asymptotic behavior of solutions of second order differential equations with integrable coefficients the differential equation x a t f x 0 t 0 is considered under the condition that lim x fl a s ds exists and is finite and necessary and or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions cl c2t

monotone positive solutions of second order nonlinear differential equations we obtain an existence theorem for monotone positive solutions of nonlinear second order ordinary differential equations by using the schauder tikhonov fixed point theorem the result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in r n n greater than or equal to 3

nonparametric probabilistic approach of uncertainties for elliptic boundary value problem the paper is devoted to elliptic boundary value problems with uncertainties such a problem has already been analyzed in the context of the parametric probabilistic approach of system parameters uncertainties or for random media model uncertainties are induced by the mathematical physical process which allows the boundary value problem to be constructed from the design system if experiments are not available the bayesian approach cannot be used to take into account model uncertainties recently a nonparametric probabilistic approach of both the model uncertainties and system parameters uncertainties has been proposed by the author to analyze uncertain linear and non linear dynamical systems nevertheless the use of this concept that has to be developed for dynamical systems cannot directly be applied for elliptic boundary value problem for instance for a linear elastostatic problem relative to an elastic bounded domain we then propose an extension of the nonparametric probabilistic approach in order to take into account model uncertainties for strictly elliptic boundary value problems the theory and its validation are presented

a probabilistic construction of model validation we describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty the proposed approach is a functional analysis based probabilistic approach for which we represent random quantities using polynomial chaos expansions pces the approach permits the formulation of the uncertainty assessment in validation a significant component of the process as a problem of approximation theory it has two essential parts first a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model based measurements such a calibration technique employs pces to represent the inherent uncertainty of the model parameters based on the asymptotic behavior of the statistical parameter estimator the associated pce coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information second a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem the above validation path is implemented for the case of dynamical system validation challenge exercise

non gaussian positive definite matrix valued random fields for elliptic stochastic partial differential operators this paper deals with the construction of a class of non gaussian positive definite matrix valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled the properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given such a matrix valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters the class of random fields which can be approximated is presented and their experimental identification is analyzed an example is given in three dimensional linear elasticity for which the fourth order elasticity tensor valued random field is constructed for a random non homogeneous anisotropic elastic material

matrix variate distributions for probabilistic structural dynamics matrix variate distributions are proposed to quantify uncertainty in the mass stiffness and damping matrices arising in linear structural dynamics the proposed approach is based on the so called wishart random matrices it is assumed that the mean of the system matrices are known a new optimal wishart distribution is proposed to model the random system matrices the optimal wishart distribution is such that the mean of the matrix and its inverse produce minimum deviations from their respective deterministic values the method proposed here gives a simple nonparametric approach for uncertainty quantification and propagation for complex aerospace structural systems the new method is illustrated using a numerical example it is shown that wishart random matrices can be used to model uncertainty across a wide range of excitation frequencies

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

stochastic inversion in acoustic scattering a novel approach is presented for the probabilistic characterization of the solution in inverse acoustic scattering problems the approach consists of representing the unknown parameters of the scatterer as stochastic processes with spatially random fluctuations the stochastic processes are represented by their polynomial chaos decomposition the deterministic coefficients in this parametrization are then computed by solving a deterministic optimization algorithm with an appropriate objective function that captures the probabilistic significance of these coefficients the general procedure is demonstrated by its application to a two dimensional problem with the hidden object a hard obstacle being immersed in a heterogeneous medium the formulation provides as well a measure of validity of a particular predictive model at matching experimental evidence

asymptotic to polynomials solutions for nonlinear differential equations abstract this article is concerned with solutions that behave asymptotically like polynomials for nth order n 1 nonlinear ordinary differential equations for each given integer m with 1 m n 1 sufficient conditions are presented in order that for any real polynomial of degree at most m there exists a solution which is asymptotic at to this polynomial conditions are also given which are sufficient for every solution to be asymptotic at to a real polynomial of degree at most n 1 the application of the results obtained to the special case of second order nonlinear differential equations leads to improved versions of the ones contained in the recent paper by lipovan glasg math j 45 2003 179 and of other related results existing in the literature

a comprehensive overview of a non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics abstract in structural dynamics a predictive model is constructed by developing a mathematical mechanical model of a designed system in order to predict the response of the real system which is the manufactured system realized from the designed system the mathematical mechanical modelling process of the designed system introduces two fundamental types of uncertainties the data uncertainties and the model uncertainties uncertainties have to be taken into account for improving the predictability of the model model uncertainties cannot be modelled by using the usual parametric probabilistic approach recently a general non parametric probabilistic approach of model uncertainties for dynamical systems has been proposed using the random matrix theory this paper gives a comprehensive overview of this approach in developing its foundations in simple terms and in illustrating all the concepts and the tools introduced in the general theory by using a simple example this paper deals with 1 notions of designed systems real systems mean models as predictive models errors and uncertainties 2 the definition of a simple example in linear elastodynamics 3 a comprehensive overview of the non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics 4 a summary of the random matrix ensembles which are necessary for the non parametric modelling of random uncertainties 5 the estimation of the dispersion parameters of the non parametric probabilistic model using experimental data 6 the method to solve the stochastic equation of the dynamical system with non parametric probabilistic model of random uncertainties 7 a numerical simulation and the validation for the simple example

parameterized algorithms for d hitting set the weighted case we are going to analyze search tree algorithms for weightedd hitting set although the algorithms that we develop are fairly simple their analysis is technically involved we compare the weighted case with the previously analyzed unweighted one exhibiting that the advantage of the unweighted case dwindles with growing d

independent component analysis to enhance performances of karhunen loeve expansions for non gaussian stochastic processes application to uncertain systems abstract in the context of engineering systems an essential step in uncertainty quantification is the development of accurate and efficient representation of the random input parameters for such input parameters modeled as stochastic processes karhunen loeve expansion is a classical approach providing efficient representations using a set of uncorrelated but generally statistically dependent random variables the dependence structure among these random variables may be difficult to estimate statistically and is thus ignored in many practical applications this simplifying assumption of independence may lead to considerable errors in estimating the variability in the system state thus limiting the effectiveness of karhunen loeve expansion in certain cases in this paper independent component analysis is exploited to linearly transform the random variables used in karhunen loeve expansion resulting into a set of random variables exhibiting higher order decorrelation the stochastic wave equation is investigated for numerical illustration whereby the random stiffness coefficient is modeled as a non gaussian stochastic process under the assumption of independence among the random variables used in the karhunen loeve expansion and independent component analysis representations the latter provides more accurate statistical characterization of the output process for the specific cases examined

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

parameterized algorithms for feedback set problems and their duals in tournaments the parameterized feedback vertex arc set problem is to find whether there are k vertices arcs in a given graph whose removal makes the graph acyclic the parameterized complexity of this problem in general directed graphs is a long standing open problem we investigate the problems on tournaments a well studied class of directed graphs we consider both weighted and unweighted versions we also address the parametric dual problems which are also natural optimization problems we show that they are fixed parameter tractable not just in tournaments but in oriented directed graphs where there is at most one directed arc between a pair of vertices more specifically the dual problem we show fixed parameter tractable are given an oriented directed graph is there a subset of k vertices arcs that forms an acyclic directed subgraph of the graph our main results include an o 2 4143 kn 1 algorithm for weighted feedback vertex set problem and an o 2 415 kn algorithm for weighted feedback arc set problem in tournaments an o e2k k kk2 min m ig n n2 algorithm for the dual of feedback vertex set problem maximum vertex induced acyclic graph in oriented directed graphs and an o 4kk m algorithm for the dual of feedback arc set problem maximum arcinduced acyclic graph in general directed graphs we also show that the dual of feedback vertex set is w 1 hard in general directed graphs and the feedback arc set problem is fixed parameter tractable in dense directed graphs our results are the first non trivial results for these problems

uncertainty in structural dynamics experimental validation of a wishart random matrix model in the modeling of complex dynamical systems high resolution finite element models are routinely adopted to reduce the discretization error such an approach may fail to enhance the confidence in simulation based predictions when the dynamical systems exhibit significant variability in the parameterization and description of the mathematical model due to parametric data uncertainty and model uncertainty in contrast to parametric uncertainty model uncertainty poses significant challenges as no explicit parameter is available a priori to characterize its behavior in the current investigation a set of sprung mass oscillators randomly attached to a baseline system namely a homogeneous cantilever plate gives rise to model uncertainty contrary to model parametric uncertainty traditionally tackled using stochastic finite element method sfem the model uncertainty arising from randomly attached sprung masses gives rise to a new variant of dynamical system for every sample the feasibility of adopting a wishart random matrix to represent such ensemble of dynamical systems derived from model perturbation is investigated in this paper the experiments conducted on a vibrating plate with randomly attached sprung mass oscillators demonstrate that the wishart random matrix model may provide a reasonable representation of model uncertainty in linear dynamical systems in the medium and high frequency region

vibration isolation from irregularity in a nearly periodic structure theory and measurements this article describes the theory and a simple experiment carried out to demonstrate the phenomenon of anderson localization in an acoustical context this is an effect whereby the propagation of vibration in a structure which is not entirely regular is impeded by the irregularities giving rise on the average to an exponential decay of vibration level away from the driving point even in the absence of any dissipation the structure used in the experiment was a stretched string with masses attached to it this string was studied with regular spacings of the masses and after the masses had been moved in a controlled way to provide a small degree of irregularity in both cases the transmission of energy from end to end of the string was measured as a function of frequency and also the mode shapes in the second and fourth passbands were measured so as to demonstrate the underlying physics of the localization phenomenon in which the individual modes making up each passband change from being extended throughout the structure in the regular case to being localized in specific areas of the structure in the presence of irregularity all measurements yielded satisfactory agreement with the theoretical predictions

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

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

asymptotic decay of solutions of a nonlinear second order differential equation with deviating argument on etudie l existence et le comportement asymptotique des solutions decroissantes des equations differentielles fonctionnelles du type p t h x x t f t x g t

a fixed parameter algorithm for the directed feedback vertex set problem the parameterized feedback vertex set problem on directed graphs i e the dfvs problem is defined as follows given a directed graph g and a parameter k either construct a feedback vertex set of at most k vertices in g or report that no such a set exists it has been a well known open problem in parameterized computation and complexity whether the dfvs problem is fixed parameter tractable that is whether the problem can be solved in time f k no 1 for some function f in this article we develop new algorithmic techniques that result in an algorithm with running time 4k ke no 1 for the dfvs problem therefore we resolve this open problem

integral inequalities of gronwall bellman bihari type and asymptotic behavior of certain second order nonlinear differential equations abstract several integral inequalities similar to gronwall bellmann bihari inequalities are obtained these inequalities are used to discuss the asymptotic behavior of certain second order nonlinear differential equations

reduced models in the medium frequency range for general external structural acoustic systems this paper presents a theoretical approach for constructing a reduced model in the medium frequency range in the area of structural acoustics for a general three dimensional dissipative structure made of an anisotropic inhomogeneous viscoelastic bounded medium coupled with an external acoustic fluid all the results presented can be used if the structure is made of beams plates and shells the boundary value problem in the frequency domain and its variational formulation are presented for a fixed medium frequency band an energy operator related to the structural acoustic system is introduced this operator is symmetric positive definite and has a countable set of positive eigenvalues its dominant eigensubspace allows a reduced model to be constructed using the ritz galerkin method a finite dimension approximation of the three dimensional continuous case is presented and an effective construction of the dominant subspace using the subspace iteration method is developed as an example the reduced model is used for constructing the time stationary random response of the structural acoustic system submitted to a random wall pressure field finally the theory is validated for a finite length circular cylindrical shell coupled with several dashpots and springs and immersed in a gas air and in a liquid water

on the asymptotic integration of nonlinear differential equations the aim of the present paper is twofold firstly the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations the class of second order equations with bihari like nonlinearity secondly some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations

propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach abstract this paper presents an efficient procedure for characterizing the solution of evolution equations with stochastic coefficients these typically model the behavior of physical systems whose properties are modeled as spatially or temporally varying stochastic processes described within the framework of probability theory the concepts of projection orthogonality and weak convergence are exploited in a manner which directly mimics deterministic finite element solutions except that in the stochastic case inner products refer to expectation operations specifically the karhunen loeve expansion is used to discretize these processes into a denumerable set of random variables thus providing a denumerable function space in which the problem is cast the polynomial chaos expansion is then used to represent the solution in this space and the coefficients in the expansion are evaluated as generalized fourier coefficients via a galerkin procedure in the hilbert space of random variables

hardness of subgraph and supergraph problems in c tournaments problems like the directed feedback vertex set problem have much better algorithms in tournaments when compared to general graphs this motivates us to study a natural generalization of tournaments named c tournaments and see if the structural properties of these graphs are helpful in obtaining similar algorithms c tournaments are simple digraphs which have directed paths of length at most c 1 between all pairs of vertices we study the complexity of feedback vertex set and r dominating set in c tournaments we also present hardness results on some graph editing problems involving c tournaments

global existence of solutions with prescribed asymptotic behavior for second order nonlinear differential equations where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel we always assume without mentioning that the function f t u v is continuous in the domain d t u v t t0 u v r where t0 1 it is supposed also that all equations and inequalities involving t hold for t t0 unless otherwise speci9ed the symbols o and o have their standard meanings as t and we use the notation r def 0 eq 1 is often used for mathematical modelling of various physical chemical and biological systems and attracts constant interest of researchers a great deal of papers published during the last three decades are concerned with local and global existence of solutions of eq 1 and its numerous particular cases uniqueness of solutions

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

reduced models for the medium frequency dynamics of stochastic systems in this paper a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium frequency range in this frequency range both traditional modal analysis and statistical energy analysis sea procedures well suited for low and high frequency vibration analysis respectively lead to computational and conceptual difficulties the uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical the methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods in particular the karhunen loeve and polynomial chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest