asymptotic properties of solutions of difference equations with several delays and volterra summation equations we study a scalar linear difference equation with several delays by transforming it to a system of volterra equations without delays the results obtained for this system are then used to establish oscillation criteria and asymptotic properties of solutions of the considered equation

stability and asymptoticity of volterra difference equations a progress report we survey some of the fundamental results on the stability and asymptoticity of linear volterra difference equations the method of z transform is heavily utilized in equations of convolution type an example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty it is shown that the two notions are equivalent if the kernel decays exponentially for equations of nonconvolution type liapunov functions are used to find explicit criteria for stability moreover the resolvent matrix is defined to produce a variation of constants formula the study of asymptotic equivalence for difference equations with infinite delay is carried out in section 6 finally we state some problems

stability and synchronization analysis of neural networks via halanay type inequality in this paper not only some novel sufficient criteria for stability of discrete delay neural networks but also a new discriminant method for self synchronization of hopfield neural networks are considered some novel stability criteria for discrete delay neural networks are obtained if the time average of the difference for two coefficients on some fixed lengths is lower bounded by some positive numbers moreover a new discrete halanay inequality is given which extends the existing results based on the halanay inequality a novel sufficient criterion on self synchronization of hopfield neural networks with time delay is obtained two numerical examples are given to demonstrate the effectiveness of proposed methods

structural damage assessment using a generalized minimum rank perturbation theory recently the authors proposed computationally attractive algorithms to determine the location and extent of structural damage for undamped structures assuming damage results in a localized change in stiffness properties the algorithms make use of a finite element model and a subset of measured eigenvalues and eigenvectors the developed theories approach the damage location and extent problem in a decoupled fashion first a theory is developed to determine the location of structural damage with location determined a damage extent theory is then developed the damage extent algorithm is a minimum rank perturbation which is consistent with the effects of many classes of structural damage on a finite element model in this work the concept of the minimum rank perturbation theory mrpt is adopted to determine the damage extent on the mass properties of an undamped structure in addition the mrpt is extended to the case of proportionall y damped structures for proportionally damped structures the mrpt is used to find the damage extent in any two of the three structural property matrices mass damping or stiffness finally illustrative case studies using both numerical and actual experimental data are presented he advent of the space shuttle has prompted considerable attention to the design and control of large space structures due to the large size and complexity of envisioned structures as well as the use of advanced materials to reduce structural weight it may become necessary to develop a structural health monitoring system to detect and locate structural damage as it occurs from experience gained in the machinery health monitoring field one would expect the vibration signature of the structure either frequency response functions and or modal parameters to provide useful information in determining the location and extent of structural damage assume that a refined finite element model fem of the structure has been developed before damage has occurred by refined we mean that the measured and analytical modal properties are in agreement next assume that at a later date some form of structural damage has occurred if significant the damage will result in a change in the structures modal parameters the question is can the discrepancy between the original fem modal properties and postdamage modal properties be used to ascertain structural damage most prior work in damage detection has used the general framework of fem refinement system identification in the development of damage assessment algorithms the motivation behind the development of fem refinement techniques is based on the need to validate engineering fems before their acceptance as the basis for final design analysis the standard problem has been to seek a refined fem that is as close to the original fem and whose modal properties are in agreement with those that are measured subject to various constraints such as symmetry and sparsity preservation a considerable amount of work in this area has been

optimum approximation for residual stiffness in linear system identification a new method of system identification uti 1 i zes projector m atrix theory and the moorepenrose generalized inverse to derive an analytical stiffness matrix which when combined with the analytical mass matrix will more closely p redict modal test results weighting matrices a re used to enforce connectivity and make weighted corrections to the original analytical stiffness matrix a simple and straightforward mathematical formulation is obtained the method is compared and contrasted with other methods found in the literature and a simple numerical example is presented

updating finite element dynamic models using an element by element sensitivity methodology a sensitivity based methodology for improving the finite element model of a given structure using test modal data and a few sensors is presented the proposed method searches for both the location and sources of the mass and stiffness errors and does not interfere with the theory behind the finite element model while correcting these errors the updating algorithm is derived from the unconstrained minimization of the squared 2 norms of the modal dynamic residuals via an iterative two step staggered procedure at each iteration the measured mode shapes are first expanded assuming that the model is error free then the model parameters are corrected assuming that the expanded mode shapes are exact the numerical algorithm is implemented in an element by element fashion and is capable of zooming on the detected error locations several simulation examples which demonstrate the potential of the proposed methodology are discussed i introduction e ven though the field of finite element analysis of structural dynamic problems has witnessed tremendous progress in the last two decades greater confidence is still placed in experimental data therefore the finite element model of a given structure is often updated to reflect the results of a particular experiment usually the computed eigenmodes are compared with the measured frequencies and mode shapes if both sets are not in agreement the finite element model is refined via a two step updating procedure which 1 locates the errors and 2 corrects them clearly the first step is the most challenging of the two once the location of the errors is known it is relatively easy to correct them especially if the error sources can be identified however locating and identifying these errors can be a difficult task for the following reasons in general only a few experimental modes are available and these may be contaminated with random and systematic measuring errors moreover only a subset of the degrees of freedom dof in the finite element model can be monitored and as for practical and economical reasons only a few sensors can be utilized

asymptotic behavior of volterra difference equations abstract in this paper we investigate boundedness and stability properties of some classes of discrete volterra equations the main tool in this work is the use of a representation formula which allows us to express the solution of discrete volterra equations in terms of the resolvent matrix of the corresponding system of volterra difference equations

a new razumikhin theorem for delay difference equations abstract in this paper we established a new razumikhin type theorem on stability for delay difference equations

structural model updating and prediction variability using pareto optimal models a multi objective identification method for structural model updating based on modal residuals is presented the method results in multiple pareto optimal structural models that are consistent with the experimentally measured modal data and the modal residuals used to measure the discrepancies between the measured and model predicted modal characteristics these pareto optimal models are due to uncertainties arising from model and measurement errors the relation between the multi objective identification method and the conventional single objective weighted modal residuals method for model updating is investigated using this relation an optimally weighted modal residuals method is also proposed to rationally select the most preferred model among the alternative multiple pareto optimal models for further use in structural model prediction studies computational issues related to the reliable solution of the resulting multi objective and single optimization problems are addressed the model updating methods are compared and their effectiveness is demonstrated using experimental results obtained from a three story laboratory structure tested at a reference and a mass modified configuration the variability of the pareto optimal models and their associated response prediction variability are explored using two structural model classes a simple 3 dof model class and a higher fidelity 546 dof finite element model class it is demonstrated that the pareto optimal structural models and the corresponding response and reliability predictions may vary considerably depending on the fidelity of the model class and the size of measurement errors

minimum correlation in construction of multivariate distributions in this paper we present an algorithm for exact generation of multivariate samples with prespecified marginal distributions and a given correlation matrix based on a mixture of fr echet hoeffding bounds and marginal products in the bivariate case the algorithm can accommodate any among the theoretically possible correlation coefficients and explicitly provides a connection between simulation and the minimum correlation attainable for different distribution families we calculate the minimum correlations in several common distributional examples including in some that have not been looked at before as an illustration we provide the details and results of implementing the algorithm for generating three dimensional negatively and positively correlated beta random variables making it the only noncopula algorithm for correlated beta simulation in dimensions greater than two this work has potential for impact in a variety of fields where simulation of multivariate stochastic components is desired

boundedness of discrete volterra equations abstract investigation of the boundedness of the solutions of volterra equations with discrete time by means of the direct liapunov method and comparison theorems is performed boundedness conditions of solutions of some particular classes of equations are derived using such conditions stability of direct quadrature methods for second kind volterra equations is analyzed

asymptotic constancy of solutions of linear parabolic volterra difference equations abstract in this paper we obtain some sufficient conditions under which every solution of a class of linear parabolic volterra difference equations of neutral type tends to a constant as n we also consider the asymptotic constancy of solutions for a class of linear ordinary volterra difference equations of neutral type the results obtained improve and generalize some known results

stability and instability for dynamic equations on time scales in this paper we examine the stability and instability of the equilibrium solution x 0 to the first order system of dynamic equations x d f t x t t 0 x d r n where t is from a so called time scale t with t 0 tand d is a compact set our methods involve the existence of a positive definite liapunov function v such that its delta derivative v d satisfies certain integral definite or semidefinite sign properties finally we use liapunov functions to develop an invariance principle regarding solutions to the above dynamic equation

partial differential equations on time scales discrete and continuous formulations of partial differential operators are unified by a time scale formulation of partial differential operators results include an euler lagrange equation for double integral variational problems on time scales and a picone identity which implies a sturm picone comparison theorem for second order elliptic pdes on time scales

correlations and copulas for decision and risk analysis the construction of a probabilistic model is a key step in most decision and risk analyses typically this is done by defining a joint distribution in terms of marginal and conditional distributions for the model s random variables we describe an alternative approach that uses a copula to construct joint distributions and pairwise correlations to incorporate dependence among the variables the approach is designed specifically to permit the use of an expert s subjective judgments of marginal distributions and correlations the copula that underlies the multivariate normal distribution provides the basis for modeling dependence but arbitrary marginals are allowed we discuss how correlations can be assessed using techniques that are familiar to decision analysts and we report the results of an empirical study of the accuracy of the assessment methods the approach is demonstrated in the context of a simple example including a study of the sensitivity of the results to the assessed correlations

invariant and attracting sets of volterra difference equations with delays will be concerned with the invariant and asymptotic properties of volterra differ ence equations with delays sufficient conditions for determining the invariant and attracting sets of the equations are obtained examples are given to illustrate the obtained results 2003 elsevier science ltd all rights reserved

introduction to the bayesian approach applied to elastic constants identification the basic formulation of the least squares method based on the l2 norm of the residuals is still widely used today for identifying elastic constants of aerospace materials from experimental data while this method often works well methods that can benefit from statistical information such as the bayesian method may sometimes be more accurate we seek situations with significant difference between the material properties identified by the two methods for a simple three bar truss example we illustrate three situations in which the bayesian approach systematically leads to more accurate results different sensitivity of the measured response to the parameters to be identified different uncertainty in the measurements and correlation among response components when all three effects add up the bayesian approach can be much more accurate furthermore the bayesian approach has the additional advantage of providing the uncertainty in the identified parameters we also compare the two methods for a more realistic problem of identification of elastic constants from natural frequencies of a composite plate

on asymptotic stability of delay difference systems in this paper we obtain some criteria for determining the asymptotic stability of the zero solution for some classes of delay difference systems by using a discrete version of the second lyapunov method our results can be well suited for computational purposes

linearized stability analysis of discrete volterra equations abstract various different types of stability are defined in a unified framework for discrete volterra equations of the type x n f n n j 0 k n j x n n 0 under appropriate assumptions stability results are obtainable from those valid in the linear case k n j x n b n j x j and a linearized stability theory is studied here by using the fundamental and resolvent matrices several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of banach space b the elements of which are sequences of vectors x n f n e d b n j e d e d n j 0 etc we show that the theory including a number of new results as well as results already known can be presented in a systematic framework in which results parallel corresponding results for classical volterra integral equations of the second kind

stiffness matrix adjustment using mode data a procedure is introduced that uses in addition to mode data structural connectivity information to optimally adjust deficient stiffness matrices the adjustments performed are such that the percentage change to each stiffness coefficient is minimized the physical configuration of the analytical model is preserved and the adjusted model will exactly reproduce the modes used in the identification the theoretical development is presented and the procedure is demonstrated by numerical simulation of a test problem

discrete time sis epidemicmodel in a seasonal environment we study the combined effects of seasonal trends and diseases on the extinction and persistence of discretely reproducing populations we introduce the epidemic threshold parameter r0 for predicting disease dynamics in periodic environments typically in periodic environments r0 1 implies disease persistence on a cyclic attractor while r0 1 implies disease extinction we also explore the relationship between the demographic equation and the epidemic process in particular we show that in periodic environments it is possible for the infective population to be on a chaotic attractor while the demographic dynamics is nonchaotic

on the recovery of joint distributions from limited information the paper considers the role of entropy and other information theoretic concepts in the description and formation of joint distributions for m random variables the discussion includes a review of methods for constructing discrete and continuous joint distributions from the component marginal distributions we then propose a minimum cross entropy approach that recovers continuous joint distributions from the joint and marginal moments and the marginal densities the large sample properties of the associated estimator are outlined and a simple demonstration problem that highlights the advantages and drawbacks of the proposed method is presented

stability and periodicity in dynamic delay equations let t be an arbitrary time scale that is unbounded above by means of a variation of lyapunov s method and contraction mapping principle this paper handles asymptotic stability of the zero solution of the completely delayed dynamic equations x d t a t x d t d d t moreover if t is a periodic time scale then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation

stability of finite delay difference systems the stability of finite delay difference systems has recently been discussed in l in this paper we consider the delay difference systems with infinite delay of the following form x n 1 g n x s s 1 1 1 n g n x 1 where 1 is an integer or w we assume g n 0 0 forn l l l so that 1 has the zero solution x n 0 clearly for any given integer n l 1 and a given bounded initial function 4 l 1 1 no rk there is a unique solution of l denoted by x n n 4 such that it satisfies 1 for all integers n 2 n and x n no 4 444 forn f l l no for any integer m 2 co let 2 m m 1 i and 2 z the integer set if m a for any n e z define ii411 tlnl sup l s 15 s 5 n and for any h 0 let c n 4 z rk i nl h in the sequel we always assume that the variables n r and s take integer values and the corresponding intervals are discrete intervals

asymptotic equivalence between two linear volterra difference systems abstract we study asymptotic equivalence between the solutions of linear volterra difference system x n 1 a n x n s n 0 n b n s x s and its perturbed system y n 1 a n y n s n 0 n b n s y s f n

stability of difference volterra equations direct liapunov method and numerical procedure abstract a procedure to construct liapunov functionals for discrete volterra equations is proposed using this procedure stability conditions are derived for general volterra difference equations some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for volterra integro differential equations are presented

l2 spaces and boundary value problems on time scales abstract in this paper we consider a second order sturm liouville type boundary value operator of the form l u p u q u on an arbitrary bounded time scale t for suitable functions p q together with suitable boundary conditions operators of this type on time scales have normally been considered in a setting involving banach spaces of continuous functions on t in this paper we introduce a space l 2 t of square integrable functions on t and sobolev type spaces h n t n 1 consisting of l 2 t functions with nth order generalised l 2 t type derivatives we prove some basic functional analytic results for these spaces and then formulate the operator l in this setting in particular we allow p h 1 t while q l 2 t this generalises the usual conditions that p c rd 1 t q c rd 0 t 2 we give some immediate applications of the functional analytic results to l such as positivity injectivity invertibility and compactness of the inverse we also construct a green s function for l the analogues of these results on real intervals are well known and are fundamental to the usual sturm liouville theory on such intervals

the asymptotic behavior of the solutions of a volterra difference equation the aim of the present paper is to study the asymptotic behavior of the sum difference equation of volterra type an xn bn xn cn n l 0n 1gl h n xn xn as n

secant method adjustment for structural models on orbit testing will be required for final tuning and validation of any mathematical model of large space structures identification methods using limited response data to produce optimally adjusted property matrices seem ideal for this purpose but difficulties exist in the application of previously published methods to large space truss structures this article presents new stiffness matrix adjustment methods that generalize optimal update secant methods found in quasi newton approaches for nonlinear optimization many aspects of previously published methods of stiffness matrix adjustment may be better understood within this new framework of secant methods one of the new methods preserves realistic structural connectivity with minimal storage requirements and computational effort a method for systematic compensation for errors in measured data is introduced that also preserves structural connectivity two demonstrations are presented to compare the new methods results to those of previously published techniques

uniform asymptotic stability in nonlinear volterra discrete systems abstract we employ the notion of total stability to obtain new criteria for uniform asymptotic stability of the zero solution of a nonlinear volterra discrete system resolvent equation methods are employed and a summability criterion on the resolvent kernel is obtained also we obtain a new difference equation that the resolvent r n s satisfies

qualitative analysis of nonlinear volterra integral equations on time scales using resolvent and lyapunov functionals in this paper we use the notion of the resolvent equation and lyapunov s method to study boundedness and integrability of the solutions of the nonlinear volterra integral equation on time scales x t a t t 0 t c t s g s x s s t t 0 t in particular the existence of bounded solutions with various lp properties are studied under suitable conditions on the functions involved in the above volterra integral equation at the end of the paper we display some examples on different time scales

stability and boundedness in difference systems with finite delay abstract in this paper stability and boundedness theorems for delay difference systems with the condition v n x n w 2 x n are given where is the backward difference operator some known results are generalized

double integral calculus of variations on time scales we consider a version of the double integral calculus of variations on time scales which includes as special cases the classical two variable calculus of variations and the discrete two variable calculus of variations necessary and sufficient conditions for a local extremum are established among them an analogue of the euler lagrange equation

on the existence of solutions of linear volterra difference equations asymptotically equivalent to a given sequence abstract schauder s fixed point technique is applied to asymptotical analysis of solutions of a linear volterra difference equation x n 1 a n b n x n i 0 n k n i x i where n n 0 x n 0 r a n 0 r k n 0 n 0 r and b n 0 r 0 is periodic in the paper sufficient conditions are derived for the validity of a property of solutions that for every admissible constant c r there exists a solution x x n such that x n c i 0 n 1 a i i 1 n where n j 0 n 1 b j for n and inequalities for solutions are derived relevant comparisons and illustrative examples are given as well

direct least squares formulation of a stiffness adjustment method a direct least squares formulation of the stiffness matrix adjustment method known as the kma method kabe a m stiffness matrix adjustment using mode data aiaa journal vol 23 no 9 1985 pp 1431 1436 is presented the kma method belongs to a class of procedures that refine dynamic models using test measured modes and structural connectivity viewed as a constrained minimization problem most of these methods have applied lagrange multiplier techniques in their development it is shown that these two approaches result in linear systems of equations that are algebraically equivalent and therefore have identical solutions by virtue of this equivalence the direct least squares formulation of the kma method is shown to be equivalent to its original lagrange multiplier formulation the direct least squares versions of other model refinement methods that preserve structural connectivity are also discussed in this respect this approach provides a rigorous framework for unifying these optimal update procedures and indicates how their solutions should compare numerical results are presented that illustrate the equivalence between the two formulations and also the conditions under which the various methods yield identical solutions the results also indicate that because of improved numerical conditioning the direct least squares approach yields more accurate computational solutions and generally requires less computer storage than methods that were developed using lagrange multipliers

on exponential dichotomy bohl perron type theorems and stability of difference equations abstract for a delay difference equation x n 1 k d n a n k x k f n in a banach space the following result is proved if for any f l p the solution is x l p then the solution of the homogeneous equation f 0 is exponentially stable this result is applied to obtain new explicit conditions for exponential stability of a scalar nonautonomous delay difference equation

boundedness of infinite delay difference systems x n 1 g n x s s i 1 1 n g n x 1 where i is an integer or co g is a volterra functional determined by the integer n and the values of x s for s i 1 1 n and taking values in rk the stability results of 1 have been recently established in 11 however in order to explore the existence of periodic solutions of 1 we need to know not only the stability but also the boundedness of solutions for the linear case of 1 for the relevant results we refer to 2 while for the corresponding results of infinite delay differential systems see 3 41 in this paper we are going to establish theorems on boundedness for 1 in terms of both liapunov functionals and functions with the so called razumikhin techniques before doing this we give the following notations and definitions for any integer m 2 00 let z m m 1 and z z the integer set if m a trivially for any ie e z and a given bounded initial function 4 i i 1 no rk there is a unique solution of l denoted by x n n d such that it satisfies 1 for all integers 17 2 n and x n n 4 d n forn l l l n

improvement of a large analytical model using test data a method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure the method directly identifies without iteration a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements an application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure the resulting changes in the model are judged to be small compared to expectations of error in the analysis thus the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics in addition it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure nomenclature analytical matrix matrix of changes identity matrix full improved stiffness and mass matrices n x n full analytical k m matrices n x n partitions of ka ma corresponding to test coordinates partitions of ka ma corresponding to coupling elements partitions of ka ma corresponding to unmea sured coordinates number of measured modes number of degrees of freedom in the model measures of changes eqs 15 17 matrix norm sum of the squares of all the elements rectangular modal matrix normalized n x m th mode th column of measured and unmeasured partitions of diagonal matrix of measured natural frequencies m xm natural frequency of th mode 12 17 sum of the squares of all elements of matrix

finite element model update via bayesian estimation and minimization of dynamic residuals an algorithm is presented for updating finite element models based upon a minimization of dynamic residuals the dynamic residual of interest is the force unbalance in the homogeneous form of the equations of motion arising from errors in the model s mass and stiffness when evaluated with the identified modal parameters the present algorithm is a modification and extension of a previously developed sensitivity based element by element sb ebe method for damage detection and finite element model up dating in the present algorithm sb ebe has been generalized to minimize a dynamic displacement residual quantity which is shown to improve test analysis mode correspondence furthermore the algorithm has been modified to include bayesian estimation concepts and the underlying nonlinear optimization problem has been consistently linearized to improve the convergence properties the resulting algorithm is demonstrated via numerical and experimental examples to be an efficient and robust method for both localizing model errors and estimating physical parameters

attenuant cycles in periodically forced discrete time age structured population models in discrete time age structured population models a periodic environment is not always deleterious we show that it is possible to have the average of the age class populations over an attracting cycle in a periodic environment not less than the average of the carrying capacities in a corresponding constant environment in our age structured model a periodic environment does not increase the average total biomass no resonance however a periodic environment is disadvantageous for a population whenever there is no synchrony between the number of age classes and the period of the environment as in periodically forced models without age structure we show that periodically forced age structured population models support multiple attractors with complicated structures

periodicity and stability of linear volterra difference systems abstract we give sufficient conditions for a system of volterra difference equations to have a unique periodic solution then a criterion for uniform asymptotic stability is established

properties of certain iterated dynamic integrodiffetential equation on time scales abstract the main objective of this paper is study the existence uniqueness and other properties of solution of some iterated dynamic integrodifferential on time scales the main tools employed are banach fixed point theorem and an inequality with explicit estimates are used for proving our results

asymptotic property of linear volterra difference systems by using the resolvent matrix and the comparison principle we investigate the asymptotic behavior of linear volterra difference systems

a family of matrices the discretized brownian bridge and distance based regression abstract the investigation of a distance based regression model using a one dimensional set of equally spaced points as regressor values and x y as a distance function leads to the study of a family of matrices which is closely related to a discrete analog of the brownian bridge stochastic process we describe its eigenstructure and several properties recovering in particular well known results on tridiagonal toeplitz matrices and related topics

general theorems for stability and boundedness for nonlinear functional discrete systems abstract we consider the nonlinear functional discrete system x n 1 g n x s 0 s n def g n x and obtain sufficient conditions for stability properties of the zero solution and for the boundedness of solutions

asymptotically ordinary linear volterra difference equations with infinite delay abstract a class of linear volterra difference equations with infinite delay is considered it is shown that if the coefficient matrices are sufficiently small then the volterra difference equation is asymptotically equivalent to a linear ordinary difference equation at infinity the coefficient matrix of the ordinary difference equation is a solution of an associated matrix equation which can be obtained by successive approximations the eigenvalues of the approximating matrices converge exponentially to the characteristic roots of the volterra difference equation as a corollary an efficient new method is obtained for the computation of the characteristic roots with an explicit error estimate

periodic solutions of difference equations abstract in this paper we discuss the existence of periodic solutions of periodic difference equations including x n 1 f n x n n z where x and f are d vectors and z denotes the set of integers as well as scalar autonomous difference equations including x n f x n 1 n z first we state some recent results concerning periodic solutions of periodic difference equations then we show the existence of periodic solutions of scalar autonomous difference equations by using schauder s fixed point theorem we illustrate the results with some examples

on a family of multivariate copulas for aggregation processes we introduce a family of multivariate copulas a special type of n ary aggregation operations depending on a univariate function this family is used in the construction of a special aggregation operation that satisfies a lipschitz condition several examples are provided and some statistical properties are studied