sum rules and the domain after the last node of an eigenstate it is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable at any boundstate energy the schrodinger equation has two linearly independent solutions one of which is normalizable while the other is not in the domain after the last node of a boundstate eigenfunction the unnormalizable linearly independent solution has a simple form which enables the construction of functions analogous to green s functions that lead to certain sum rules one set of sum rules gives conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates such sum rules may be useful in the study of properties of confining potentials the exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure the relations between one of the sum rules and two particle densities and a construction based on supersymmetric quantum mechanics are discussed

subspace based fault detection robust to changes in the noise covariances the detection of changes in the eigenstructure of a linear time invariant system by means of a subspace based residual function has been proposed previously while enjoying some success in its applicability in particular in the context of vibration monitoring the robustness of this framework against changes in the noise properties has not been properly addressed yet in this paper a new robust residual is proposed and the robustness of its statistics against changes in the noise covariances is shown the complete theory for hypothesis testing for fault detection is derived and a numerical illustration is provided

convergence of the modified mann s iteration method for asymptotically strict pseudo contractions abstract let c be a closed convex subset of a real hilbert space h and assume that t is an asymptotically strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the modified mann s algorithm generates a sequence x n via the formula x n 1 n x n 1 n t n x n n 0 it is proved that if the control sequence n is chosen so that n 1 for some 0 1 then x n converges weakly to a fixed point of t we also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence

iterative construction of fixed points of asymptotically nonexpansive mappings abstract let t be a completely continuous and asymptotically nonexpansive self mapping of a nonempty closed bounded and convex subset of a hilbert space then under certain conditions the sequence defined by xn 1 ntn xn 1 n x n converges strongly to some fixed point of t a similar result is obtained for asymptotically pseudocontractive mappings whose iterates are lipschitz continuous with respect to some common lipschitz constant

generating converging bounds to the complex discrete states of the p2 ix3 iαx hamiltonian the eigenvalue moment method emm is applied to the h p2 ix3 i x hamiltonian enabling the algebraic numerical generation of converging bounds to the complex energies of the l2 states as argued through asymptotic methods by delabaere and trinh the robustness of the formalism and its computational implementation suggest that the present non negativity formulation implicitly contains the key algebraic relations by which to prove bessis conjecture that the eigenenergies of the h0 hamiltonian are real the required algebraic analysis of the emm procedure pertaining to this problem will be presented in a forthcoming paper

detection and diagnosis of abrupt changes in modal characteristics of nonstationary digital signals corresp new instrumental tests for detecting and diagnosing changes in the poles of a signal having unknown time varying zeros are proposed numerical results for nonstationary scalar signals are given the extension of these tests to the vector case may be used for vibration monitoring

changing spectrum estimation abstract an autoregressive ar model with time varying coefficients is used for the modeling of non stationary covariance time series a stochastically perturbed linear constraint model is presented as a model of time varying ar coefficients the overall model for the time series is fitted by using the kalman filter and akaike s aic criterion an estimate of a changing spectrum is obtained by the fitted model and the fixed interval smoother examples are given to illustrate the procedure

realization and reduction of markovian models from nonstationary data the realization of markovian models for nonstationary processes generated by time invariant linear systems is considered a model is obtained by constructing an orthogonal finite step predictor for the process optimal model approximation by order reduction is naturally defined in this framework the construction and reduction of markovian models from multiple data records and the numerical considerations involved are illustrated by examples

output only identification and dynamic analysis of time varying mechanical structures under random excitation a comparative assessment of parametric methods this article addresses the problem of parametric time domain identification and dynamic analysis for time varying tv mechanical structures under unobservable random excitation the methods presented are based on time dependent autoregressive moving average tarma models and are classified according to the mathematical structure imposed on the tv parameter evolution as unstructured parameter evolution stochastic parameter evolution and deter ministic parameter evolution the features and relative merits of each class are outlined a representativ e method from each is then assessed through its application to the identification and dynamic analysis o f a laboratory tv structure consisting of a beam with a mass moving on it the results are mutually compared and contrasted to those obtained through frozen configuration multiple experiment bas eline identification

merging sensor data from multiple measurement set ups for non stationary subspace based modal analysis processing sensor data from multiple non simultaneously recorded measurement set ups for structural analysis is often achieved by merging identification results obtained from records corresponding to different sensor pools since pole matching and eigenvector gluing might not be consistent in some cases the question arises to merge the data first and then process them globally subspace identification algorithms have proven efficient for performing output only modal analysis of mechanical systems subject to uncontrolled unmeasured and non stationary excitation the purpose of this paper is to investigate as to how subspace based output only modal analysis algorithms can be adapted to handle the multi patch measurements set up numerical results obtained on both laboratory and real application examples are reported which show the relevance and usefulness of the proposed algorithm

complex square well a new exactly solvable quantum mechanical model recently a class ofpt invariant quantum mechanical models described by the non hermitian hamiltonianhdp2 x2 ix was studied it was found that the energy levels for this theory are real for all 0 here the limit as 1is examined it is shown that in this limit the theory becomes exactly solvable a generalization of this hamiltonian h d p 2 x 2m ix md 1 2 3 is also studied and thispt symmetric hamiltonian becomes exactly solvable in the large limit as well in effect what is obtained in each case is a complex analogue of the hamiltonian for the square well potential expansions about the large limit are obtained

a model selection rule for sinusoids in white gaussian noise the model selection problem for sinusoidal signals has often been addressed by employing the akaike 1974 information criterion aic and the minimum description length principle mdl the popularity of these criteria partly stems from the intrinsically simple means by which they can be implemented they can however produce misleading results if they are not carefully used the aic and mdl have a common form in that they comprise two terms a data term and a penalty term the data term quantifies the residuals of the model and the penalty term reflects the desideratum of parsimony while the data terms of the aic and mdl are identical the penalty terms are different in most of the literature the aic and mdl penalties are however both obtained by apportioning an equal weight to each additional unknown parameter be it phase amplitude or frequency by contrast we demonstrate that the penalties associated with the amplitude and phase parameters should be weighted differently than the penalty attached to the frequencies following the bayesian methodology we derive a model selection criterion for sinusoidal signals in gaussian noise which also contains the log likelihood and the penalty terms the simulation results disclose remarkable improvement in our selection rule over the commonly used mdl and aic

estimation of stationary structural system parameters from non stationary random vibration data a locally stationary model method abstract stationary structural system building natural frequency and damping parameters are estimated from a long non stationary ambient vibration time series record the non stationary vibration record is decomposed into continuous not necessarily equal duration stationary time series segments by a locally stationary autoregressive ar model method estimates of the stationary structural system parameters are extracted from each of the locally stationary ar models the final estimate of the structural system parameters is a time duration weighted average of the estimates in each stationary segment under reasonable conditions the theoretical variance of the structural system parameter estimates decreases with the duration of the long non stationary record the results of the analysis of a long duration non stationary record are compared with the results of an earlier approach in which the long duration non stationary record was decomposed into relatively short fixed duration vibration records and analyzed as stationary time series 1 the variances of the damping parameter estimates achieved by the current method are one half to one quarter of those obtained by the earlier method the sample variance of the structural system parameter estimates decreases with the reciprocal of the duration of the overall non stationary record

principal components algorithms for arma spectrum estimation principal components algorithms are presented for the problem of fitting an arma model to a given segment of a sample sequence of a discrete time stochastic process and the model is used to estimate the process power spectrum to reduce the effects of finite wordlength errors balanced state pace parameterization of the arma model is used instead of the more popular difference equation parameterization model identification is formulated as a problem of selecting a partial state to span approximately an apparently large dimensional information interface between the past and the future of the process different criteria are used to measure the quality of the approximation which leads to principal components algorithms for the problem that are based on singular value decomposition

statistical model based damage detection and localization subspace based residuals and damage to noise sensitivity ratios the vibration based structural health monitoring problem is addressed as the double task of detecting damages modeled as changes in the eigenstructure of a linear dynamic system and localizing the detected damages within a fem of the monitored structure the proposed damage detection algorithm is based on a residual generated from a stochastic subspace based covariance driven identification method and on the statistical local approach to the design of detection algorithms this algorithm basically computes a global test which performs a sensitivity analysis of the residuals to the damages relative to uncertainties and noises how this residual relates to some residuals for damage localization and model updating is discussed damage localization is stated as a detection problem this problem is addressed by plugging aggregated sensitivities of the modes and mode shapes w r t fem structural parameters in the above setting this results in directional tests which perform the same type of damage to noise sensitivity analysis of the residual as for damage detection how the sensitivity aggregation mechanism relates to sub structuring is outlined numerical results obtained on one example are reported

generalized pt symmetry and real spectra the fact that eigenvalues of pt symmetric hamiltonians h can be real for some values of ap arameter and complex for others is explained by showing that the matrix elements ofh a ndhence the secular equation are real not only for pt but also for an ya ntiunitary operator a satisfying a 2k 1w ith k odd the argument is illustrated by a 2 2m atrix hamiltonian and two examples of the generalization are given

a smoothness priors time varying ar coefficient modeling of nonstationary covariance time series a smoothness priors time varying ar coefficient model approach for the modeling of nonstationary in the covariance time series is shown smoothness priors in the form of a difference equation constraint excited by an independent white noise are imposed on each ar coefficient the unknown white noise variances are hyperparameters of the ar coefficient distribution the critical computation is of the likelihood of the hyperparameters of the bayesian model this computation is facilitated by a state space representation kalman filter implementation the best difference equation order best ar model order best hyperparameter model locally in time is selected using the minimum aic method also an instantaneous spectral density is defined in terms of the instantaneous ar model coefficients and a smoothed estimate of the instantaneous time series variance an earthquake record is analyzed the changing spectral analysis of the original data and of simulations from a time varying ar coefficient model of that data are shown

input output versus output only data processing for structural identification application to in flight data analysis the problem of structural model identification under both known and unknown input excitations is addressed in flight data analysis is an important instance of that problem input output and output only eigenstructure identification methods are described and compared within two classes of methods subspace based and prediction error in particular different types of relevant projections for handling the known and unknown inputs are discussed the relevance of the methods is emphasized through numerical results obtained on real flight test data sets

extension of a spectral bounding method to complex rotated hamiltonians with application to p2 i x3 we show that a recently developed method for generating bounds for the discrete energy states of the non hermitian ix3 potential handy c r 2001 j phys a math gen 34 l271 is applicable to complex rotated versions of the hamiltonian this has important implications for extension of the method in the analysis of resonant states regge poles and general bound states in the complex plane bender and boettcher 1998 phys rev lett 80 5243

conjecture on the interlacing of zeros in complex sturm liouville problems the zeros of the eigenfunctions of self adjoint sturm liouville eigenvalue problems interlace for these problems interlacing is crucial for completeness for the complex sturm liouville problem associated with the schrodinger equation for a non hermitian pt symmetric hamiltonian completeness and interlacing of zeros have never been examined this paper reports a numerical study of the sturm liouville problems for three complex potentials the large n limit of a ix n potential a quasiexactly solvable x4 potential and an ix3 potential in all cases the complex zeros of the eigenfunctions exhibit a similar pattern of interlacing and it is conjectured that this pattern is universal understanding this pattern could provide insight into whether the eigenfunctions of complex sturm liouville problems form a complete set

comparison of some instrumental variable methods consistency and accuracy aspects a general analysis of various instrumental variable methods is given the investigation includes methods for open loop operation as well as methods designed for closed loop operation the instrumental variables are formed as different combinations of inputs delayed inputs delayed outputs filtered inputs and external setpoint variations necessary conditions as well as sufficient conditions for consistency are explicitly derived and discussed the paper also includes some simple counter examples to general consistency for some common instrumental variable methods also asymptotic accuracy properties are reviewed it is e g studied how the use of filters will influence the covariance matrix of the parameter estimates

approximation of common fixed points of a countable family of nonexpansive mappings in a banach space in this paper to find a common fixed point of a family of nonexpansive mappings we introduce a halpern type iterative sequence then we prove that such a sequence converges strongly to a common fixed point of nonexpansive mappings moreover we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings and the problem of finding a zero of an accretive operator

quantifying uncertainty in material damage from vibrational data the response of a vibrating beam to a force depends on many physical parameters including those determined by material properties damage caused by fatigue or cracks results in local reductions in stiffness parameters and may drastically alter the response of the beam data obtained from the vibrating beam are often subject to uncertainties and or errors typically modeled using probability densities the goal of this paper is to estimate and quantify the uncertainty in damage modeled as a local reduction in stiffness using uncertain data we present various frameworks and methods for solving this parameter determination problem we also describe a mathematical analysis to determine and compute useful output data for each method we apply the various methods in a specified sequence that allows us to interface the various inputs and outputs of these methods in order to enhance the inferences drawn from the numerical results obtained from each method numerical results are presented using both simulated and experimentally obtained data from physically damaged beams

equilibrium programming using proximal like algorithms we compute constrained equilibria satisfying an optimality condition important examples include convex programming saddle problems noncooperative games and variational inequalities under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization in the main algorithm each stage of computation requires two proximal steps possibly using bregman functions one step serves to predict the next point the other helps to correct the new prediction to enhance practical applicability we tolerate numerical errors

exercises in exact quantization the formalism of exact 1d quantization is reviewed in detail and applied to the spectral study of three concrete schrodinger hamiltonians d2 dq2 v q on the half line q 0 with a dirichlet or neumann condition at q 0 emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters we first discuss the homogeneous potential v q qn as n versus its solvable n limit an infinite square well useful distinctions are established between regular and singular behaviours of spectral quantities various identities among the square well spectral functions are unravelled as limits of finite n properties the second model is the quartic anharmonic oscillator the zero energy spectral determinants det d2 dq2 q4 vq2 are explicitly analysed in detail revealing many special values algebraic identities between taylor coefficients and functional equations of a quartic type coupled to asymptotic v properties of airy type the third study addresses the potentials v q qn vqn 2 1 of even degree their zero energy spectral determinants prove computable in closed form and the generalized eigenvalue problems with v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case corresponding to n 2 these results partly reflect the presence of quasi exactly solvable potentials in the family above

on strong convergence by the hybrid method for families of mappings in hilbert spaces abstract let c be a nonempty closed convex subset of a real hilbert space and let t n be a family of mappings of c into itself such that the set of all common fixed points of t n is nonempty we consider a sequence x n generated by the hybrid method in mathematical programming we give the new conditions of t n under which x n converges strongly to a common fixed point of t n and generalize the unified result for families of nonexpansive mappings k nakajo k shimoji w takahashi strong convergence theorems by the hybrid method for families of nonexpansive mappings in hilbert spaces taiwanese j math 10 2006 339 360 and the results for asymptotically nonexpansive mappings and semigroups t h kim h k xu strong convergence of modified mann iterations for asymptotically nonexpansive mappings and semigroups nonlinear anal 64 2006 1140 1152 s plubtieng and k ungchittrakool strong convergence of modified ishikawa iteration for two asymptotically nonexpansive mappings and semigroups nonlinear anal 67 2007 2306 2315

weak and strong convergence theorems for countable lipschitzian mappings and its applications abstract we use mann s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of lipschitzian mappings finally we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings

modular subspace based system identification from multi setup measurements subspace identification algorithms are efficient for output only eigenstructure identification of linear mimo systems the problem of merging sensor data obtained from moving and nonsimultaneously recorded measurement setups under varying excitation is considered to address the problem of dimension explosion when retrieving the system matrices of the complete system a modular and scalable approach is proposed adapted to a large class of subspace methods observability matrices are normalized and merged to retrieve global system matrices

nonperturbative square well approximation to a quantum theory the possibility of expressing the solution to a 2p quantum field theory as a series in powers of 1 p is proposed such a series would be nonperturbative in its dependence on the fundamental parameters of the theory such as the mass and the coupling constant the first term in such a series describes a field in an infinite dimensional square well potential in this paper the quantum mechanical hamiltonian h p2 q2p is studied as a model calculation and the expansion of the energy levels as series in powers of 1 p is examined the method of matched asymptotic expansions to determine the first five terms in the series for all energy levels is used the results are compared with extensive numerical calculations of the ground state energy and it is found that the series is extremely accurate when p 2 the five term series has a relative error of 6 when p 10 the relative error is 0 009 and when p 200 the relative error is 3 4 10 9

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

strong convergence theorems for a countable family of quasi lipschitzian mappings and its applications abstract we use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi lipschitzian mappings as a consequence several convergence theorems for quasi nonexpansive mappings and asymptotically strict pseudo contractions are deduced we also establish strong convergence of iterative sequences for finding a common element of the set of fixed point the set of solutions of an equilibrium problem and the set of solutions of a variational inequality with an appropriate setting we obtain the corresponding results due to tada takahashi and nakajo shimoji takahashi

on board component fault detection and isolation using the statistical local approach we describe both the key principles and real application examples of a unified theory which allows us to perform the on board incipient fault detection and isolation tasks involved in monitoring for condition based maintenance this theory is known under the name of local approach and it is especially suited to component faults it aims at transforming complex detection problems concerning a parameterized stochastic process into the universal problem of monitoring the mean of a gaussian vector based on a small deviation assumption the key tools are the first order taylor expansion and the asymptotic gaussianity of a convenient parameter estimating function ml ls iv and subspace identification methods are addressed in this perspective in the case of dynamic processes given in state space form the approach may also call for observer based state estimation or state elimination experiments concerning both simulated and real data for linear and nonlinear dynamical processes are reported how the key principles and features of the local approach compare with those of other approaches is discussed

some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant comparison between the exact value of the spectral zeta function zh 1 5 6 5 3 2cos 5 2 1 5 3 5 and the results of numeric and wkb calculations supports the conjecture by daniel bessis 1995 private communication that all the eigenvalues of this pt invariant hamiltonian are real for one dimensional schrodinger operators with complex potentials having a monotonic imaginary part the eigenfunctions and the imaginary parts of their logarithmic derivatives have no real zeros

comment on a recent paper by mezincescu it has been conjectured that for 0 the entire spectrum of the non hermitian symmetric hamiltonian hn p2 x2 ix where n 2 is real strong evidence for this conjecture for the special case n 3 was provided in a recent paper by mezincescu mezincescu g a 2000 j phys a math gen 33 4911 in which the spectral zeta function z3 1 for the hamiltonian h3 p2 ix3 was calculated exactly here the calculation of mezincescu is generalized from the special case n 3 to the region of all n 2 0 and the exact spectral zeta function zn 1 for hn is obtained using zn 1 it is shown that to extremely high precision about three parts in 1018 the spectrum of hn for other values of n such as n 4 is entirely real

scaling corrections site percolation and ising model in three dimensions using finite size scaling techniques we obtain accurate results for critical quantities of the ising model and the site percolation in three dimensions we pay special attention to parametrizing the corrections to scaling which is necessary to bring the systematic errors below the statistical ones

single sample modal identification of a nonstationary stochastic process gauss markov processes excited by nonstationary noises are encountered in the modeling of vibrating systems we prove that the classical instrumental variable method as well as the ho kalman realization algorithm for identifying the pole part modal characteristics of the model are consistent when used on a single sample of the nonstationary signal

a comment on the gelfand yaglom theorem zeta functions and heat kernels for pt symmetric hamiltonians we propose a numerical method for computing zeta functions for general classes of hamiltonians based on the gelfand yaglom theorem which computes the determinant without needing to compute the eigenvalues we verify that this agrees with known analytic results for certain hermitean hamiltonians and we present numerical evidence that it may extend also to non hermitean pt symmetric hamiltonians

nonstationary consistency of subspace methods in this paper we study ldquononstationary consistencyrdquo of subspace methods for eigenstructure identification i e the ability of subspace algorithms to converge to the true eigenstructure despite nonstationarities in the excitation and measurement noises note that such nonstationarities may result in having time varying zeros for the underlying system so the problem is nontrivial in particular likelihood and prediction error related methods do not ensure consistency under such situation because estimation of poles and estimation of zeros are tightly coupled we show in turn that subspace methods ensure such consistency our study carefully separates statistical from nonstatistical arguments therefore enlightening the role of statistical assumptions in this story

state space formulation a solution to modal parameter estimation abstract as an alternative to fourier techniques based on transfer function fitting modal parameters identification methods derived from a state space formulation of the dynamic equation have been recently studied and developed this formulation leads to the solution of the modal problem via multi dimensional arma model parameters identification in this paper the transition from the dynamic equation to the arma model is described and the algorithms of identification are explained for the very common case of uncontrolled and unmeasured excitation these techniques the qualities of which have been established during extensive processing of offshore structures measurements have been generalized to more complex excitation models such as those of turbo alternators several examples illustrate these different application domains

calculation of the hidden symmetry operator in symmetric quantum mechanics in a recent paper it was shown that if a hamiltonian h has an unbroken symmetry then it also possesses a hidden symmetry represented by the linear operator the operator commutes with both h and the inner product with respect to is associated with a positive norm and the quantum theory built on the associated hilbert space is unitary in this paper it is shown how to construct the operator for the non hermitian symmetric hamiltonian h 1 2p2 1 2x2 ix3 using perturbative techniques it is also shown how to construct the operator for h 1 2p2 1 2x2 x4 using nonperturbative methods

extension of a spectral bounding method to the pt invariant states of the ix n non hermitian potential the eigenvalue moment method emm developed by handy j phys a math gen 2001 34 l271 j phys a math gen 2001 34 5065 handy et al j phys a math gen 2001 34 5593 and handy and xiao qian wang j phys a math gen 2001 34 8297 which generates converging lower and upper bounds to the complex discrete state energies is extended to the case of discrete states with non real support in particular bender and boettcher phys rev lett 1998 80 5243 have argued on the reality of the discrete state spectrum for the ix n potential for n integer 4 such pt invariant solutions can only exist on appropriate complex contours we develop and apply the necessary emm formalism to such cases in particular the restriction of emm to the anti stokes angles significantly increases the convergence rate of the bounds