are contained in a one parameter subfamily of the quartet diagrams mentioned above the parameter being the spin s0 further we give the classification of the irreps presented in a diagrammatic way which makes easy the derivation of all character formulae the paper concludes with a speculation on the possible applications of the character formulae to integrable models

quantum supergroup structure of 1 1 dimensional quantum superplane its dual and its differential calculus we show that the 1 1 dimensional quantum superplane introduced by manin is a quantum supergroup according to the faddeev reshetikhin takhtajan approach when it is extended by the inverse of the bosonic variable we then give its supermatrix element its corresponding r matrix and its hopf structure this new point of view allows us first to realize its dual hopf superalgebra starting from postulated initial pairings second we construct a right invariant differential calculus on it and then deduce the corresponding quantum lie superalgebra which as a commutation superalgebra appears classical and as hopf structure is a non cocommutative q deformed one

robustness of pole location in perturbed systems abstract in this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients a geometric approach in coefficient space is exploited to derive maximal deviations in a given class of admissible perturbations of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane it is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane this last result can be considered an extension of the result of the well known theorem of kharitonov it is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations

an interval kalman filtering with minimal conservatism abstract the interval kalman filtering ikf can handle parametric interval uncertainties of the system matrices and it computes the lower and upper boundaries of the estimated states in this paper we propose an alternative form of interval kalman filtering to reduce the conservatism inherent to the existing interval kalman filtering first we address why the existing interval kalman filtering scheme induces conservatism in the boundary estimation then to remove the conservatism we derive noise covariance matrices taking into account of interval uncertainties as well as process and measurement noises following the typical derivation process of the standard kalman filtering a new recursive form of interval kalman filtering is derived through numerical simulations the superiority of the new algorithm over existing ikf is illustrated

on combined standard nonstandard or hybrid q h deformations combined q h deformations proposed by kupershmidt and ballesteros herranz parashar are studied in each case a transformation is shown to lead to an equivalent standard q deformation we briefly indicate that appropriate singular limits of the same type of transformations can however lead from standard biparametric p q deformations to nonhybrid but biparametric nonstandard g h ones finally a case of hybrid q h deformation is recalled related to the superalgebra gl 1 1

stabilisation of matrix polynomials a state feedback is proposed to analyse the stability of a matrix polynomial in closed loop first it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable following the ideas of the routh hurwitz stability procedure for scalar polynomials certain continued fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the euclid s division algorithm which leads to an extension of the well known routh hurwitz stability criteria but this time in terms of matrix coefficients after that stability of the closed loop matrix polynomial is guaranteed based on a corollary of a lyapunov theorem the sufficient stability conditions are i the matrices of one column of the presented array must be symmetric and positive definite and ii the matrices of the cascade realisation must satisfy a commutative condition these stability conditions are also necessary for matrix polynomial of second order

brief an lmi condition for robust stability of polynomial matrix polytopes a sufficient lmi condition is proposed for checking robust stability of a polytope of polynomial matrices it hinges upon two recent results a new approach to polynomial matrix stability analysis and a new robust stability condition for convex polytopic uncertainty numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact np hard robust stability results

finite dimensional representations of the quantum group glp q 2 using the exponential map from up q gl 2 using the fronsdal galindo formula for the exponential mapping from the quantum algebra up q gl 2 to the quantum group glp q 2 we show how the 2j 1 dimensional representations of glp q 2 can be obtained by exponentiating the well known 2j 1 dimensional representations of up q gl 2 for j 1 3 2 j 1 2 corresponds to the defining two dimensional t matrix the earlier results on the finite dimensional representations of glq 2 and slq 2 or suq 2 are obtained when p q representations of uq q 2 q in c mod r and uq 2 q in r mod 0 are also considered the structure of the clebsch gordan matrix for up q gl 2 is studied the same clebsch gordan coefficients are applicable in the reduction of the direct product representations of the quantum group glp q 2

comments on a necessary and sufficient condition for the stability of interval matrices in a recent paper by bialas a new proof of a known result concerning interval polynomials was given this result is summarized here and a much stronger result is mentioned bialas main theorem on the stability of interval matrices is shown to be false

a less conservative lmi condition for robust cal d stability of polynomial matrix polytopes a projection approach this technical note investigates the problem of checking robust d stability of a polytope of polynomial matrices finsler s projection lemma is taken to derive a new sufficient condition in terms of a linear matrix inequality lmi feasibility problem the basic idea behind this condition is to lift the existing stability condition into the one of larger space by means of finsler s lemma and based on it to introduce additional decision variables examples are given to show that the proposed condition can yield less conservative results than the previous one

duality for multiparametric quantum gl n we show that the hopf algebra uuq dual to the multiparameter matrix quantum group gluq n may be realized a la sudbery 1990 i e tangent vectors at the identity furthermore we give the cartan weyl basis of uuq and show that this is consistent with the duality we show that as a commutation algebra uuq equivalent to uu sl n c x uu z where z is one dimensional and uu z is a central algebra in uuq however as a co algebra uuq cannot be split in this way and depends on all parameters

models of q algebra representations tensor products of special unitary and oscillator algebras this paper begins a study of one and two variable function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions the algebras considered are the quantum algebra uq su2 and a q analog of the oscillator algebra not a quantum algebra in each case a simple one variable model of the positive discrete series of finite and infinite dimensional irreducible representations is used to compute the clebsch gordan coefficients it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed it is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations it is also demonstrated that general q hypergeometric functions can occur as

estimation in the cox proportional hazards model with left truncated and interval censored data nous montrons que l estimateur non parametrique du maximum de vraisemblance npmle du coefficient de regression obtenu a partir de la vraisemblance jointe du coefficient de regression et de la survie de base est correct pour le modele de cox a hasards proportionnels avec des donnees tronquees a gauche et censurees par intervalle mais le npmle pent sous estimer la survie de base deux alternatives sont egalement etudiees premierement la vraisemblance marginale en etendant satten 1996 aux donnees tronquees la survie de base etant eliminee comme un parametre nuisible deuxiemement l estimateur monotone du maximum de vraisemblance qui maximise la vraisemblance jointe en supposant que la fonction de hasard est non decroissante pour la distribution de base ceci a ete initialement propose pour resoudre la sous estimation de la survie par le npmle dans le cas de donnees tronquees a gauche sans covariate tsai 1988 le boostrap est utilise a des fins d inference et des simulations sont realisees pour evaluer leur performance les methodes sont appliquees pour evaluer leur performance les methodes sont appliquees aux donnees de l etude du massachusetts health care panel afin de comparer chez les seniors en fonction du sexe les probabilites de perte d independance fonctionnelle

relating the approaches to quantised algebras and quantum groups this paper constructs two representations of the quantum groupu q g by exploiting its quotient structure and the quantum double construction here the quantum group is taken as the dual to the quantised algebrau q g a one parameter deformation of the universal enveloping algebra of the lie algebra g as in drinfel d 6 and jimbo 10 from the two representations the hopf structure of the quantised algebrau q g is reexpressed in a matrix format this is the very structure given by faddeev et al 7 in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated lie group to g

duality for the jordanian matrix quantum group we find the hopf algebra dual to the jordanian matrix quantum group as an algebra it depends only on the sum of the two parameters and is split into two subalgebras with three generators and with one generator the subalgebra is a central hopf subalgebra of the subalgebra is not a hopf subalgebra and its co algebra structure depends on both parameters we discuss also two one parameter special cases g h and g h the subalgebra is a hopf algebra and coincides with the algebra introduced by ohn as the dual of the subalgebra is isomorphic to u sl 2 as an algebra but has a nontrivial co algebra structure and again is not a hopf subalgebra of

on irreducible p q representations of gl 2 the theory of irreducible p q representations of the complex lie algebra gl 2 is developed we construct a one variable model of irreducible p q representations of the gl 2 in terms of p q derivative operator and derive a generating function based on it

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients