axisymmetric vibration of a circular plate with double linear variable thickness abstract the rayleigh ritz method has been employed to study the axisymmetric transverse vibration of a circular plate with two different linear variations in thickness one in the central core and the other in the outer annular region the boundary has been taken either clamped or simply supported results for linearly varying thickness and uniform thickness have been obtained as special cases and are compared with those available in the literature the effects of the various parameters on the frequencies mode shapes and the nodal radii have been examined through extensive numerical results given in the form of graphs and tables

vibrations and elastic stability of thin circular plates with variable profile abstract circular plates of thicknesses varying to the functional relationho 1 r a n wherenis a positive integer are studied in the present paper uniform elastic boundary restraints are considered and the first two natural frequency coefficients corresponding to axisymmetric modes and the buckling in plane pressure are determined using simple polynomial co ordinate functions which identically satisfy the boundary conditions and the classical rayleigh ritz method an independent solution is also obtained by means of the differential quadrature technique

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

a fast automatic algorithm for image denoising by a regularization method a new fast iterative algorithm is proposed to denoise images affected by gaussian additive noise by solving a constrained optimization problem a family of functionals parameterized by a few hyperparameters already proposed in the literature to solve denoising problems is considered the algorithm estimates jointly the image and the hyperparameters therefore providing an automatic method each member of the family is made up of a coherence with the data term and a term enforcing a roughness penalty and preserving jump discontinuities assuming we know the noise variance an adequacy constraint is also considered the algorithm computes a member of the family and a minimizer of it which satisfies the constraint a convergence proof is provided we then consider a heuristic version of the algorithm which gives restorations of comparable quality whose computational complexity is a linear function of the pixels number experimental results on synthetic and real data are presented moreover numerical comparisons with several fast denoising methods are provided

free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule the generalized differential quadrature rule proposed recently by these authors is applied here to the free vibration analyses of solid circular plates with radially varying thickness and elastic restraints the thickness can vary radially in any given continuous form and it varies exponentially and linearly in this work two regularity conditions corresponding to the plate center are expressed in explicit formulae since they have been inexactly or even wrongly expressed in the literature such errors are pointed out in the application of the differential quadrature techniques and in the expression of the regularity conditions at the plate center numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

brief accuracy enhancing methods for balancing related frequency weighted model and controller reduction we consider the solution of the balancing related frequency weighted model and controller reduction problems using accuracy enhanced numerical algorithms we propose first new stability enforcing choices of the frequency weighted grammians which can guarantee the stability of reduced models for two sided frequency weights then we show that for the frequency weighted controller reduction problems with standard stability and performance enforcing frequency weights the computation of the frequency weighted grammians can be done by solving reduced order lyapunov equations for both frequency weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their cholesky factors this allows the extension of the square root and balancing free accuracy enhancing techniques to the frequency weighted case

on the free vibration analysis of circular plates with stepped thickness over a concentric region by the differential quadrature element method it can be seen the dqem retains all advantages of the differential quadrature method extends the applicability range of the dq method and is more convenient for treating various boundary conditions

a new approach to the differential quadrature method for fourth order equations a generalized and more complete methodology for treating boundary conditions in the differential quadrature method dqm is presented this improved approach eliminates the deficiencies of the type grid arrangement which represents an approximation by applying the boundary conditions exactly two kinds of basis functions chebyshev and lagrange are used for concept demonstration it is found that the new approach cures most deficiencies of the current dqm 1997 by john wiley sons ltd

axisymmetric vibration of a circular plate with exponential thickness variation abstract the rayleigh ritz method has been used to find the first four frequencies mode shapes and nodal radii for axisymmetric vibration of a circular plate when the thickness varies exponentially with the radial distance results are given for a clamped as well as simply supported boundary for various values of the parameter controlling the thickness variation convergence is ensured by working out a sufficient number of approximations comparison has been made with existing results for uniform thickness and linearly varying thickness

free vibration of plates by the high accuracy quadrature element method in this paper a highly accurate and rapidly converging hybrid approach is presented for the quadrature element method qem solution of plate free vibration problems the hybrid qem essentially consists of a collocation method in conjunction with a galerkin finite element technique to combine the high accuracy of the differential quadrature method dqm for the efficient solution of differential equations with the generality of the finite element formulation this results in superior accuracy with fewer degrees of freedom than conventional fem or fdm a series of numerical tests is conducted to assess the performance of the quadrature plate element in free vibration problems anisotropic and stepped thickness plates are investigated as well as mixed boundary conditions and point supports at the edges in all cases the results obtained are quite accurate

transverse vibrations of circular plates and membranes with intermediate supports the present paper deals with an approximate solution of the title problem in the case of a a concentric circular support and b a secant support fundamental frequency coefficients are determined by means of the rayleigh minimization procedure the modal shapes been approximated by simple polynomial co ordinate functions in the case of a circular plate with a secant support an independent solution is obtained by means of a finite element code and very good agreement is shown to exist

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

analysis of the rational krylov subspace and adi methods for solving the lyapunov equation for large scale problems an effective approach for solving the algebraic lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation although krylov subspaces have been used for a long time only more recent developments have shown that rational krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit adi recurrence in this paper we develop a convergence analysis of the rational krylov subspace method rksm based on the kronecker product formulation and on potential theory moreover we propose new enlightening relations between this approach and the adi method our results provide solid theoretical ground for recent numerical evidence of the superiority of rksm over adi when the involved parameters cannot be computed optimally as is the case in many practical application problems

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

interpolatory weighted h2 model reduction this paper introduces an interpolation framework for the weighted h 2 model reduction problem we obtain a new representation of the weighted h 2 norm of siso systems that provides new interpolatory first order necessary conditions for an optimal reduced order model the h 2 norm representation also provides an error expression that motivates a new weighted h 2 model reduction algorithm several numerical examples illustrate the effectiveness of the proposed approach

frequency weighted model reduction via optimal projection the problem of order reduction with frequency weighting is considered necessary conditions characterizing the reduced order model which minimizes a quadratic error criterion in the frequency domain with either input or output frequency weighting are given the problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input when the weighting matrix is singular at infinity the reduced order model has a direct transmission term even if the original system is strictly proper this additional degree of freedom can be used to obtain better approximation over a prespecified frequency range

h2 tf optimality conditions for a finite time horizon abstract in this paper we establish the interpolatory model reduction framework for optimal approximation of mimo dynamical systems with respect to the h 2 norm over a finite time horizon denoted as the h 2 t f norm using the underlying inner product space we derive the interpolatory first order necessary optimality conditions for approximation in the h 2 t f norm then we develop an algorithm which yields a locally optimal reduced model that satisfies the established interpolation based optimality conditions we test the algorithm on various numerical examples to illustrate its performance

a new frequency weighted balanced truncation method and an error bound in this paper a new frequency weighted model reduction method with an a priori error bound is proposed the method is a generalization of enns technique 1984 and yields stable models even when both input and output weightings are included the proposed method is compared with other existing methods using numerical examples

application of generalized differential quadrature rule in blasius and onsager equations the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to third order non linear differential equations of the blasius type and to sixth order linear onsager differential equations high 3rd order differential equations in fluid mechanics are dealt with without using point techniques the half space domain is simplified in a practical way accurate results are obtained for both kinds of problems the wide applicability of the gdqr in high order differential equations is manifested further through this work copyright 2001 john wiley sons ltd

transverse vibrations of a circular annular plate with an intermediate circular support and a free inner edge the present study deals with the determination of the fundamental frequency of transverse vibration

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

vibrating circular plates with stepped thickness over a concentric circular region a general approximate solution natural frequencies corresponding to axi and antisymmetric modes of vibration of the structural elements described in the title are studied using the optimized rayleigh ritz approach it is assumed that the plates are elastically restrained against translation and rotation the plate amplitude is approximated in terms of polynomial coordinate functions that satisfy the boundary conditions each one containing an undetermined exponential parameter that allows for the minimization of the eigenvalues under investigation

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

exact vibration solutions for circular mindlin plates with multiple concentric ring supports this paper presents an analytical approach to investigate the vibration behaviour of circular mindlin plates with multiple concentric ring supports a circular plate is first divided into multiple annular segments and a circular segment at the locations of the ring supports the governing differential equations and the solutions of these equations for the annular and circular segments based on the mindlin plate theory are presented a homogenous equation system governing the vibration of circular mindlin plates with ring supports is derived by imposing the essential and natural boundary and segment interface conditions the first known exact vibration frequencies for circular mindlin plates with multiple concentric ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented the influence of the ring support locations plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed

noise estimation in magnitude mr datasets estimating the noise parameter in magnitude magnetic resonance mr images is important in a wide range of applications we propose an automatic noise estimation method that does not rely on a substantial proportion of voxels being from the background specifically we model the magnitude of the observed signal as a mixture of rice distributions with common noise parameter the expectation maximization em algorithm is used to estimate all parameters including the common noise parameter the algorithm needs initializing values for which we provide some strategies that work well the number of components in the mixture model also needs to be estimated en route to noise estimation and we provide a novel approach to doing so our methodology performs very well on a range of simulation experiments and physical phantom data finally the methodology is demonstrated on four clinical datasets

a note on closed loop balanced truncation closed loop balanced truncation is a model reduction method that aims at preserving the closed loop properties of the original controller we show that this method is equivalent to frequency weighted balanced truncation with certain weighting and that for observer based controllers a significant amount of computation can be saved because of the observer state feedback structure of the controller

vibration analysis of beams using the generalized differential quadrature rule and domain decomposition abstract this study dealt with domain decomposition in the recently proposed generalized differential quadrature rule in detail the authors concentrated on the free vibration of multispan and stepped euler beams and beams carrying an intermediate or end concentrated mass since compatibility conditions should be implemented in a strong form at the junction of the subdomains concerned the fem techniques used for internal moments and shear forces must not be used compatibility conditions and their differential quadrature expressions were explicitly formulated a peculiar phenomenon was found in differential quadrature applications that equal length subdomains gave more accurate results than unequal length ones using the same number of subdomain grids various examples were presented and very accurate results have been obtained

application of differential quadrature to transport processes the methodology and numerical solution of problems concerning transport processes via the method of differential quadrature are presented application of the method is demonstrated by solving a simple one dimensional time dependent transient diffusion process involving an irreversible reaction without any flux across the end boundary in addition the same technique is used for the first time to the authors knowledge to solve a steady state problem for this purpose a convection diffusion problem involving an irreversible reaction is considered the demonstration is carried out in two ways 1 using the bellman et al technique which employs approximation formulas for higher order partial derivatives derived by iterating the linear quadrature approximation for the first order partial derivative and 2 using individual quadratures to approximate the partial derivatives of first as well as higher orders as suggested by mingle both approaches give the same results however the latter saves an appreciable amount of iterative computing effort despite the fact that it requires separate weighting coefficients for each individual quadrature since the technique of differential quadrature can produce solutions with sufficient accuracy even when using as few as three discrete points both the programming task and computational effort are alleviated considerably for these reasons the differential quadrature approach appears to be very practical in solving a variety of problems related to transport phenomena

on optimality of approximate low rank solutions of large scale matrix equations abstract in this paper we discuss some optimality results for the approximation of large scale matrix equations in particular this includes the special case of lyapunov and sylvester equations respectively we show a relation between the iterative rational krylov algorithm and a riemannian optimization method which recently has been shown to locally minimize a certain energy norm of the underlying lyapunov operator moreover we extend the results for a more general setting leading to a slight modification of irka by means of some numerical test examples we show the efficiency of the proposed methods

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

balanced performance preserving controller reduction two methods for controller reduction based on the closed loop system performance preserving criterion are proposed they are the closed loop block balanced and the closed loop structurally balanced truncating methods reduced order controllers are obtained directly by solving the closed loop system reduction problem with performance preserving weights

numerical solution for differential equations of duffing type non linearity using the generalized differential quadrature rule the generalized differential quadrature rule gdqr proposed recently by the authors is applied for the first time to second and fourth order initial value differential equations with duffing type non linearity procedures are given in detail to convert these non linear differential equations into a set of linear algebraic equations in an iterative loop using the frechet derivative the effectiveness of the gdqr for obtaining the periodic solution of the duffing equation has been demonstrated through a number of examples it is also shown that the use of the frechet derivative makes it easier for the gdqr to handle non linearity the wide applicability of the gdqr is manifested further through this work

synthetic magnetic resonance imaging revisited synthetic magnetic resonance mr imaging is an approach suggested in the literature to predict mr images at different design parameter settings from at least three observed mr scans however performance is poor when no regularization is used in the estimation and otherwise computationally impractical to implement for 3 d imaging methods we propose a method which accounts for spatial context in mr images by the imposition of a gaussian markov random field mrf structure on a transformation of the spin lattice relaxation time the spin spin relaxation time and the proton density at each voxel the mrf structure is specified through a matrix normal distribution we also model the observed magnitude images using the more accurate but computationally challenging rice distribution a one step late expectation maximization approach is adopted to make our approach computationally practical we evaluate predictive performance in generating synthetic mr images in a clinical setting our results indicate that our suggested approach is not only computationally feasible to implement but also shows excellent performance

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

high accuracy plane stress and plate elements in the quadrature element method in this paper a high accuracy and rapid convergence hybrid approach is developed for the quadrature element method qem solution of two dimensional plane stress and plate bending problems the hybrid qem essentially consists of a collocation method in conjunction with a galerkin finite element technique to combine the high accuracy of the differential quadrature method dqm with the generality of finite element formulations this results in superior accuracy with fewer degrees of freedom than conventional fem or fdm the present method also extends the general application of the collocation numerical approach to fourth order governing equation systems here the influence of collocation point location is investigated a series of numerical tests is conducted in order to assess the performance of the quadrature plane stress and plate elements in static problems

near optimal frequency weighted interpolatory model reduction abstract this paper develops an interpolatory framework for weighted h 2 model reduction of mimo dynamical systems a new representation of the weighted h 2 inner products in mimo settings is introduced and used to derive associated first order necessary conditions satisfied by optimal weighted h 2 reduced order models equivalence of these new interpolatory conditions with earlier riccati based conditions given by halevi is also shown an examination of realizations for equivalent weighted h 2 systems leads then to an algorithm that remains tractable for large state space dimension several numerical examples illustrate the effectiveness of this approach and its competitiveness with frequency weighted balanced truncation and an earlier interpolatory approach the weighted iterative rational krylov algorithm

on the lateral vibration of thin annular and circular composite plates subject to certain complicating effects abstract series comprising orthogonally generated polynomial functions are used in the rayleigh ritz method to obtain an eigenvalue equation for the flexural vibration of thin flat annular and circular plates the plates may be of isotropic or polar orthotropic material of uniform or radially varying thickness and may pass over intermediate concentric ring supports numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

a bayesian approach to synthetic magnetic resonance imaging summary synthetic magnetic resonance imaging involves the estimation based on a set of measured images with noise of three basic physical quantities that are nonlinearly related to the observations the methods currently available for this ill conditoned inverse problem either do not provide sufficiently accurate estimates or require time consuming data collection we formulate this nonlinear problem in a bayesian framework taking into account knowledge about the physics of the magnetic resonance imaging experiment statistical properties of the experimental noise and prior information about the underlying physical quantities modelled by a suitable markov random field a new multilayer markov random field is proposed inference is drawn by means of markov chain monte carlo methods or iterated conditional modes some examples are included to demonstrate how synthetic magnetic resonance imaging by this approach can be performed in an accurate and reliable way

vibration of non homogeneous circular mindlin plates with variable thickness free vibration analysis of non homogeneous circular plates of variable thickness has been presented using the first order shear deformation plate theory of mindlin chebyshev collocation technique has been employed to obtain the first three natural frequencies and mode shapes for clamped simply supported and free plate with four digit exactitude the effect of structural parameters such as taper non homogeneity and density and that of edge conditions on the natural frequencies has been investigated comparison of results for some special cases with published results obtained from other approximate methods has been presented which shows an excellent agreement

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

the generalized differential quadrature rule for initial value differential equations abstract the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to solve initial value differential equations of the 2nd to 4th order differential quadrature expressions are derived based on the gdqr for these equations the hermite interpolation functions are used as trial functions to obtain the explicit weighting coefficients for an easy and efficient implementation of the gdqr the numerical solutions for example problems demonstrate that the gdqr has high efficiency and accuracy a detailed discussion on the present method is presented by comparing with other existing methods the present method can be extended to other types of differential equation systems

differential quadrature for multi dimensional problems the technique of differential quadrature for the solution of partial differential equations introduced by bellman et al is extended and generalized to encompass partial differential equations involving multiple space variables approximation formulae for a variety of first and second order partial derivatives and typical weighting coefficients are presented application of these formulae is demonstrated on the solution of the convection diffusion equation for the two and three dimensional space dependent cases and for both the transient and steady state dispersion of inert neutrally buoyant pollutants from continuous sources into an unbounded atmosphere

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

application of differential quadrature to static analysis of structural components the numerical technique of differential quadrature for the solution of linear and non linear partial differential equations first introduced by bellman and his associates is applied to the equations governing the deflection and buckling behaviour of one and two dimensional structural components separate transformations are used for higher order derivatives as suggested by mingle thus extending the method to treat fourth order equations and to include multiple boundary conditions in the respective co ordinate directions results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods the application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort

the generalized differential quadrature rule for fourth order differential equations the generalized differential quadrature rule gdqr proposed here is aimed at solving high order differential equations the improved approach is completely exempted from the use of the existing point technique by applying multiple conditions in a rigorous manner the gdqr is used here to static and dynamic analyses of bernoulli euler beams and classical rectangular plates numerical error analysis caused by the method itself is carried out in the beam analysis independent variables for the plate are first defined the explicit weighting coefficients are derived for a fourth order differential equation with two conditions at two different points it is quite evident that the gdqr expressions and weighting coefficients for two dimensional problems are not a direct application of those for one dimensional problems the gdqr are implemented through a number of examples good results are obtained in this work copyright 2001 john wiley sons ltd

a generalized differential quadrature rule for bending analysis of cylindrical barrel shells the generalized differential quadrature rule gdqr proposed recently by the present authors is applied here to solve an eighth order boundary value differential equation with four boundary conditions at each boundary which governs the bending analysis of circular cylindrical barrel shells thus multiple boundary conditions are implemented exactly at boundaries without applying the approximate point non boundary point technique both hermite interpolation functions and gdqr s explicit weighting coefficients are obtained while the latter are the most important for an accurate implementation of the gdqr the analytical results for various cylindrical barrel shells are derived and calculated here in order to verify the accuracy of the calculated gdqr results in implementation of multiple conditions at one point the gdqr gains a clear advantage over the existing point technique in the differential quadrature applications and manifests a great potential for applications in many fields of applied mathematics

a new algorithm for l 2 optimal model reduction in this paper the quadratically optimal model reduction problem for single input single output systems is considered the reduced order model is determined by minimizing the integral of the magnitude squared of the transfer function error it is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and on this basis a two step iterative algorithm is developed combining a least squares solver with a gradient minimizer convergence of the proposed algorithm to stationary values of the quadratic cost function is proved the formulation is extended to handle the frequency weighted optimal model reduction problem three examples demonstrate the optimization algorithm

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix