the clebsch gordan coefficients for the two parameter quantum algebra sup q 2 in the lowdin shapiro approach the structure of the irreducible representations of the two parameter quantum algebra sup q 2 is studied the projection operators for this algebra are constructed in the lowdin shapiro form the explicit analytical expressions for the sup q 2 clebsch gordan coefficients are obtained with the help of these projection operators there are clear perspectives to elaborate the theory of tensor operators universal r matrix 6j 9j symbols etc for the sup q 2 algebra using the same tools

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

transverse vibration of a rectangular plate with bidirectional thickness variation abstract the transverse vibrations of a rectangular plate of variable thickness have been investigated with different combinations of boundary conditions at the four edges the thickness variation is bidrectional and is the cartesian product of linear variations along two concurrent edges of the plate successive approximations have been worked out by the rayleigh ritz method using basis functions satisfying essential boundary conditions convergence is ensured by taking a sufficient number of approximations extensive numerical work has been carried out to obtain the first three frequencies for some selected cases the mode shapes have also been plotted comparison is made with results available in the literature for speacial cases

q deformed harmonic and clifford analysis and the q hermite and laguerre polynomials we define a q deformation of the dirac operator inspired by the one dimensional q derivative this implies a q deformation of the partial derivatives by taking the square of this dirac operator we find a q deformation of the laplace operator this allows us to construct q deformed schrodinger equations in higher dimensions the equivalence of these schrodinger equations with those defined on q euclidean space in quantum variables is shown we also define the m dimensional q clifford hermite polynomials and show their connection with the q laguerre polynomials these polynomials are orthogonal with respect to an m dimensional q integration which is related to integration on q euclidean space the q laguerre polynomials are the eigenvectors of an suq 1 1 representation

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

realizations of su 1 1 and uq su 1 1 and generating functions for orthogonal polynomials positive discrete series representations of the lie algebra su 1 1 and the quantum algebra uq su 1 1 are considered the diagonalization of a self adjoint operator the hamiltonian in these representations and in tensor products of such representations is determined and the generalized eigenvectors are constructed in terms of orthogonal polynomials using simple realizations of su 1 1 uq su 1 1 and their representations these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials the relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials or to poisson kernels in particular a group theoretical derivation of the poisson kernel for meixner pollaczek and al salam chihara polynomials is obtained

robust synchronization of chaotic horizontal platform systems with phase difference this paper studies a robust synchronization of non autonomous chaotic systems with parameter mismatch in the synchronization scheme a linear state error feedback control is used to couple the master and slave horizontal platform systems excited by harmonic external forces between which there exists a phase mismatch a new definition of global synchronization with error bound is introduced using lyapunov s stability theory the sufficient synchronization criteria for the scheme are proven and the corresponding synchronization error bound is estimated the synchronization criteria are further optimized by optimally designing a quadratic lyapunov function to more precisely estimate the synchronization error bound the illustrative simulations verify the effectiveness of these criteria the estimated synchronization error bound is compared with numerical one in the examples

intertwiners of uq sl 2 representations and the vector valued big q jacobi transform linear operators r are introduced on tensor products of evaluation modules of uq sl 2 obtained from the complementary and strange series representations the operators r satisfy the intertwining condition on finite linear combinations of the canonical basis elements of the tensor products infinite sums associated with the action of r on six pairs of tensor products are evaluated for two pairs the sums are related to the vector valued big q jacobi transform of the matrix elements defining the operator r in one case the sums specify the action of r on the irreducible representations present in the decomposition of the underlying indivisible sum of uqsl 2 tensor products in both cases bilinear summation formulae for the matrix elements of r provide a generalization of the unitarity property corresponding results are given for the remaining pairs

master slave chaos synchronization criteria for the horizontal platform systems via linear state error feedback control global chaos synchronization of two identical non autonomous horizontal platform systems coupled by linear state error feedback controller is investigated the sufficient criteria for global chaos synchronization are deduced based on the stability theory of linear time varied systems and lyapunov s direct method of which the criteria related to general coupling matrix are first proved and applied to derive the ones related to some special coupling matrices in the examples the coupling strengths are designed by the obtained criteria and the appearances of chaos synchronization are verified it is analytically and numerically examined that the synchronization criteria based on lyapunov s direct method are sharper than the criteria based on the stability theory of linear time varied systems

free vibration analysis of orthotropic rectangular plates with variable thickness and general boundary conditions a discrete method is developed for analyzing the free vibration problem of orthotropic rectangular plates with variable thickness the green function which is obtained by transforming the differential equations into integral equations and using numerical integration is used to establish the characteristic equation of the free vibration the effects of the aspect ratios boundary conditions the variation of the thickness on the frequencies are considered by comparing the numerical results obtained by the present method with the those previously published the efficiency and accuracy of the present method are investigated the frequency parameters are obtained for the orthotropic plates with general boundary conditions and variable thickness in one or two directions the model shapes are given for some of the square plates with three kinds of thickness variations

chaos synchronization of a horizontal platform system chaos and chaos synchronization of the horizontal platform system are studied in this paper because of the non linear terms of the systems the systems exhibit both regular and chaotic motions by applying various numerical results such as phase portraits poincare maps time history and power spectrum analysis the behaviors of the periodic and chaos synchronization are presented the effects of the change of parameters in the system can be found in the bifurcation diagrams chaos synchronization of feedback methods in two coupled systems has been studied by lyapunov exponent and coupling strength besides phase effect of external excitations and the transient time in unidirectional synchronization also have been researched

on q orthogonal polynomials dual to little and big q jacobi polynomials we derive discrete orthogonality relations for polynomials dual to little and big q jacobi polynomials this derivation essentially requires use of bases consisting of eigenvectors of certain self adjoint operators which are representable by a jacobi matrix recurrence relations for these polynomials are also given

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

the free vibrations of tapered rectangular plates using a new set of beam functions with the rayleigh ritz method abstract in19991999 academic pressacademic pressthis paper the free vibrations of a wide range of non uniform rectangular plates in one or two directions are considered the domain of the plate is bounded by copyright x 1a a and y 1b b in rectangular co ordinates the thickness of the plate is continuously varying and proportional to the power functionxsyt a variety of tapered rectangular plates can be described by giving the taper factors s and t different values s and t may be given arbitrary real numbers if both 1 0 and 1 0 or arbitrary non negative numbers if 1 0 or 1 0 the uniform rectangular plate is a special case by letting both s and t equal to zero a new set of admissible functions which are the static solutions of the tapered beam or a strip taken from the tapered rectangular plate under an arbitrary static load expanded into a taylor series is developed unlike conventional admissible functions the set of static beam functions will vary appropriately with the thickness variation of the plate the eigenfrequency equation is obtained by the use of the rayleigh ritz method a general computer program has been compiled and some numerical results are tabulated on the basis of comparison with available results in the literature it is shown that the first few eigenfrequencies can be obtained with good accuracy by using only a small number of terms of the static beam functions

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

solutions of the quantum yang baxter equation with extra nonadditive parameters we present a systematic technique to construct solutions to the yang baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters these extra parameters enter the yang baxter equation in a similar way to the spectral parameter but in a non additive form we exploit the fact that quantum non compact algebras such as uq su 1 1 and type i quantum superalgebras such as uq gl 1 mod 1 and uq gl 2 mod 1 are known to admit non trivial one parameter families of infinite dimensional and finite dimensional irreps respectively even for generic q we develop a technique for constructing the corresponding spectral dependent r matrices as examples we work out the the r matrices for the three quantum algebras mentioned above in certain representations

design and implementation of fuzzy sliding mode controllers for generalized projective synchronization of chaos horizontal platform systems this paper presents robust chattering free fuzzy sliding mode control fsmc as a way to achieve generalized projective synchronization gps of two horizontal chaos platform vibration systems with system uncertainty and external disturbance an appropriate sliding surface is acquired through an smc design such that the error state trajectories of a master slave system will reach such a sliding surface and approaches to the origin based on the lyapunov stability theory we address the design schemes of integration fuzzy sliding mode control where the reaching law is proposed by a set of linguistic rules and the control input is chattering free as a consequence the system is simulated and experimented to demonstrate that the error state tends to be stabilized and the chattering problem can be thus resolved implying that the slave horizontal platform system hps is gps with the master hps

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

transverse vibration of triangular plates with variable thickness transverse vibration of a triangular plate with thickness varying as a linear function of co ordinates in the plane of the plate has been solved by working out several approximations in the rayleigh ritz method the basis functions are chosen so that the essential boundary conditions are satisfied three parameters control the three types of boundary conditions at the edges two parameters control the thickness variation and two parameters control the shape of the plate a general procedure has been laid down to compute the frequencies and the associated mode shapes for any set of these parameters convergence has been ensured by working out a sufficiently large number of approximations so that at least the first three frequencies converge to three significant digits the results are tabulated for various values of the parameters and comparisons have been made with known results in special cases three dimensional plots of the mode shapes have also been given in some cases

free vibration analysis of rectangular plates with variable thickness abstract an approximate method for analyzing the free vibration of thin and moderately thick rectangular plates with arbitrary variable thickness is proposed the approximate method is based on the green function of a rectangular plate the green function of a rectangular plate with arbitrary variable thickness is obtained as a discrete form solution for deflection of the plate with a concentrated load the discrete form solution is obtained at each discrete point equally distributed on the plate it is shown that the numerical solution for the green function has good convergency and accuracy by applying the green function the free vibration problem of the plate is translated into the eigenvalue problem of the matrix the convergency and accuracy of the numerical solutions for the natural frequency parameter calculated by the proposed method are investigated and the frequency parameters and their modes of free vibration are shown for some rectangular plates

representations of the q deformed algebras uq so2 1 and uq so3 1 representations of the algebra uq so2 1 are studied this algebra is a q deformation of the universal enveloping algebra u so2 1 of the lie algebra of the group so0 2 1 and differs from the quantum algebra uq su1 1 classifications of irreducible representations and of infinitesimally unitary irreducible representations of uq so2 1 with simple and discrete spectrum of one of generators are derived the set of infinitesimally unitary representations of uq so2 1 does not coincide with that for the algebra uq su1 1 the sets of irreducible representations and of infinitesimally unitary irreducible representations of the algebra uq so3 1 are given we also consider representations of uq son 1 which are of class 1 with respect to the subalgebra uq son

free vibration analysis of tapered rectangular plates by differential quadrature method a semi analytical approach abstract the particular type of thin rectangular plate configurations considered in the present work are the ones having two opposite edges simply supported and are of varying thickness between the other two edges the boundary conditions at these two edges may be quite general the plates may isotropic or specially orthotropic for the numerical solution to evaluate the free vibration characteristics of such plates a semi analytical approach is adopted first by using the trigonometric sine functions for the mode shape between the simply supported edges the eigenvalue partial differential equation is reduced to an ordinary differential equation the solution of the reduced equation is then carried out by the differential quadrature method on the basis of comparison with the available results in the published literature it is believed that this solution method is capable of yielding highly accurate results moreover the very small computation times required for the evaluation of the vibration characteristics demonstrate the high computational efficiency of the method and point to its possible use in the real time analysis and design of vibrating plate systems

diagonalization of operators and one parameter families of nonstandard bases for representations of suq 2 nonstandard bases for finite dimensional irreducible representations of the quantum algebra suq 2 are constructed by diagonalizing one parameter families of the operators qj3 4 j j qj3 4 cqj3 and iqj3 4 j j qj3 4 cqj3 c we derive explicit expressions for the eigenfunctions and the corresponding eigenvalues of these operators in an arbitrary irreducible representation of suq 2 it is shown that the matrix elements of the intertwining operator aj c which is a q extension of the classical su 2 operator aj j1aj ajj3 are expressed in terms of the dual q krawtchouk polynomials diagonalization of some other operators associated with the dual q hahn polynomials is also examined

tensor products of uq sl 2 modules and the big q2 jacobi function transform four tensor products of evaluation modules of the quantum affine algebra uq sl 2 obtained from the negative and positive series the complementary and the strange series representations are investigated linear operators r z satisfying the intertwining property on finite linear combinations of the canonical basis elements of the tensor products are described in terms of two sets of infinite sums r t r t z 0 and r t r t z 0 involving big q2 jacobi functions or related nonterminating basic hypergeometric series inhomogeneous recurrence relations can be derived for both sets evaluations of the simplest sums provide the corresponding initial conditions for the first set of sums the relations entail a big q2 jacobi function transform pair an integral decomposition is obtained for the sum r t a partial description of the relation between the decompositions of the tensor products with respect to uqsl 2 or with respect to its complement in uq sl 2 can be formulated in terms of askey wilson

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

a semi analytical solution of the flexural vibration of orthotropic plates of variable thickness a semi analytical method based on the finite strip technique in conjunction with the transition matrix is applied to investigate the flexural vibration of orthotropic rectangular plates of variable thickness in one direction the effect of orthotropy aspect ratio and taper ratio on the natural frequencies of the vibration of such plates is investigated the efficiency of the proposed technique is illustrated by comparing the results with those available in the literature for several examples the results show excellent agreement and stable convergence for all plate configurations considered in this paper

nonlinear analysis and control of the uncertain micro electro mechanical system by using a fuzzy sliding mode control design this study analyzes the chaotic behavior of a micromechanical resonator with electrostatic forces on both sides and investigates the control of chaos a phase portrait maximum lyapunov exponent and bifurcation diagram are used to find the chaotic dynamics of this micro electro mechanical system mems to suppress chaotic motion a robust fuzzy sliding mode controller fsmc is designed to turn the chaotic motion into a periodic motion even when the mems has system uncertainties

harmonics on the quantum euclidean space we study harmonic polynomials on the quantum euclidean space enq generated by quantum coordinates xi i 1 2 n on which the quantum group soq n acts they are defined as solutions of the equation qp 0 where q is the q laplace operator on enq we construct a q analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup soq n 2 the associated spherical polynomials constitute an orthogonal basis of the spaces of homogeneous harmonic polynomials they are represented as products of polynomials depending on q radii and xj xj j n j 1 this representation is in fact a q analogue of the classical separation of variables

quintic splines in the study of transverse vibrations of non uniform orthotropic rectangular plates an analysis and numerical results are presented for free transverse vibrations of orthotropic rectangular plates of linearly varying thickness along one direction and resting on an elastic foundation of the winkler type on the basis of classical plate theory following the levy approach i e two parallel edges being simply supported the fourth order differential equation governing the motion of such plates has been solved by using the quintic splines interpolation technique for three different combinations of clamped simply supported and free boundary conditions at the other two edges the effect of the elastic foundation together with the orthotropy aspect ratio and thickness variation on the natural frequencies of vibration is illustrated for the first three modes of vibration normalized displacements are presented for two different values of the taper constant keeping other plate parameters fixed for all the three boundary conditions a comparison of the results with those available in literature is presented

vibration of elastically restrained cross ply laminated plates with variable thickness abstract the natural frequencies of symmetrically laminated plates of variable thickness are analyzed using the finite strip transition matrix technique in this paper the natural frequencies of such plates are determined for edges with being elastically restrained against both rotation and transition or both a successive conjunction of the classical finite strip method and the transition matrix method is applied to develop a new modification of the finite strip method to reduce the complexity of the problem the displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction using the new transition matrix after necessary simplification and the satisfaction of the boundary conditions yields a set of simultaneous equations that leads to the characteristic matrix of vibration the mode shapes and the frequency parameters for different combinations of elastic or translational restraint coefficients have been presented and compared with those available from other methods in the literature also the effect of the tapered ratio and the aspect ratio on the natural frequencies and the mode shapes of the plates are presented the good agreement with other methods demonstrates the validity and the reliability of the proposed method

the quantum algebra uq su2 and q krawtchouk families of polynomials the aim of this paper is to study quantum and affine q krawtchouk polynomials by means of operators of irreducible representations of the quantum algebra uq su2 we diagonalize a certain operator i in such a representation and show that elements of the transition matrix from the initial canonical basis to the basis consisting of eigenfunctions of the operator i are expressed in terms of quantum q krawtchouk polynomials then we find an explicit form of the operator in the basis of the eigenfunctions of i in which it has the form of a jacobi matrix normalizing this basis and using the operator we thus derive the orthogonality relations for quantum and affine q krawtchouk polynomials we show that affine q krawtchouk polynomials are dual to quantum q krawtchouk polynomials a biorthogonal system of functions with respect to the scalar product in the representation space is also derived

more on uq su 1 1 with q a root of unity highest weight representations of uq su 1 1 with q exp i n are investigated the structures of the irreducible highest weight modules are discussed in detail the clebsch gordan decomposition for the tensor product of two irreducible representations is discussed by using the results a representation of sl 2 r uq su 2 is also presented in terms of holomorphic sections which also have uq su 2 index furthermore zn graded supersymmetry is realized in terms of the representation an explicit realization of osp 1 2 via the highest weight representation of uq su 1 1 with q2 1 is given

a laplace operator and harmonics on the quantum complex vector space the aim of this article is to study the q laplace operator and q harmonic polynomials on the quantum complex vector space generated by elements zi wi i 1 2 n on which the quantum group glq n or uq n acts the q harmonic polynomials are defined as solutions of the equation qp 0 where p is a polynomial in zi wi i 1 2 n and the q laplace operator q is determined in terms of q derivatives the q laplace operator q commutes with the action of glq n the projector hm m am m hm m is constructed where am m and hm m are the spaces of homogeneous of degree m in zi and of degree m in wi polynomials and homogeneous q harmonic polynomials respectively by using these projectors a q analog of the classical zonal spherical and associated spherical harmonics are constructed they constitute an orthogonal basis of hm m a q analog of separation of variables is given the quantum algebra uq gln acting on hm m determines an irreducible representation of uq gln this action is explicitly con

on the spectrum of a hamiltonian defined on suq 2 and quantum optical models analytical expressions are given for the eigenvalues and eigenvectors of a hamiltonian with suq 2 dynamical symmetry the relevance of such an operator in quantum optics is discussed as an application the ground state energy in the dicke model is studied through suq 2 perturbation theory

a q analogue of bargmann space and its scalar product a q analogue of bargmann space is defined using the properties of coherent states associated with a pair of q deformed bosons the space consists of a class of entire functions of a complex variable z and has a reproducing kernel on this space the q boson creation and annihilation operators are represented as multiplication by z and q differentiation with respect to z respectively a q integral analogue of bargmann s scalar product is defined involving the q exponential as a weight function associated with this is a completeness relation for the q coherent states

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

representations of the quantum algebra uq su1 1 all irreducible representations of the quantum group uq su1 1 are given they are determined by two complex numbers infinitesimally unitary representations are separated from the set of irreducible representations it is shown that the symmetric operators of irreducible infinitesimally unitary representations corresponding to generators of the lie algebra su1 1 admit self adjoint extensions

a classification of unitary highest weight modules of the quantum analogue of the symmetric pair an an 1 abstract we study the irreducibility and unitarity of highest weight modules of the quantized enveloping algebra associated to s u n 1 we obtain a classification of these modules analogous to that of ehw in the classical context we also compute all commutation relations between root vectors in u q s u n 1

spectra eigenvectors and overlap functions for representation operators of q deformed algebras operators of representations corresponding to symmetric elements of theq deformed algebrasuq su1 1 uq so2 1 uq so3 1 uq son and representable by jacobi matrices are studied closures of unbounded symmetric operators of representations of the algebrasuq su1 1 anduq so2 1 are not selfadjoint operators for representations of the discrete series their deficiency indices are 1 1 bounded symmetric operators of these representations are trace class operators or have continuous simple spectra eigenvectors of some operators of representations are evaluated explicitly coefficients of transition to eigenvectors overlap coefficients are given in terms ofq orthogonal polynomials it is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq deformed algebras

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

big q laguerre and q meixner polynomials and representations of the quantum algebra uq su1 1 diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra uq su1 1 is studied spectrum and eigenfunctions of this operator are explicitly found these eigenfunctions when normalized constitute an orthonormal basis in the representation space the initial uq su1 1 basis and the basis of these eigenfunctions are interconnected by a matrix with entries expressed in terms of big q laguerre polynomials the unitarity of this connection matrix leads to an orthogonal system of functions which are dual with respect to big q laguerre polynomials this system of functions consists of two separate sets of functions which can be expressed in terms of q meixner polynomials mn x b c q either with positive or negative values of the parameter b the orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix as a consequence one obtains an orthogonality relation for the q meixner polynomials mn x b c q with b 0 a biorthogonal system of functions with respect to the scalar product in the representation space is also derived