an algebraic construction of generalized coherent states for shape invariant potentials generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able to a supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system continuity of labelling resolution of unity temporal stability and action identity b reproduce results already known for shape invariant systems such as harmonic oscillator double anharmonic poschl teller and self similar potentials and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems

structural modal interaction with combination internal resonance under wide band random excitation in this paper the non linear interaction of a three degree of freedom structural model subjected to a wide band random excitation is examined the non linearity of the system results in different critical regions of internal resonance and this has a significant effect on the response statistics with reference to the combination internal resonance of the summed type the system response is analyzed by using the fokker planck equation approach together with a non gaussian closure scheme the non gaussian closure is based on the cumulant properties of order greater than three as a first order approximation the scheme yields 209 first order differential equations in first through fourth order joint moments of the response co ordinates the analysis is carried out with the aid of the computer algebra software macsyma the response statistics are determined numerically in the time and frequency internal detuning domains contrary to the gaussian closure scheme the non gaussian closure solution yields a strictly stationary response in addition to a number of complex response characteristics not previously reported in the literature of the area of non linear random vibration these include multiple solutions and jump phenomena jump and collapse in the response mean squares at internal detuning slightly shifted from the exact internal resonance condition at exact internal resonance the system response possesses a unique limit cycle in a stochastic sense the regions of multiple solutions are defined in terms of system parameters damping ratios and non linear coupling parameter and excitation spectral density level

raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

dynamics of a single degree of freedom structure with quadratic cubic and quartic non linearities to a harmonic resonance the solution of a non linear equation having often encountered in the study of structural dynamically and in which both parametric and external excitations are present have been obtained analytically and numerically the only considered case is when both the parametric and external excitations have the same frequency the method of multiple time scale is applied to determine the equations that describe up to and including the second order both the modulation of the amplitude and phase these equations are used to determine the fixed points and their stability frequency response curves are presented graphically discussion of the results is given numerical solutions are presented for the effects of the different equation parameters on system stability response and chaos different cases of resonance are investigated and reported

shape invariance raising and lowering operators in hypergeometric type equations the schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self adjoint operators defined in the theory of hypergeometric type equations the fundamental formulae occurring in a supersymmetric approach to these hamiltonians are consequences of some formulae concerning the general theory of associated special functions we use this connection in order to obtain a general theory of schrodinger equations exactly solvable in terms of associated special functions and to extend certain results known in the case of some particular potentials

theoretical and experimental study of modal interaction in a two degree of freedom structure abstract fpr a two degree of freedom structure an experimental and theoretical investigation has been made of the primary resonances of the system which occur when the frequency of excitation is near one of the natural frequencies 1 and 2 the additional constraint of 2 2 1 has been included and its profound influence on the response studied the investigation has revealed some characteristics of the response that are the result of both non linearity and more than one degree of freedom the principal one of these being saturation as far as is known this is the first time that saturation has beeb observed and studied in detail in a physical structure

modal interaction of resonantly forced oscillations of two degree of freedom structure an analysis is presented for simultaneous primary resonances of two degree of freedom system with quadratic and cubic nonlinearities to external excitations having the frequencies w 1 and w 2 the relations between the external frequencies of the system are such that they are excited into external resonances in two cases the first case without internal resonances and the second case in the presence of different possibilities of internal resonances one to one two to one three to one one to two and one to three the method of multiple scales mms is used to determine four first order ordinary differential equations which describe the modulation of the amplitudes and phases of both modes in order to use the mms properly we introduce new detuning parameters of both modes it is found analytically that two mode responses can occur at each primary resonance in all cases the steady state solutions and their stability are determined numerical results depicting the resonances are presented

generalized coherent states for associated hypergeometric type functions we introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric type operators this makes possible the definition of generalized coherent states for the associated eigenfunctions these generalized coherent states satisfy the relevant properties i e label continuity overcompleteness temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system

non linear vibrations of free edge thin spherical shells modal interaction rules and 1 1 2 internal resonance this paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement the analog for thin shallow shells of von karman s theory for large deflection of plates is used the validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution the specific case of a free edge is considered the governing partial differential equations are expanded onto the natural modes of vibration of the shell the problem is replaced by an infinite set of coupled second order differential equations with quadratic and cubic non linear terms analytical expressions of the non linear coefficients are derived and a number of them are found to vanish as a consequence of the symmetry of revolution of the structure then for all the possible internal resonances a number of rules are deduced thus predicting the activation of the energy exchanges between the involved modes finally a specific mode coupling due to a 1 1 2 internal resonance between two companion modes and an axisymmetric mode is studied 2004 elsevier ltd all rights reserved

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

non linear beam oscillations excited by lateral force at combination resonance abstract non linear oscillations of a beam subjected to a periodic force at a combination resonance are considered using the galerkin method a partial differential equation of oscillations is reduced to a system of ordinary differential equations with a small parameter a system of three autonomous differential equations is derived the multiple scales method being used qualitative properties of trajectories are analyzed the naimark sacker bifurcations at the combination resonance are analyzed by the center manifold method almost periodic oscillations of a beam arise due to these bifurcations these oscillations are investigated qualitatively and numerically

a general theory of phase space quasiprobability distributions we present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are finite dimensional lie groups the family of distributions on a phase space is postulated to satisfy the stratonovich weyl correspondence with a generalized traciality condition the corresponding family of the stratonovich weyl kernels is constructed explicitly in the presented theory we use the concept of generalized coherent states that brings physical insight into the mathematical formalism

period one motions of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness the dynamic behavior of a piecewise nonlinear mechanical oscillator with parametric and external excitations is investigated the viscously damped single degree of freedom oscillator is subjected to a periodically time varying piecewise nonlinear restoring function defined by a clearance surrounded by continuously nonlinear quadratic and cubic regions typical applications represented by this oscillator are highlighted a multi term harmonic balance formulation is used in conjunction with discrete fourier transforms and a parametric continuation scheme to determine steady state period 1 motions of the system due to both parametric and external excitations the accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions floquet theory is applied to determine the stability of the steady state harmonic balance solutions at the end detailed parametric studies are presented to quantify the combined influence of clearance quadratic and cubic nonlinearities within reasonable ranges of all other system parameters a comparison between time varying and time invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise nonlinear system

the growth of bargmann functions and the completeness of sequences of coherent states the growth of bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of bargmann functions of an order not exceeding and of a type not exceeding if of order these results generalize known results on the completeness of von neumann lattices the practical significance of this formalism in the context of quantum optics is also discussed

combination parametric resonance leading to periodic and chaotic response in two degree of freedom systems with quadratic non linearities the dynamic response of a parametrically excited non linear system with two degrees of freedom is studied the system of equations which describes the compliant motion of a robotic manipulator exhibits internal resonance and is excited at a frequency near a primary resonance the effect of coupling is studied in detail non zero periodic motions are observed to co exist with a stable equilibrium state over a range of detuning near the resonant frequency loss of stability of the periodic response is observed at points of vertical tangency corresponding to a jump in the response while for some of the cases hopf bifurcation in the response corresponding to amplitude modulated motions is observed in the latter case period doubling of the amplitude of response is observed with a monotonic change in detuning eventually leading to chaotic motion in the system near exact tuning to the parametric resonance a quenching of the chaotic motion is found leaving only the stable zero equilibrium state for the system over some detuning interval

temporally stable coherent states for infinite well and poschl teller potentials this article is a direct illustration of a construction of coherent states which has been recently proposed by two of us jpg and jk we have chosen the example of a particle trapped in an infinite square well and also in poschl teller potentials of the trigonometric type in the construction of the corresponding coherent states we take advantage of the simplicity of the solutions which ultimately stems from the fact they share a common su 1 1 symmetry a la barut girardello many properties of these states are then studied both from mathematical and from physical points of view

a non linear analysis of the interactions between parametric and external excitations abstract solutions of a non linear equation having a form often encountered in the study of structural dynamics and in which both parametric and external excitation is present have been obtained analytically and numerically all the asymptotic analytical results can be verified by numerical integration for small values of some other solutions obtained numerically cannot be approximated by conventional asymptotic methods both types of solutions can exist simultaneously in these situations the initial conditions determine which solution describes the actual response of the system the solutions that can be predicted only by numerical means exhibit period multiplications i e the response has a period that is an integral multiple of the period of the excitation and chaos in fact in a certain range of one set of initial conditions can lead to period doubling in the response while another set can lead to a response very close to the asymptotic prediction very flexible structures can exhibit non linear characteristics in their response at very small amplitudes of the excitation the present results serve to illustrate some of the extraordinary behavior that such structures can exhibit and suggest that it may be extremely difficult to develop control systems for them

explicit wavefunctions for shape invariant potentials by operator techniques the authors obtain explicit expressions for the wavefunctions of all the known shape invariant potentials by using the operator method

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

generalized hypergeometric coherent states we introduce a large class of holomorphic quantum states by choosing their normalization functions to be given by generalized hypergeometric functions we call them generalized hypergeometric states in general and generalized hypergeometric coherent states in particular if they allow a resolution of unity depending on the domain of convergence of the generalized hypergeometric functions we distinguish generalized hypergeometric states on the plane the open unit disc and the unit circle all states are eigenstates of suitably defined lowering operators we then study their photon number statistics and phase properties as revealed by the husimi and pegg barnett phase distributions on the basis of the generalized hypergeometric coherent states we introduce new analytic representations of arbitrary quantum states in bargmann and hardy spaces as well as generalized hypergeometric husimi distributions and corresponding phase distributions

analytic approach to the problem of convergence of truncated levy flights towards the gaussian stochastic process an analytic expression for characteristic function defining a truncated l evy flight is derived it is shown that the characteristic function yields results in agreement with recent simulations of truncated l evy flights by mantegna and stanley phys rev lett 73 2946 1994 with the analytic expression for the characteristic function the convergence of the l evy process towards the gaussian is demonstrated without simulations in the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures

coherent states for transparent potentials the darboux transformation operator method is applied to the investigation of coherent states of transparent multisoliton potentials an isometric correspondence between hilbert spaces of the states of a free particle and a particle moving in the soliton potential is established it is shown that the darboux transformation operator being unbounded and closed cannot realize an isometric mapping between hilbert spaces a quasispectral representation of transformation operators in terms of continuous basis sets is obtained different families of coherent states of the multisoliton potential are introduced measures that realize the resolution of the unity in terms of the projectors on the coherent states vectors are calculated these measures are defined by ordinary smooth functions for the states obtained with the help of bounded transformation operators and by generalized ones otherwise

coherent states for systems with discrete and continuous spectrum for a system with one degree of freedom coherent states that are parametrized by classical canonical action angle variables are introduced these states also possess continuity of labelling a resolution of unity and temporal stability the insistence on canonical action angle variables strongly restricts any remaining arbitrariness in the coherent state definition such states are introduced for semibounded hamiltonian operators having either a discrete or a continuous spectrum hamiltonians that have both discrete and continuous parts in their spectrum are also discussed

q ν deformation of generalized basic hypergeometric states we provide with a q deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions these states are eigenstates of suitably defined deformed lowering operators we study the domain of convergence of the corresponding normalization function on the basis of these states we investigate generalized basic hypergeometric husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in bargmann and hardy spaces the quantum statistical properties of the states such as photon counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics

factorization of sturm liouville operators solvable potentials and underlying algebraic structure in this paper a general method of factorization of sturm liouville sl operators is provided a method to solve sl eigenvalue problems is presented new classes of exactly solvable potentials are obtained the supersymmetry and shape invariance approaches are generalized to the sl operators it is shown that the sl shape invariance potentials have an underlying algebraic structure this algebra is in general infinite dimensional the condition of finite algebra is obtained

hardening softening behaviour in non linear oscillations of structural systems using non linear normal modes the definition of a non linear normal mode nnm as an invariant manifold in phase space is used in conservative cases it is shown that normal form theory allows one to compute all nnms as well as the attendant dynamics onto the manifolds in a single operation then a single mode motion is studied the aim of the present work is to show that too severe truncature using a single linear mode can lead to erroneous results using single non linear mode motion predicts the correct behaviour hence the nonlinear change of co ordinates allowing one to pass from the linear modal variables to the normal ones linked to the nnms defines a framework to properly truncate non linear vibration pdes two examples are studied a discrete system a mass connected to two springs and a continuous one a linear euler bernoulli beam resting on a non linear elastic foundation for the latter a comparison is given between the developed method and previously published results

tuneable non degenerated nonlinear parametrically excited amplifier abstract the proposed parametric amplifier scheme can be tuned to amplify a wide range of input frequencies by altering the parametric excitation with no need to physically modify the oscillator parametric amplifiers had been studied extensively although most of the work focused on amplifiers that are parametrically excited at a frequency twice the amplifier s natural frequency these amplifiers are confined to amplifying predetermined frequencies the proposed parametric amplifier s bandwidth is indeed tuneable to nearly any input frequency not bound to be an integer multiple of a natural frequency in order to tune the stiffness and induce a variable frequency parametric excitation a digitally controlled electromechanical element must be incorporated in the realization we introduce a novel parametric amplifier with nonlinearity duffing type hardening that bounds the otherwise unlimited amplitude moreover we present a multi degree of freedom system in which a utilization of the proposed method enables the projection of low frequency vector forces on any eigenvector and corresponding natural frequency of the system and thus to transform external excitations to a frequency band where signal levels are considerably higher using the method of multiple scales analytical expressions for the responses have been retrieved and verified numerically parametric studies of the amplifiers gain sensitivities and spatial projection of the excitation on the system eigenvectors were carried out analytically the results demonstrate the advantage of the proposed approach over existing schemes practical applications envisaged for the proposed method will be outlined

experimental investigation of structural linear and autoparametric interactions under random excitation when the first normal mode is externally excited by a band limited random excitation the system mean square response is found to be linearly proportional to the excitation spectal density up to a certain level above which the two normal modes exhibit discontinuity when the second normal mode is externally excited for small levels of excitation spectral density the response is dominated by the second normal mode for higher levels of excitation spectral density the first normal mode attends and interacts with the second normal mode

tempered fractional diffusion equations for pricing multi asset options under cgmye process abstract we concentrate on the analytical study on the pricing model for options written on multi assets these assets are assumed to follow a multi variate cgmye process through linear combination we show that a european option satisfies a tempered fractional partial differential equation for which the fundamental solution is given in an explicit form for american options the decomposition formula and the integral equation for the optimal exercise boundary are also established furthermore parameter estimations are implemented for the multi dimensional problem and several numerical simulations are provided for a european basket put

algebraic nature of shape invariant and self similar potentials self similar potentials generalize the concept of shape invariance which was originally introduced to explore exactly solvable potentials in quantum mechanics in this paper it is shown that the previously introduced algebraic approach to the latter can be generalized to the former the infinite lie algebras introduced in this context are shown to be closely related to the q algebras the associated coherent states are investigated

stochastic response of nonlinear structures with parameter random fluctuations the random response of a nonlinear structural system is examined when its parameters are experiencing random fluctuations with time the treatment is based on the recent developments in the mathematical theory of stochastic differential equations these include the ito stochastic calculus and the fokker planck equation approach to derive a general differential equation that describes the evolution of the statistical moments of the response coordinates the differential equation is found to constitute an infinite coupled set of differential equations that are closed via two different closure schemes the system response is determined in the neighborhood of internal resonance condition and for various random intensities of the system parameters it is found that the random modal interaction is governed mainly by the internal resonance ratio and the stiffness fluctuation intensity the effect of the random damping fluctuation on the system response is found to be very small compared to the stiffness fluctuation effect

forcing induced asymmetry on dynamical systems with cubic non linearities abstract the present work investigates the dynamics of a class of two degree of freedom oscillators with cubic non linearity in the restoring forces these oscillators are under the action of an external load including constant and harmonic components initially a perturbation analysis is applied to the equations of motion demonstrating the effect of the asymmetry induced by the constant loading component on the classical 1 1 and 1 3 internal resonances as well as on the possibility of the appearance of a first order 1 2 internal resonance next sets of slow flow equations governing the amplitudes and phases of vibration are derived for the special case of no internal resonance and for the most complicated case corresponding to 1 1 internal resonance the analytical findings are then complemented by numerical results obtained by examining the dynamics of a two degree of freedom mechanical system first the effect of certain system parameters on the existence and stability of constant and periodic solutions of the slow flow equations is illustrated by presenting a sequence of response diagrams finally the dynamics of the system used as an example is investigated further by direct integration of the slow flow equations this shows the existence of a period doubling sequence culminating into a continual interchange between quasiperiodic and chaotic response it also demonstrates a new transition scenario from phase locked to phase entrained and drift response