generalized laguerre polynomials and quantum mechanics we comment on serious errors in a recently published paper el sayed a m a 1999 j phys a math gen 32 8647 54

technical communique a note on fractional order derivatives of periodic functions in this paper it is shown that the fractional order derivatives of a periodic function with a specific period cannot be a periodic function with the same period the fractional order derivative considered here can be obtained based on each of the well known definitions grunwald letnikov definition riemann liouville definition and caputo definition this concluded point confirms the result of a recently published work proving the non existence of periodic solutions in a class of fractional order models also based on this point it can be easily proved the absence of periodic responses in a wider class of fractional order models finally some examples are presented to show the applicability of the paper achievements in the solution analysis of fractional order systems

synthesis of complete orthonormal fractional basis functions with prescribed poles in this paper fractional orthonormal basis functions that generalize the well known fixed pole rational basis functions are synthesized for a range of noninteger differentiation orders and under mild restrictions on the choice of the basis poles the synthesized basis functions are shown to be complete in the space of functions which are analytic on the open right half plane and square integrable on the imaginary axis this presents an extension of the completeness results for the fractional laguerre and kautz bases to fractional orthonormal bases with prescribed pole locations

on the generalized laguerre polynomials of arbitrary fractional orders and quantum mechanics the generalized laguerre polynomials l x of arbitrary order have been defined by the author el sayed 1997 math sci res hot line 1 7 14 1999 to appear in the latter reference it is proved that they are continuous as functions of and some other properties that generalize interpolate those of the classical laguerre polynomials ln x n 1 2 have been proved here we prove that l x are orthogonal in l2 0 and are particular solutions of the differential equation generalizing the one for ln x n 1 2 also some applications in quantum mechanics are discussed

maximum number of frequencies in oscillations generated by fractional order lti systems in this paper relation between the inner dimension of a fractional order lti system and the maximum number of frequencies which exist in oscillations generated by the system is investigated the considered system is defined in pseudo state space form and the orders of its involved fractional derivatives are rational numbers between zero and one first an upper bound is derived for the maximum number of frequencies then using the restricted difference bases concept a new method is introduced to design a multifrequency oscillatory fractional order system finally based on the proposed method some lower bounds are derived for the maximum number of frequencies obtainable in solutions of a fractional order system having a fixed inner dimension

simultaneous confidence bands for linear regression and smoothing suppose we observe y i f x i varepsilon i i 1 ldots n we wish to find approximate 1 alpha simultaneous confidence regions for f x x in mathscr x our regions will be centered around linear estimates hat f x of nonparametric or nonparametric f x there is a large amount of previous work on this subject substantial restrictions have been usually placed on some or all of the dimensionality of x the class of functions f that can be considered the class of linear estimates hat f and the region mathscr x the method we present is an approximation to the tube formula dn can be used for multidimensional x and a wide class of linear estimates by considering the effect of bias we are able to relax assumptions on the class of functions f which are considered simultaneous and numerical computations are used to illustrate the performance

a novel class of fractionally orthogonal quasi polynomials and new fractional quadrature formulas a novel class of quasi polynomials orthogonal with respect to the fractional integration operator has been developed in this paper the related gaussian quadrature formulas for numerical evaluation of fractional order integrals have also been proposed by allowing the commensurate order of quasi polynomials to vary independently of the integration order a family of fractional quadrature formulas has been developed for each fractional integration order including novel quadrature formulas for numerical approximation of classical integer order integrals a distinct feature of the proposed quadratures is high computational efficiency and flexibility as will be demonstrated in the paper as auxiliary results the paper also presents methods for lagrangian and hermitean quasi polynomial interpolation and hermitean fractional quadratures the development is illustrated by numerical examples

simple and robust experiment design for system identification using fractional models this paper tackles the problems of simple and robust experiment design for system identification using elementary fractional models it is based on a frequency domain approach and allows to determine the best sine input signal maximizing d optimality criterion of the parameters inverse covariance matrix in different contexts first a single parameter any of the parameters of the elementary fractional model is assumed to be unknown next any combination of two and then three parameters are supposed to be unknown finally the problem of robust experiment design is treated when a bounded interval of the estimated parameters is known in the same contexts

large amplitude time periodic solutions of a semilinear wave equation abstract this paper studies the existence of periodic solutions for a family of semilinear wave equations where the restoring force is independent of time monotone and grows at a more rapid rate than linear near infinity with appropriate technical assumptions it is shown that there is an unbounded sequence of such free vibrations i e there are solutions of arbitrarily large amplitude if the restoring force is independent of x the monotonicity assumption can be omitted

system identification with generalized orthonormal basis functions a least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function where the expansion is in terms of recently introduced generalized basis functions the basis functions are orthogonal in h2 and generalize the pulse laguerre and kautz bases one of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion this leads to accurate approximate models with only a few coefficients to be estimated explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates

confidence bands for least squares support vector machine classifiers a regression approach this paper presents bias corrected 100 email protected simultaneous confidence bands for least squares support vector machine classifiers based on a regression framework the bias which is inherently present in every nonparametric method is estimated using double smoothing in order to obtain simultaneous confidence bands we make use of the volume of tube formula we also provide extensions of this formula in higher dimensions and show that the width of the bands are expanding with increasing dimensionality simulations and data analysis support its usefulness in practical real life classification problems

stability and bifurcation of delayed fractional order dual congestion control algorithms in this technical note fractional order congestion control systems are introduced for the first time in comparison with the conventional integer order dual congestion control algorithms the fractional control algorithms are more accurate and versatile bifurcation theory in fractional order differential equations is still an outstanding problem sufficient conditions for the occurrence of hopf bifurcations are extended from integer order dynamical systems to fractional order cases then these conditions are used to establish the existence of hopf bifurcations for the delayed fractional order model of dual congestion control algorithms proposed in this note finally the onsets of bifurcations are identified where hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium illustrative examples are also provided to demonstrate the theoretical results

brief paper synthesis of fractional laguerre basis for system approximation fractional differentiation systems are characterized by the presence of non exponential aperiodic multimodes although rational orthogonal bases can be used to model any l2 0 system they fail to quickly capture the aperiodic multimode behavior with a limited number of terms hence fractional orthogonal bases are expected to better approximate fractional models with fewer parameters intuitive reasoning could lead to simply extending the differentiation order of existing bases from integer to any positive real number however classical laguerre and by extension kautz and generalized orthogonal basis functions are divergent as soon as their differentiation order is non integer in this paper the first fractional orthogonal basis is synthesized extrapolating the definition of laguerre functions to any fractional order derivative completeness of the new basis is demonstrated hence a new class of fixed denominator models is provided for fractional system approximation and identification

brief paper stability and resonance conditions of elementary fractional transfer functions elementary fractional transfer functions are studied in this paper some basic properties of elementary transfer functions of the first kind are recalled then two main results are presented regarding elementary fractional transfer functions of the second kind written in a canonical form and characterized by a commensurate order a pseudo damping factor and a natural frequency first stability conditions are established in terms of the pseudo damping factor and the commensurate order as a corollary to matignon s stability theorem they extend the previous result into conditions that are simpler to check then resonance conditions are established numerically in terms of the commensurate order and the pseudo damping factor and give interesting information on the frequency behavior of fractional systems it is shown that elementary transfer functions of the second kind might have up to two resonant frequencies moreover three abaci are given allowing to determine the pseudo damping factor and the commensurate order for respectively a desired normalized gain at each resonance a desired phase at each resonance and a desired normalized first or second resonant frequency

dynamic siso and miso system approximations based on optimal laguerre models a general procedure based on the knowledge of the input and the output signals is proposed to approximate the prescribed linear time invariant lti systems by means of optimal laguerre models the main contribution of this paper is to apply the newton raphson s iterative technique to compute the so called optimal laguerre pole in a continuous time case or optimal time scale factor in a discrete time case and especially to show that the gradient and the hessian can be expressed analytically moreover the excitations used are not limited to the ones that ensure the orthogonality of the outputs of laguerre filters i e dirac delta or white noise as is usually done in existing methods however persistently exciting input signal s are used the proposed procedure will be directly formulated for siso systems siso systems being a special case with the number of inputs equal to one the proposed algorithm has direct applications in system identification model reduction and noisy modeling

a note on ℒ p norms of fractional systems this paper discusses p norm 1 p finiteness property of the impulse response of a stable fractional system cascaded with a pure fractional integrator the established results are used to choose a performance index for evaluating the output feedback control system performances

brief paper a proof for non existence of periodic solutions in time invariant fractional order systems the aim of this note is to highlight one of the basic differences between fractional order and integer order systems it is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals as a result a limit cycle cannot be expected in the solution of these systems our investigation is based on caputo s definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

two class support vector data description support vector data description svdd is a data description method that can give the target data set a spherically shaped description and be used to outlier detection or classification in real life the target data set often contains more than one class of objects and each class of objects need to be described and distinguished simultaneously in this case traditional svdd can only give a description for the target data set regardless of the differences between different target classes in the target data set or give a description for each class of objects in the target data set in this paper an improved support vector data description method named two class support vector data description tc svdd is presented the proposed method can give each class of objects in the target data set a hypersphere shaped description simultaneously if the target data set contains two classes of objects the characteristics of the improved support vector data descriptions are discussed the results of the proposed approach on artificial and actual data show that the proposed method works quite well on the 3 class classification problem with one object class being undersampled severely

parameter and differentiation order estimation in fractional models this paper deals with continuous time system identification using fractional differentiation models an adapted version of the simplified refined instrumental variable method is first proposed to estimate the parameters of the fractional model when all the differentiation orders are assumed known then an optimization approach based on the use of the developed instrumental variable estimator is presented two variants of the algorithm are proposed either all differentiation orders are set as integral multiples of a commensurate order which is estimated or all differentiation orders are estimated the former variant allows to reduce the number of parameters and can be used as a good initial hit for the latter variant the performances of the proposed approaches are evaluated by monte carlo simulation analysis finally the proposed identification algorithms are used to identify thermal diffusion in an experimental setup

a unifying construction of orthonormal bases for system identification this paper develops a general and very simple construction for complete orthonormal bases for system identification this construction provides a unifying formulation of many previously studied orthonormal bases since the common fir and recently popular laguerre and two parameter kautz model structures are restrictive special cases of the construction presented here however in contrast to these special cases the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided

on strongly indefinite functionals with applications recently in their remarkable paper critical point theory for indefinite functionals v benci and p rabinowitz gave a direct approach avoiding finitedimensional approximationsto the existence theory for critical points of indefinite functionals in this paper we develop under weaker hypotheses a simpler but more general theory for such problems in the second part of the paper the abstract results are applied to a class of resonance problems of the landesman and lazer type and moreover they are illustrated by an application to a wave equation problem 0 introduction this paper is devoted to the critical point theory of strongly indefinite functionals in a real hilbert space h here a strongly indefinite functional is a cl map h r which is neither bounded from above or below not even modulo a finite dimensional subspace such functionals arise for example in the study of periodic solutions of the one dimensional wave equation in the study of periodic solutions of hamiltonian systems of ordinary differential equations in the existence theory for systems of elliptic equations etc to find the critical points of strongly indefinite functionals different methods were used for example rabinowitz studied the restrictions of the functional to finite dimensional subspaces 1 2 and then he tried to find the critical points of the original problem by passing to the limit for a suitable subsequence of the critical points of the restricted problems amann 3 reduced the indefinite functionals by means of a saddlepoint reduction to a finite dimensional problem of course this is only possible under some strong hypotheses under certain assumptions the functional can be replaced by a dual variational problem having some better properties see for example 4 or 5 the first direct approach was given by benci and rabinowitz 6 the essential in their paper is the observation that more detailed information about the special form of the deformations a substitute for the gradient flow associated to the functional is needed they constructed deformations having special representations by solving appropriate differential equations approximately by time discretisation this method has some disadvantages because one needs strong hypotheses in order to get the approximations uniformly on bounded sets a main part of our paper will be the construction of flows and deformations having special representations by solving appropriate differential equations exactly received by the editors march 30 1981 and in revised form november 30 1981 1980 mathematics subject classification primary 58e05 secondary 47h25 35l05

brief paper analytical computation of the h2 norm of fractional commensurate transfer functions 2 norm or impulse response energy of any fractional commensurate transfer function is computed analytically a general expression depending on transfer function coefficients and differentiation orders is established then more concise expressions are given for elementary fractional transfer functions unlike stable rational transfer functions it is proven that the 2 norm of stable fractional transfer functions may be infinite finiteness conditions are established in terms of transfer function relative degree moreover it is proven that the 2 norm of a fractional transfer function with a proper integrator of order less than 0 5 may be finite the obtained results are used to evaluate the integral squared error of closed loop control systems