eigenderivative analysis of asymmetric non conservative systems in general the damping matrix of a dynamic system or structure is such that it can not be simultaneously diagonalized with the mass and stiffness matrices by any linear transformation for this reason the eigenvalues and eigenvectors and consequently their derivatives become complex expressions for the first and second order derivatives of the eigenvalues and eigenvectors of these linear non conservative systems are given traditional restrictions of symmetry and positive definiteness have not been imposed on the mass damping and stiffness matrices the results are derived in terms of the eigenvalues and left and right eigenvectors of the second order system so that the undesirable use of the first order representation of the equations of motion can be avoided the usefulness of the derived expressions is demonstrated by considering a non proportionally damped two degree of freedom symmetric system and a damped rigid rotor on flexible supports copyright 2001 john wiley sons ltd

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

particularities of newton s method in space frame force determination utilising eigenpair functions abstract redundant space frame dynamics are modelled through the structural eigenproblem which is formulated to account for the geometric stiffness due to self equilibrating membrane forces the resulting mathematical eigenvalues and eigenvectors respectively indicative of natural frequencies and modes of oscillation are then assumed to be continuous functions of a scalar factor upon a normalised set of forces in equilibrium newton s method or sensitivity analysis provides a means for minimising the difference between mathematical and physically observed eigenvalues whereupon the axial forces are inferred from the model issues of convergence and root uniqueness are anticipated eigenvalue coalescence in which eigenvalues transitorily coalesce to permute their arbitrarily numerical ordering is seen to define non smooth eigenvalue functions and hence encumber newton s method mode tracing strategies to overcome this are discussed the existence of various degrees of eigenvalue degeneracy owing to frame cyclic symmetry or otherwise is anticipated in terms of the influence upon newton s method if the degree of frame static redundancy is greater than one and there exist a number of linearly independent force distributions then the required force determination is a multidimensional newton method problems encountered in these manifold dimensions including those arising from frame spatial periodicity and the adversity of newton s method are overcome emphasis is placed upon minimising the dimensionality of newton s method through enforcement of equilibrium constraints illustrative numerical simulations are given and some applications of the method are proposed

the bando calabi futaki character and its lifting to a group character the bando calabi futaki character of a compact kahler manifold is an obstruction to the existence of kahler metrics with constant scalar curvature which is a generalization of the futaki character of a fano manifold in this paper we shall establish an interpretation of the bando calabi futaki character as a secondary characteristic class by using this interpretation we shall also prove that the bando calabi futaki character can be lifted to a group character

sensitivity analysis of non conservative eigensystems an expression for the derivatives of eigenvalues and eigenvectors of non conservative systems is presented contrary to previous methods that use state space form 2n space to consider damping proposed method solves the eigenpair derivatives of damped system explicitly the computation size of n order is maintained and the eigenpair derivatives are obtained simultaneously from one equation so that it is efficient in cpu time and storage capacity moreover this method can be extended to asymmetric non conservative damped systems although additional problems are generated contrary to the eigenpair sensitivity methods of symmetric systems in asymmetric case an algebraic method for the eigenpair derivatives can be obtained through similar procedure the proposed expression is derived by combining the differentiations of the eigenvalue problem and normalization condition into one linear algebraic equation the numerical stability is proved by showing non singularity of the proposed equation and the efficiency of the derived expression is illustrated by considering a cantilever beam with lumped dampers and a whirling beam

testing for stationarity ergodicity and for comovements between nonlinear discrete time markov processes in this paper we introduce a class of nonlinear data generating processes dgps that are rst order markov and can be represented as the sum of a linear plus a bounded nonlinear component we use the concepts of geometric ergodicity and of linear stochastic comovement which correspond to the linear concepts of integratedness and cointegratedness to characterize the dgps we show that the stationarity test due to kwiatowski et al 1992 journal of econometrics 54 159 178 and the cointegration test of shin 1994 econometric theory 10 91 115 are applicable in the current context although the shin test has a di erent limiting distribution we also propose a consistent test which has a null of linear cointegration comovement and an alternative of n c22

a direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems in general the derivative of an eigenvector of a vibrating symmetric system is the solution of a singular problem further complications are encountered in dealing with asymmetric damped systems for which the left and right eigenvectors and their derivatives become distinct and complex several approaches have been proposed to overcome this singularity such as nelson s method and the modal method in the present work a new approach is presented for calculating simultaneously the derivatives of the eigenvalues and their associated derivatives of the left and right eigenvectors for asymmetric damped systems with the proposed method the exact eigenderivatives can be obtained by solving a first order linear algebraic system of equations the method is applied on a 104 dof ventilator rotor system which is used as an example of an asymmetric damped system with distinct eigenvalues the diameter of the shaft has been chosen as the design parameter the comparison of the computational time shows that the proposed method is more efficient than both nelson s approach and the modal method copyright 2006 john wiley sons ltd

hilbert series and obstructions to asymptotic semistability abstract given a polarized manifold there are obstructions for asymptotic chow semistability described as integral invariants which can be regarded as characters of the lie algebra of holomorphic vector fields in this paper we show that on toric fano manifolds the linear span of those lie algebra characters coincides with the derivatives of the laurent series of the hilbert series

varieties of long memory models long memory is defined as a series having a slowly declining correlogram or equivalently an infinite spectrum at zero frequency fractional integrated processes have such properties but here it is pointed out that a number of other processes can also be long memory including generalized fractionally integrated models arising from aggregation time changing coefficient models and possibly nonlinear models it seems that there are many classes of processes that deserve further study the relevance of long memory is illustrated using absolute returns from a daily stock market index

derivation and significance of second order modal design sensitivities if the system is considered to be unsatisfactory in some way e g a natural frequency to the system coinciding with a constant shaft speed then it is of interest to find the design change which will give maximum benefit with minimum disturbance to the system this may be quantified in terms of design sensitivities of the structure fox and kapoor derived an expression for the sensitivity of an eigenvalue for self adjoint systems i e where the left and right eigenvectors are simple transposes typical of conservative systems and duncan forms garg extended this result to apply to non self adjoint systems e g where dissipative and gyroscopic effects are present the analysis for eigenvectors was presented by plaut and huseyin for the general case most recently nelson has provided an alternative approach to eigenvector sensitivity the sensitivities derived by all of the above methods are expressed in the form of first derivatives and hence are properties local to the original solution of the system in a recent paper by vanhonacker there is a clear implication that these first derivatives may be extrapolated for gross system changes clearly this assumption can be valid only if the value of the second derivative is negligible

comment on derivation and significance of second order modal design sensitivities b derivation of the second order modal design sensitivities leads to results identical to the fir stand second order terms obtained from rayleigh s classical perturbation theory for variations in stiffness but the second order terms are in error for variations in mass rayleigh s methods are well known in the literature his formulation is general and valid whether the eigenvalue problem is in matrix differential or integral equation form brandon s perturbation eigenvalue problem in matrix form is

the bando calabi futaki character as an obstruction to semistability the purpose of this note is to correct some errors in b and c of main theorem in mn and to make an additional observation in particular we shall give a first nonvanishing example of the obstruction obstrasymp m l introduced in m2

central kahler metrics the determinant of the ricci endomorphism of a kahler metric is called its central curvature a notion well defined even in the riemannian context this work investigates two types of kahler metrics in which this curvature potential gives rise to a potential for a gradient holornorphic vector field these metric types generalize the kahler einstein notion as well as that of bando and mabuchi 1986 whenever possible the central curvature is treated in analogy with the scalar curvature and the metrics are compared with the extremal kahler metrics of calabi an analog of the futaki invariant is employed both invariants belonging to a family described in the language of holomorphic equivariant cohomology it is shown that one of the metric types realizes the minimum of an l 2 functional defined on the space of kahler metrics in a given kahler class for metrics of constant central curvature results are obtained regarding existence uniqueness and a partial classification in complex dimension two consequently on a manifold of fano type such metrics and kahler einstein metrics can only exist concurrently an existence result for the case of non constant central curvature is slated and proved in a sequel to this work

second order eigensensitivity analysis of asymmetric damped systems using nelson s method first order eigensensitivity analysis using nelson s method is first reviewed then nelson s approach is extended to the computation of the second order derivatives of the eigenvalues and eigenvectors of symmetric and asymmetric damped systems the computation of second order derivatives may be required for large variation of design parameters and for some optimization algorithms nelson s method has the advantage of requiring only the knowledge of the eigenvector to be differentiated only systems with distinct eigenvalues are considered in this paper two numerical examples have been selected to illustrate the application and the utility of the derived expressions of the first and second order eigensensitivities a four degree of freedom symmetric damped system is chosen for the first example whereas a finite element asymmetric 20 dof damped rotor model is considered in the second example

some generalizations on the algebra of i 1 processes abstract a number of techniques have recently appeared in the literature utilizing instantaneous nonlinear transformations of stochastic processes as in economics most variables appear to be nonstationary it is interesting to investigate the temporal properties of nonlinear transformations of these processes using a unified approach based on hermite polynomials this paper provides some insight for polynomial exponential periodic and neural network transformations of first order integrated processes

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

multiplier ideal sheaves and integral invariants on toric fano manifolds we extend nadel s results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author applying our extension we can determine the multiplier ideal subvarieties on toric del pezzo surfaces which do not admit kahler einstein metrics we also show that one can define multiplier ideal sheaves for kahler ricci solitons and extend the result of nadel using the holomorphic invariant defined by tian and zhu

derivatives of complex eigenvectors with distinct and repeated eigenvalues in this paper we investigate the computation of the first order derivatives of complex eigenvectors for general non defective eigensystems a new normalization condition is proposed with which we can compute unique first order derivatives of arbitrary differentiable eigenvectors of systems with distinct and repeated eigenvalues we also present an efficient algorithm to compute the particular solutions to the governing equations of the first order derivatives of eigenvectors finally numerical examples are included to demonstrate the validity of the proposed method copyright 2007 john wiley sons ltd

second order design sensitivities to assess the applicability of sensitivity analysis a quality criterion based on the cauchy ratio test for convergence of infinite series is suggested to examine the liability of systems to perform poorly under extrapolation

a new method for calculating derivatives of eigenvalues and eigenvectors for discrete structural systems abstract a new method is presented for calculating the derivatives of eigenvalues and eigenvectors for discrete structural systems in the cases of distinctive eigenvalues proportional damping is assumed and the exact eigen pair derivatives of arbitrary order are calculated for a repeated eigenvalue its first order derivatives are calculated by solving a reduced eigenvalue problem of order equal to the eigenvalue s geometric multiplicity the new method is computationally efficient and numerically robust examples are presented to illustrate the use and effectiveness of the new method

some asymptotic results for learning in single hidden layer feedforward network models abstract we investigate the properties of a recursive estimation procedure the method of back propagation for a class of nonlinear regression models single hidden layer feedforward network models recently developed by cognitive scientists the results follow from more general results for a class of recursive m estimators obtained using theorems of ljung 1977 and walk 1977 for the method of stochastic approximation conditions are given ensuring that the back propagation estimator converges almost surely to a parameter value that locally minimizes expected squared error loss provided the estimator does not diverge and that the back propagation estimator is asymptotically normal when centered at this minimizer this estimator is shown to be statistically inefficient and a two step procedure that has efficiency equivalent to that of nonlinear least squares is proposed practical issues are illustrated by a numerical example involving approximation of the henon map