on the two q analogue logarithmic functions there is a simple multi sheet riemann surface associated with s inverse function for a principal sheet for can be defined however the topology of the riemann surface for changes each time q increases above the collision point of a pair of the turning points of there is also a power series representation for an infinite product representation for is used to obtain the ordinary natural logarithm and the values of the sum rules for the zeros of for where the values of the sum rules for the q trigonometric functions and are q deformations of the usual bernoulli numbers

new q deformed coherent states with an explicitly known resolution of unity we construct a new family of q deformed coherent states zq where 0 q 1 these states are normalizable on the whole complex plane and continuous in their label z they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem and expressed in terms of one of the two jacksons s q exponentials they also permit exact evaluation of matrix elements of physically relevant operators we use this to show that the photon number statistics for the states is sub poissonian and that they exhibit quadrature squeezing as well as an enhanced signal to quantum noise ratio over the conventional coherent state value finally we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

on the zeros of the q analogue exponential function an asymptotic formula for the zeros zn of the entire function eq x for q 1 is obtained as q increases above the first collision point at q1 approximately 0 14 these zeros collide in pairs and then move off into the complex z plane they move off as and remain a complex conjugate pair the zeros of the ordinary higher derivatives and of the ordinary indefinite integrals of eq x vary with q in a similar manner properties of eq z for z complex and for arbitrary q are deduced for 0 or q 1 eq z is an entire function of order 0 by the hadamard weierstrass factorization theorem infinite product representations are obtained for eq z and for the reciprocal function eq 1 z if q not 1 the zeros satisfy the sum rule sigma n 1infinity 1 zn 1

predicting response bounds for friction damped gas turbine blades with uncertain friction coupling abstract friction dampers are often used to reduce high amplitude vibration within gas turbines they are a robust solution that are able to withstand extreme operating environments although the turbine blades are manufactured to tight tolerances there can be significant variability in the overall response of the assembly uncertainties associated with the frictional contact properties are a major factor contributing to this variability this paper applies a recently developed method for predicting response bounds to friction damped gas turbines when the characteristics of the friction dampers are unknown including uncertainty regarding the underlying functional form of the friction law the approach taken is to represent the frictional contact using a describing function and formulate an optimisation problem to seek upper and lower bounds on a chosen response metric such as displacement amplitude constraints are chosen that describe known properties of the frictional nonlinearity without needing to specify a particular constitutive law the method was validated by comparison with numerical and experimental results from an idealised test system the experimental test rig consisted of an array of eight beams coupled by pin contact friction dampers a modal description of this test rig formed the basis of a numerical model which uses the harmonic balance method hbm for nonlinear simulations a set of monte carlo tests was carried out numerically and experimentally for both a two beam sub assembly as well as for the full eight beam assembly comparisons with numerical results showed excellent agreement providing confident verification of the implementation and comparisons with experimental results revealed that the bounds became less conservative as the system complexity increased overall the results are promising upper and lower response bounds for an array of friction damped systems can be computed at similar cost to a single hbm simulation giving reliable bounds that are valid for both parametric and model uncertainties associated with the friction couplings

basic sturm liouville problems this paper is devoted to studying a q analogue of sturm liouville eigenvalue problems we formulate a self adjoint q difference operator in a hilbert space some of the properties of the eigenvalues and the eigenfunctions are discussed green s function is constructed and the problem in question is inverted into a q type fredholm integral operator with a symmetric kernel the set of eigenfunctions is shown to be a complete orthogonal set in the hilbert space examples involving basic trigonometric functions are involved

response bounds for complex systems with a localised and uncertain nonlinearity predicting the vibration response of complex nonlinear structures is a significant challenge the response may involve many modes of the structure nonlinearity precludes the use of efficient techniques developed for linear systems and there is often uncertainty associated with the nonlinear law even to the extent that its functional form is not always known this paper builds on a recently developed method for handling this class of problem in a novel way the method exploits the fact that nonlinearities are often spatially localised and seeks the best and worst case system response with respect to a chosen metric by regarding the internal nonlinear force as an independent excitation to the underlying linear system constraints are used to capture what is thought to be known about the nonlinearity without needing to specify a particular law this paper focuses on the case of systems with a single point nonlinearity but with arbitrarily complex underlying linear dynamics driven by a sinusoidal force excitation semi analytic upper and lower bounds are proposed for root mean square response metrics subject to constraints which specify that the nonlinearity should be a combination of a passive b displacement limited and or c force saturating the concept of equivalent linear bounds is also introduced for cases where the response metric is thought to be dominated by the same frequency as the input the bounds corresponding to a passive and displacement limited nonlinearity are compared with monte carlo experimental and numerical results from an impacting beam test rig the bounds corresponding to a passive and force saturating nonlinearity are compared with numerical results for a friction damped beam the global upper and lower bounds are satisfied for all input frequencies but are generally found to be rather conservative the equivalent linear bounds show remarkably good agreement for predicting the range of root mean square velocity responses finally the principle of maximum entropy is used to estimate the response distributions which was found to give surprisingly good agreement with experimental and numerical data

a completeness relation for the q analogue coherent states by q integration q integration is defined for the q oscillator realization of quantum groups this is used to prove a completeness relation for the q analogue of the usual coherent states these states are overcomplete

quantum diffusions and appell systems within the algebraic framework of hopf algebras random walks and associated diffusion equations master equations are constructed and studied for two basic operator algebras of quantum mechanics i e the heisenberg weyl algebra hw and its q deformed version hw q this is done by means of functionals determined by the associated coherent state density operators the ensuing master equations admit solutions given by hw and hw q valued appell systems

q deformed oscillator associated with the calogero model and its q coherent state we present the q deformed para bose oscillators associated with the two body calogero model the q deformed coherent state is also constructed and its resolution of unity is demonstrated

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

completeness of coherent states associated with self similar potentials and ramanujan s integral extension of the beta function a decomposition of identity is given as a complex integral over the coherent states associated with a class of shape invariant self similar potentials there is a remarkable connection between these coherent states and ramanujan s integral extension of the beta function

generating function for clebsch gordan coefficients of the suq 2 quantum algebra some methods have been developed to calculate the suq 2 clebsch gordan coefficients cgc here the authors develop a method based on the calculation of clebsch gordan generating functions through the use of quantum algebraic coherent states calculating the suq 2 cgc by means of this generating function is an easy and straightforward task

quantum deformation of kdv hierarchies and their infinitely many conservation laws a reasonable q deformed differential is defined a set of operation rules are constructed for the q deformed formal pseudo differential operators the complete procedure of constructing the q deformed kdv hierarchies and their infinite conservation laws is given as an important example he obtains a detailed structure of the simplest 3 2 system i e the q deformed kdv equations

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

an algebraic construction of generalized coherent states associated with q deformed models for primary shape invariant systems generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able a to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system b to reproduce as particular cases results already known for shape invariant systems such as standard harmonic oscillator and p schl teller potentials as well as quantum deformed harmonic oscillator models and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems deformed or not deformed

maximum principles and bounds in a class of fourth order uniformly elliptic equations this paper is devoted to a class of fourth order uniformly elliptic equations posed by schaefer in 1987 we obtain maximum principles for certain functions which are defined on solutions of the elliptic equations the principles are then used to deduce some bounds on important quantities in the physical problems of interest

the number of edges in k quasi planar graphs a graph drawn in the plane is called k quasi planar if it does not contain k pair wise crossing edges it has been conjectured for a long time that for every fixed k t he maximum number of edges of a k quasi planar graph with n vertices is o n the best known upper bound is n log n o log k in the present paper we improve this bound to nlog n 2 n ck in the special case where the graph is drawn in such a way that every pair of edges meet at most once here n denotes the extremely slowly growing inverse of the ackermann function we also make further progress on the conjecture for k quasi planar graphs in which every edge is drawn as an x monotone curve extending some ideas of valtr we prove that the maximum number of edges of such graphs is at most 2 ck 6 nlog n

some realization of the quantum algebra uq su 2 for the quantum algebra uq su 2 a q analogue of the usual spin coherent state is constructed with help of coherent states the q deformed dyson and holstein primakoff realizations of the quantum algebra uq su 2 are given a transformation matrix which turns the dyson mapping onto the holstein primakoff is presented

algebraic properties of rogers szego functions i applications in quantum optics by means of a well established algebraic framework rogers szego functions associated with a circular geometry in the complex plane are introduced in the context of q special functions and their properties are discussed in detail the eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of rogers szego functions the coefficients being determined through proper eigenvalue equations in each situation furthermore a complementary study on the robertson schrodinger and symmetrical uncertainty relations for the cosine sine and nondeformed number operators is also conducted corroborating in this way certain features of q deformed coherent states

q integrals on the quantum complex plane q integration is defined for the complex quantum plane the existence of classical and grassmanian limits of q integration is proved quantum versions of cauchy s and stokes theorems are formulated

the multicomponent q coherent states of the quantum algebra slq 3 the multicomponent q coherent states associated with the quantum algebra slq 3 are presented the expansions are discussed for arbitrary operators in terms of the multicomponent q coherent states of slq 3 an inhomogeneous differential realization of slq 3 in this multicomponent q coherent state space is obtained

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

stochastic modeling of uncertainties in computational structural dynamics recent theoretical advances this paper deals with a short overview on stochastic modeling of uncertainties we introduce the types of uncertainties the variability of real systems the types of probabilistic approaches the representations for the stochastic models of uncertainties the construction of the stochastic models using the maximum entropy principle the propagation of uncertainties the methods to solve the stochastic dynamical equations the identification of the prior and the posterior stochastic models the robust updating of the computational models and the robust design with uncertain computational models we present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties and for their identification with experimental data we also present the construction of the posterior stochastic model of uncertainties using the bayesian method

the q superoscillators in supersymmetric quantum systems and their q supercoherent states in this paper the q deformed quantum harmonic oscillators in the supersymmetric quantum systems are introduced the q supercoherent states associated with the q deformed supersymmetric quantum harmonic oscillators are constructed explicitly and their properties are investigated in detail it is shown that the q supercoherent states are completeness the uncertainty relations for q supercoherent states are discussed as well

eigenkets of the q deformed creation operator by using the contour integral representation of function and the technique of integration within an ordered product of operators we point out that the q deformed creation operator possesses the eigenkets a set of new completeness and orthogonality relations composed of the kets and bras which are not mutually hermitian conjugates are derived application of the completeness relation in constructing the generalized p representation of density operator is demonstrated

path integrals for quantum algebras and the classical limit coherent states path integral formalism for the simplest quantum algebras q oscillator suq 2 and suq 1 1 is introduced in the classical limit canonical structure is derived with a modified symplectic and riemannian metric non constant deformation induced curvature for the phase spaces is obtained

physical consequences of a nonparametric uncertainty model in structural dynamics abstract one of the main claims of the nonparametric model of random uncertainty introduced by soize 2000 3 is its ability to account for model uncertainty the present paper investigates this claim by examining the statistics of natural frequencies total energy and underlying dispersion equation yielded by the nonparametric approach for two simple systems a thin plate in bending and a one dimensional finite periodic mass spring chain results for the plate show that the average modal density and the underlying dispersion equation of the structure are gradually and systematically altered with increasing uncertainty the findings for the mass spring chain corroborate the findings for the plate and show that the remote coupling of nonadjacent degrees of freedom induced by the approach suppresses the phenomenon of mode localization this remote coupling also leads to an instantaneous response of all points in the chain when one mass is excited in the light of these results it is argued that the nonparametric approach can deal with a certain type of model uncertainty in this case the presence of unknown terms of higher or lower order in the governing differential equation but that certain expectations about the system such as the average modal density may conflict with these results

a q analog of the quantum central limit theorem for suq 2 a q analog of the central limit theorem for suq 2 q 0 is studied it is shown that the limits of the moments and q exponential generating functions for coherent states give for q 1 0 q 1 the harmonic q oscillator q 1 oscillator introduced by biedenharn and macfarlane

new p q μ ν f deformed states we construct a new family of deformed states these states are normalizable on the whole complex plane and continuous in their label z under fixed values of parameters they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem in addition to the mathematical characteristics the quantum statistical properties of these states are analytically and numerically discussed in detail in the context of conventional as well as deformed quantum optics we find that for non deformed photons the states exhibit quadrature squeezing and their photon number statistics is sub poissonian on the other hand for deformed photons the states are super poissonian and no quadrature squeezing occurs

a new completeness relation in the q deformed two mode fock space in two mode fock space a new completeness relation for the q analogue of the single complex variable state is proved by virtue of some q deformed combinatorial identities which are also derived here for the first time

multifidelity importance sampling estimating statistics of model outputs with the monte carlo method often requires a large number of model evaluations this leads to long runtimes if the model is expensive to evaluate importance sampling is one approach that can lead to a reduction in the number of model evaluations importance sampling uses a biasing distribution to sample the model more efficiently but generating such a biasing distribution can be difficult and usually also requires model evaluations a different strategy to speed up monte carlo sampling is to replace the computationally expensive high fidelity model with a computationally cheap surrogate model however because the surrogate model outputs are only approximations of the high fidelity model outputs the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high fidelity model we introduce a multifidelity importance sampling mfis method which combines evaluations of both the high fidelity and a surrogate model it uses a surrogate model to facilitate the construction of the biasing distribution but relies on a small number of evaluations of the high fidelity model to derive an unbiased estimate of the statistics of interest we prove that the mfis estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself the mfis method can be used with any type of surrogate model such as projection based reduced order models and data fit models furthermore the mfis method is applicable to black box models i e where only inputs and the corresponding outputs of the high fidelity and the surrogate model are available but not the details of the models themselves we demonstrate on nonlinear and time dependent problems that our mfis method achieves speedups of up to several orders of magnitude compared to monte carlo with importance sampling that uses the high fidelity model only

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

a novel method for predicting the response variability of friction damped gas turbine blades abstract predicting the response of gas turbine blades with underplatform friction dampers is challenging due to the combination of frictional nonlinearity and system uncertainty a traditional monte carlo approach to predicting response distributions requires a large number of nonlinear simulations which is computationally expensive this paper presents a new approach based on the principle of maximum entropy that provides an estimate of the response distribution that is approximately two orders of magnitude faster than monte carlo harmonic balance method simulations the premise is to include the concept of computational uncertainty incorporating lack of knowledge of the solution as part of the uncertainty on the basis that there are diminishing returns in computing precise solutions to an uncertain system to achieve this the method uses a describing function approximation of the friction damped part of the system chooses an ignorance prior probability density function for the complex value of the describing function based on coulombs friction law updates the distribution using an estimate of the mean solution the admissible domain of solutions and the principle of maximum entropy then carries out a linear monte carlo simulation to estimate the response distribution the approach is validated by comparison with hbm simulations and experimental tests using an idealised academic system consisting of a periodic array of beams with controllable uncertainty coupled by single point friction dampers comparisons with two and eight blade systems show generally good agreement predicting the response statistics of the maximum blade amplitude reveals specific well understood circumstances when the method is less effective predictions of the overall blade response statistics agree with monte carlo hbm extremely well across a wide range of excitation amplitudes and uncertainty levels critically experimental comparisons reveal the care that is needed in accurately characterising uncertainty in order to obtain agreement of response percentiles the new method allowed fast iteration of uncertainty parameters and correlations to achieve good agreement which would not have been possible using traditional methods

a q deformed osp 1 2 superalgebra and its two component coherent state representations a q deformed osp 1 2 superalgebra is defined by the use of a pair of q boson annihilation and creation operators a kind of two component coherent state representation of the q deformed superalgebra is found and a q differential realization of the q deformed osp 1 2 superalgebra is obtained

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

a completeness relation for the coherent states of the p q oscillator by p q integration p q integration is defined for the p q oscillator this is used to prove a completeness relation for the coherent states of the p q oscillator the p q analogue of the bargmann fock representation is also discussed

the quantum mechanical schrodinger picture of a q oscillator a q deformed version of standard quantum mechanics in the coordinate schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so called q discrete derivative as the representative of the momentum operator the chosen q discrete derivative is symmetric with respect to the exchange of q and q 1 under the usually adopted assumptions a q deformed schrodinger equation is derived for a harmonic oscillator the complete set of eigenfunctions can be explicitly constructed as special q functions and the corresponding energy eigenvalues are identical to those obtained by biedenharn in his pioneering work 1989 this q deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions

the q supercoherent states of the q deformed su 2 superalgebra the q supercoherent states associated with the q deformed su 2 superalgebra are constructed explicitly and their properties are investigated in detail the completeness relation of the q supercoherent states is proved by the use of the q integration

generalized deformed oscillator corresponding to the modified poschl teller energy spectrum a generalized deformed oscillator with eigenvalues equal to the eigenvalues of the schrodinger equation with the modified poschl teller potential is constructed for special values of the potential depth the deformed oscillator algebra has a finite dimensional irreducible representation the polynomial representation and the associate deformed operations of integration and differentiation are studied the results of this study are general and they can be applied directly to the case of the q deformed oscillator with q being a root of unity