a framework for iterative analysis of non classically damped dynamical systems abstract in this paper we propose a general iterative framework to solve the dynamic problem for linear systems with non classical viscous damping a systematic approach is used to derive families of stationary iterative schemes that as an instance of particular interest decouple the equations of motion for numerical study of the system response for such schemes we present a detailed convergence analysis and propose several solution strategies suitable for a broad class of systems these techniques are based on spectral analysis of particular iteration matrices arising in the derivation and aim at optimizing the convergence performance of the method we demonstrate that the proposed systematic framework based on a novel application of the homotopy analysis method generalizes iterative schemes previously reported in the literature and importantly provides a unified perspective for the study of iterative solutions of dynamic problems further we establish a connection between our results and the theory of iterative schemes for algebraic linear systems thus providing insights on convergence results and applicability of the method numerical examples illustrate the effectiveness of the approach and indicate future research directions

stability of the second order matrix polynomial this note summarizes some existing stability and instability conditions of the second order matrix polynomial which arises in the formulation of classical mechanics aerodynamics and robotic systems also some sufficient conditions for stability or instability of the second order matrix polynomial are newly developed via the relevant linear matrix equation obtained from the lyapunov theory

an improved algorithm for the evaluation of cauchy principal value integrals of oscillatory functions and its application abstract a new interpolatory type quadrature rule is proposed for the numerical evaluation of cauchy principal value integrals of oscillatory kind 1 1 f x x e i x d x where 1 1 the method is based on an interpolatory procedure at clenshaw curtis points and the singular point and the fast computation of the modified moments with cauchy type singularity based on this result a new method is presented for the computation of the oscillatory integrals with logarithmic singularities too these methods enjoy fast implementation and high accuracy convergence rates on are also provided numerical examples support the theoretical analyses

abstract hardy sobolev spaces and interpolation the purpose of this work is to describe an abstract theory of hardy sobolev spaces on doubling riemannian manifolds via an atomic decomposition we study the real interpolation of these spaces with sobolev spaces and finally give applications to riesz inequalities

a characterization of hajlasz sobolev and triebel lizorkin spaces via grand littlewood paley functions abstract in this paper we establish the equivalence between the hajlasz sobolev spaces or classical triebel lizorkin spaces and a class of grand triebel lizorkin spaces on euclidean spaces and also on metric spaces that are both doubling and reverse doubling in particular when p n n 1 we give a new characterization of the hajlasz sobolev spaces m 1 p r n via a grand littlewood paley function

the derivative of the maximal function in this note we show that the local hardy littlewood maximal operator is bounded in the sobolev space thus the maximal function often has partial derivatives we also show that the maximal operator preserves the zero boundary values in sobolev s sense

on the computation of fourier transforms of singular functions abstract in this paper we consider a method for the computation of finite fourier transforms of functions having endpoint singularities of algebraic or logarithmic type this method may be considered as a modification of clenshaw curtis quadrature the nonoscillatory and nonsingular part of the integrand is replaced by a truncated chebyshev series approximation and the integral is then evaluated using recurrence relations the numerical stability of these recurrence relations is investigated

on the convergence of the method for indefinite integration of oscillatory and singular functions we consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in keller 8 to this end we have analysed the properties of the difference operator related to the difference equation for the chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients properties of this operator were never investigated before the obtained results lead us to the conclusion that the studied method is always convergent we also give a rigorous proof of the error estimates

hardy space of exact forms on mathbb r n we show that the hardy space of divergence free vector fields on r 3 has a divergence free atomic decomposition and thus we characterize its dual as a variant of bmo using the duality result we prove a div curl type theorem for b in l 2 loc r 3 r 3 sup f b u v dx is equivalent to a bmo type norm of b where the supremum is taken over all u u w 1 2 r 3 with u l 2 v l 2 1 this theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity in addition we introduce hardy spaces of exact forms on r n study their atomic decompositions and dual spaces and establish div curl type theorems on r n

computation of integrals with oscillatory singular factors of algebraic and logarithmic type in this paper we present the clenshaw curtis filon methods and the higher order methods for computing many classes of oscillatory integrals with algebraic or logarithmic singularities at the two endpoints of the interval of integration the methods first require an interpolant of the nonoscillatory and nonsingular parts of the integrands in n 1 clenshaw curtis points then the required modified moments can be accurately and efficiently computed by constructing some recurrence relations moreover for these quadrature rules their absolute errors in inverse powers of the frequency are given the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

hardy sobolev spaces on strongly lipschitz domains of rn let be a strongly lipschitz domain of rn the hardy spaces hr1 and hz1 have been introduced by miyachi studia math 95 3 1990 205 jonsson et al studia math 80 2 1984 141 and chang et al j funct anal 114 1993 286 we first investigate spaces of functions in l1 whose gradients belong to hr1 or hz1 which we call hardy sobolev spaces following strichartz coll math 60 61 1 1990 129 secondly if l a is a uniformly elliptic second order divergence operator on with measurable complex coefficients subject to the dirichlet or the neumann boundary condition we compare the norms of l1 2f and f in suitable hardy spaces on depending on the boundary condition under the assumption that the heat kernel of l satisfies suitable estimates

diagonal dominance of damping and the decoupling approximation in linear vibratory systems a common approximation in the analysis of non classically damped systems is to ignore the off diagonal elements of the modal damping matrix this procedure is termed the decoupling approximation it is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant in addition the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant it is shown numerically in this paper that over a finite range errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off diagonal elements decreasing continuously in magnitude an explanation for these unexpected drifts of decoupling errors is provided with the use of complex coupling coefficients small off diagonal elements in the modal damping matrix are not sufficient to ensure small errors due to the decoupling approximation any error criterion based solely upon the diagonal dominance of the modal damping matrix would not be accurate

asymptotic expansions and fast computation of oscillatory hilbert transforms in this paper we study the asymptotics and fast computation of the one sided oscillatory hilbert transforms of the form begin aligned h f t e i omega t x int nolimits 0 infty e i omega t frac f t t x dt quad omega 0 quad x ge 0 end aligned where the bar indicates the cauchy principal value and f is a real valued function with analytic continuation in the first quadrant except possibly a branch point of algebraic type at the origin when x 0 the integral is interpreted as a hadamard finite part integral provided it is divergent asymptotic expansions in inverse powers of omega are derived for each fixed x ge 0 which clarify the large omega behavior of this transform we then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms depending on the position of x we classify our discussion into three regimes namely x mathcal o 1 or x gg 1 0 x ll 1 and x 0 numerical experiments show that the convergence of the proposed methods greatly improve when the frequency omega increases some extensions to oscillatory hilbert transforms with bessel oscillators are briefly discussed as well

compensated compactness paracommutators and hardy spaces abstract letb1 r n r n1 r m1 b2 r n r n2 r m2andq r m2 r m1be bilinear forms which are related as follows if and satisfyb1 0 andb2 0 for some 0 then q 0 supposep 1 q 1 1 coifman lions meyer and semmes proved that ifu lp r n andv lq r n and the first order systemsb1 d u 0 b2 d v 0 hold thenu qvbelongs to the hardy spaceh1 r n provided that both i p q 2 and ii the ranks of the linear mapsbj r nj r m1are constant we apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses i ii is postulated the removal of the constant rank condition whenp q 2 involves the use of a deep result of lojasiewicz from singularity theory

when not to use an automatic quadrature routine numerical quadrature is required as a component in many large scale and small scale scientific programs frequently the scientist faces a choice either he may use one of the comparatively recent automatic quadrature routines aqr or he may use a standard rule evaluation quadrature routine rear based directly on traditional formulae the purpose of this article is to help the scientist to make an informed choice in the first half of this article we introduce some generic properties of aqrs the more straightforward of these are those concerned with reliability and economy less straightforward is the vital performance profile property the second half of this article is devoted principally to demonstrating how in some calculations the jagged performance profile associated with the aqr can cause other parts of the overall program to be far more expensive than necessary thus while in many programs the use of an aqr can save significant human time in many other programs its use can be a disaster

efficient integration for a class of highly oscillatory integrals abstract this paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval one is a filon type method based on the asymptotic expansion the other is a clenshaw curtis filon type method which is based on a special hermite interpolation polynomial and can be evaluated efficiently in o n log n operations where n 1 is the number of clenshaw curtis points in the interval of integration in addition we derive the corresponding error bound in inverse powers of the frequency for the clenshaw curtis filon type method for the class of highly oscillatory integrals the efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results

pointwise behaviour of m1 1 sobolev functions our main objective is to study hajlasz type sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure we show that a discrete maximal function is bounded in the hajlasz space with the exponent one this implies that every such function has lebesgue points outside a set of capacity zero we also show that every hajlasz function coincides with a holder continuous hajlasz function outside a set of small hausdorff content our proofs are based on sobolev space estimates for maximal functions

a singularity cancellation technique for weakly singular integrals on higher order surface descriptions accurate integration of singular and near singular functions is critical to the accuracy of the method of moments solution to surface integral equations while this problem has been widely addressed for flat geometries its extensions to higher order surface descriptions have been limited this letter provides a systematic prescription for the application of the rules for weakly singular integrals on higher order surfaces here we present implementation details and several demonstrative results that compare the accuracy and convergence of the integration rules

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

an atomic decomposition of the hajlasz sobolev space m11 on manifolds abstract several possible notions of hardy sobolev spaces on a riemannian manifold with a doubling measure are considered under the assumption of a poincare inequality the space m 1 1 defined by hajlasz is identified with a hardy sobolev space defined in terms of atoms decomposition results are proved for both the homogeneous and the nonhomogeneous spaces

eigenvalues of linear viscoelastic systems the calculation of eigenvalues of single and multiple degree of freedom linear viscoelastic systems is considered the assumed viscoelastic forces depend on the past history of motion via convolution integrals over exponentially decaying kernel functions current methods to solve this type of problem normally use the state space approach involving additional internal variables such approaches often increase the size of the eigenvalue problem to be solved and can become computationally expensive for large systems here an approximate non state space based approach is proposed for this type of problem the proposed approximations are based on certain physical assumptions which simplify the underlying characteristic equation to be solved closed form approximate expressions of the complex and real eigenvalues of the system are derived these approximate expressions are obtained as functions of the elastic eigenvalues only this enables one to approximately calculate the eigenvalues of complex viscoelastic systems by simple post processing of the elastic undamped eigenvalues representative numerical examples are given to verify the accuracy of the derived expressions

a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels based on the filon clenshaw curtis method for highly oscillatory integrals and together with the sommariva s result sommariva 2013 for clenshaw curtis quadrature rule we present a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels whose unknown function is assumed to be less oscillatory than the kernel in the proposed method the filon clenshaw curtis method is used to compute the involved oscillatory integrals which makes the proposed method very precise by solving only a small system of linear equations we can obtain a very satisfactory numerical solution the performance of the presented method is illustrated by several numerical examples compared with the method proposed by li et al 2012 this method enjoys a lower computational complexity furthermore numerical examples show that the presented method has a competitive advantage on the accuracy compared with the method in li et al 2012

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

efficient quadrature of highly oscillatory integrals with algebraic singularities in this paper we are concerned with the numerical evaluation of a class of highly oscillatory integrals containing algebraic singularities first we expand such integrals derived by two transformations t x b b 0 t 21 z email protected email protected 1 into asymptotic series in inverse powers of the frequency w then based the asymptotic series two methods are presented one is the filon type method the other is the clenshaw curtis filon type method which is based on a special hermite interpolation polynomial in the clenshaw curtis points and can be evaluated efficiently in o nlogn operations where n 1 is the number of clenshaw curtis points in the interval of integration some error and convergence analysis and robust numerical examples are used to demonstrate the accuracy and effectiveness of the proposed approaches for approximating the class of highly oscillatory singular integrals

approximate decoupling of the equations of motion of damped linear systems abstract one common procedure in solving a normalized damped linear system with non zero off diagonal damping elements is to replace the normalized damping matrix by a selected diagonal matrix the extent of approximation introduced by this method of decoupling the system is evaluated and tight upper bounds on the norm of errors are derived moreover when the normalized damping matrix is diagonally dominant it is shown that decoupling the system by neglecting the off diagonal elements indeed minimizes the error upper bound the results expounded in this paper are applicable to any damped linear dynamical system the coefficient matrices of which are symmetric and positive definite

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

div curl type estimates for elliptic systems of complex vector fields abstract in this work we extend the classical div curl lemma proved by coifman lions meyer and semmes to the setup of elliptic complexes of complex vector fields with smooth coefficients the tools include local versions of the gagliardo nirenberg estimate and rellich kondrachov compactness theorem for compactly supported distributions in the local hardy space h p r n that we prove to hold for 0 p 1

computation of integrals with oscillatory and singular integrands using chebyshev expansions we present a general method for computing oscillatory integrals of the form 1 1f x g x e i w xdx where f is sufficiently smooth on 1 1 w is a positive parameter and g is a product of singular factors of algebraic or logarithmic type based on a chebyshev expansion of f and the properties of chebyshev polynomials the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor e i w x furthermore due to numerically stable recurrence relations for the modified moments the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration numerical examples are provided to confirm our analysis

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

on the numerical approximation for fourier type highly oscillatory integrals with gauss type quadrature rules in this paper we present an improved numerical steepest descent method for the approximation of fourier type highly oscillatory integrals based on the previous numerical steepest descent method the new method used the integrand information at endpoints and stationary points the asymptotic order is given that is improved both for the case of stationary points and stationary points free several numerical examples are presented which show the high efficiency of the proposed method numerical results support our theoretical analyses