on the perturbation of the reduced minimum modulus of bounded linear operators let x and y be two banach spaces and let t x y be a bounded linear operator we study the perturbation problem of the reduced minimum modulus of t and in particular we correct a mistake in a recent paper

performance limitations of non minimum phase systems in the servomechanism problem abstract this paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non minimum phase the study extends some well known properties of perfect regulation and the perfect tracking and disturbance rejection of minimum phase systems to non minimum phase systems it is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non minimum phase whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non minimum phase system and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane furthermore this limitation provides a quantitative measure of the degree of difficulty which is inherent in the control of such non minimum phase systems

the expression of the generalized inverse of the perturbed operator under type i perturbation in hilbert spaces let h1 h2 be two hilbert spaces over the complex field c and let t h1 h2 be a bounded linear operator with the generalized inverse t let t t t be a bounded linear operator with t t 1 suppose that dim ker t dim ker t or r t r t 0 then t has the generalized inverse t i i t t 1 i t t i t i t t 1 1 1 tt 1t x i tt tt tt i tt 1 i tt ast 1tt i tt i 1 with t t 1 t t this result gives an analogue of theorem 3 9 of m z nashed generalized inverses and applications academic press new york 1976 in hilbert spaces

efficient modeling of localized material failure by means of a variationally consistent embedded strong discontinuity approach this paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite strains focus is on localized plastic deformation particularly relevant for slip bands in contrast to already existing models the proposed implementation allows to consider several interacting discontinuities in each finite element based on a proper re formulation of the kinematics an efficient parameterization of the deformation gradient is derived it permits to compute the strains explicitly that improves the performance significantly however the most important novel contribution of the present paper is the advocated variational constitutive update within this framework every aspect is naturally driven by energy minimization i e all unknown variables are jointly computed by minimizing the stress power the proposed update relies strongly on an extended principle of maximum dissipation this framework provides enough flexibility for different failure types and for a broad class of non associative evolution equations by discretizing the aforementioned continuous variational principle an efficient numerical implementation is obtained it shows in addition to its physical and mathematical elegance several practical advantages for instance the physical minimization principle itself specifies automatically and naturally the set of active strong discontinuities copyright 2011 john wiley sons ltd

some equivalent conditions of stable perturbation of operators in hilbert spaces let h 1 h 2 be two hilbert spaces over the complex field c and let t email protected h 1 h 2 be two bounded linear operators with the m p generalized inverse t if r t r t 0 we say that t is the stable perturbation of t in this paper we give five equivalent conditions that make t being the stable perturbation of t under the assumption t email protected email protected email protected 1 these equivalent conditions generalize not only the notation of rank preserving perturbations of matrices but also the notation of acute perturbations of matrices as a result we obtain the following suppose that r t r t 0 and t email protected email protected email protected 1 email protected t t email protected email protected t email protected email protected t email protected email protected email protected this result generalizes corresponding result when h 1 h 2 are all finite dimensional

geometrically non linear and consistently linearized embedded strong discontinuity models for 3d problems with an application to the dissection analysis of soft biological tissues three different finite element formulations with embedded strong discontinuities are derived on the basis of the enhanced assumed strain method according to the work by jirasek and zimmermann int j numer methods engrg 50 2001 1269 they are referred to as statically optimal symmetric sos kinematically optimal symmetric kos and statically and kinematically optimal non symmetric skon formulations the effect of the discontinuities are characterized by additional degrees of freedom on the element level modifications to the standard kos and skon formulations are proposed in order to achieve consistency with the employed type of a three field hu washizu principle under mode i condition under this condition the formulation satisfies the internal compatibility at the discontinuity i e the relation between the stress in the bulk material and the traction across the discontinuity surface which is not the case for the classical kos formulation we propose a suitable explicit expression for a transversely isotropic traction law in form of a displacement energy function and assume that softening phenomena in the cohesive zone are modeled by a damage law which depends on the maximum gap displacement of the deformation path a linearization of all quantities which are related to the non linear problem leads to new closed form expressions in particular we focus attention on the linearization of the cohesive traction vector the associated element residua and stiffness matrices are provided standard static condensation of the internal degree of freedom leads to a generalized displacement model a comparative study of the modified formulations carried out by means of two numerical examples show the performance of the individual approach we employ constant strain tetrahedral elements with a single discontinuity embedded among the known stress locking phenomena associated with the sos formulation we recognized that the non symmetric skon formulation was not able to provide meaningful results for the dissection process of an arterial layer in three dimensions on distorted meshes for both numerical examples the symmetric kos formulation seems to be most suitable for representing the embedded discontinuities

on performance limitation in tracking a sinusoid this note studies the performance limitation of a feedback system with a given linear time invariant lti plant in tracking a sinusoidal signal it continues and goes beyond some recent studies in the same topic in which it is assumed that the controller can access all the past and future values of the reference signal in this note we consider the more realistic and more difficult situation where the controller only accesses the current and past values of the reference an explicit formula of the best attainable performance is obtained for a single input single output siso system which depends on the nonminimum phase zeros of the plant and the frequency of the reference sinusoid compared to the previously studied case when the future of the reference is available this formula clearly shows the extra effort one has to pay due to the lack of the reference information a partial result for a multiple input multiple output mimo system is also given

on stable perturbations for outer inverses of linear operators in banach spaces abstract in this paper we investigate stable perturbations and their characterizations for various types of outer inverses such as generalized 2 3 2 4 2 5 1 2 3 1 2 4 moore penrose group drazin and generalized drazin inverses some known results are improved and extended

a quasi static discontinuous galerkin configurational force crack propagation method for brittle materials summary this paper presents a framework for r adaptive quasi static configurational force cf brittle crack propagation cast within a discontinuous galerkin dg symmetric interior penalty sipg finite element scheme cracks are propagated in discrete steps with a staggered algorithm along element interfaces which align themselves with the predicted crack propagation direction the key novelty of the work is the exploitation of the dg face stiffness terms existing along element interfaces to propagate a crack in a mesh independent r adaptive quasi static fashion driven by the cf at the crack tip this adds no new degrees of freedom to the data structure additionally as dg methods have element specific degrees of freedom a geometry driven p adaptive algorithm is also easily included allowing for more accurate solutions of the cf on a moving crack front further for non determinant systems we introduce an average boundary condition that restrains rigid body motion leading to a determinant system to the authors knowledge this is first time such a boundary condition has been described the proposed formulation is validated against single and multiple crack problems with single and mixed mode cracks demonstrating the predictive capabilities of the method this article is protected by copyright all rights reserved

application of material forces to hyperelastostatic fracture mechanics i continuum mechanical setting the concern of this work is a consequent exploitation of the notion of material forces for the application within hyperelastostatic fracture mechanics contrary to physical forces material forces act on the material manifold thus essentially representing the tendency of defects like cracks or inclusions to move relative to the ambient material based on the formulation of the appropriate quasi static balance laws in the material space we aim at a fresh look onto classical aspects of hyperelastostatic fracture mechanics operating throughout within the geometrically nonlinear setting we emphasize on the one hand the duality of the direct and the inverse motion description and on the other hand we re establish the classical path integrals from elementary equilibrium considerations in the material space

finite dimensional approximation settings for infinite dimensional moore penrose inverses after a brief survey of basic results about finite dimensional approximation settings such as mathematical frameworks of various projection methods including the least squares method the dual least squares method and the galerkin method for infinite dimensional moore penrose inverses this paper proceeds to a detailed study from the following aspects for projection methods we investigate convergence and weak convergence of their approximation setting to develop a unified theory on projection methods for infinite dimensional moore penrose inverses this investigation yields a fundamental convergence theorem theorem 2 2 from which the criterion for convergence the criterion for weak convergence and the generalized dual least squares method are derived we also derive general results on the least squares method by which two flaws in groestch s results are corrected for nonprojection methods whose approximation setting is a more general framework we investigate weak perfect convergence of their approximation setting and provide a necessary and sufficient condition of such convergence holding theorem 3 2 several examples are proposed as counterexamples to illustrate the differences between some important concepts or as concrete algorithms to show how the present work can help to analyze their behavior

perturbation theory for the generalized bott duffin inverse and its applications a perturbation theory for the bott duffin inverse a l of a with respect to a subspace l is developed the perturbation bound for the solution of the constrained system ax y b email protected l email protected l is established where email protected m n c a subspace email protected c n and email protected c n meanwhile this paper shows the generalized bott duffin condition number k g b d a email protected email protected email protected a l to be minimum in the inequality of error analysis

fundamental performance limitations in tracking sinusoidal signals this paper attempts to give a thorough treatment of the performance limitation of a linear time invariant multivariable system in tracking a reference signal which is a linear combination of a step signal and several sinusoids with different frequencies the tracking performance is measured by an integral square error between the output of the plant and the reference signal our purpose is to find the fundamental limit for the attainable tracking performance under any control structure and parameters in terms of the characteristics and structural parameters of the given plant as well as those of the reference signal under consideration it is shown that this fundamental limit depends on the interaction between the reference signal and the nonminimum phase zeros of the plant and their frequency dependent directional information

performance limitation of tracking control problem for a class of references this technical note deals with analysis of fundamental limitations for siso systems in tracking control problems in the existing results the fundamental limitations are analyzed based on detailed assumptions of reference signals such as the step the trigonometrical signals and so on on the other hand we define a class of reference signals in an abstract but general manner for the general class of reference inputs we give the analysis of the tracking performance limitations based on the achievable set of the outputs the analysis results can separate the contributions of the plant and the reference signal clearly

general solution of robust tracking problem in two degree of freedom control systems in this note we study linear time invariant multivariable systems with two degree of freedom compensators first we give a new parametrization of all stabilizing compensators from the viewpoint of independent design of transfer function and robustness property second utilizing this parametrization we solve a general robust tracking problem including the case of nonsquare plants and show that tracking and robustness can be independently specified in two degree of freedom control systems

a computational framework of three dimensional configurational force driven brittle crack propagation abstract we consider a variational formulation of quasi static brittle fracture and develop a new finite element based computational framework for propagation of cracks in three dimensional bodies we outline a consistent thermodynamical framework for crack propagation in elastic solids and show that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global clausius planck inequality in the sense of coleman s method consequently the crack propagation direction associated with the classical griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front the variational formulation is realized numerically by a standard spatial discretization with finite elements which yields a discrete formulation of the global dissipation in terms configurational nodal forces therefore the constitutive setting of crack propagation in the space discretized finite element context is naturally related to discrete nodes of a typical finite element mesh in a consistent way with the node based setting the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface facets of the mesh the crucial step for the success of this procedure is its embedding into an r adaptive crack facet reorientation procedure based on configurational force based indicators in conjunction with crack front constraints we propose a staggered solution procedure that results in a sequence of positive definite discrete subproblems with successively decreasing overall stiffness providing a robust algorithmic setting in the postcritical range the predictive capabilitiy of the proposed formulation is demonstrated by means of representative numerical simulations

on configurational forces in the context of the finite element method the theory of configurational forces is briefly recast together with the underlying balance laws it is shown that in the case of a homogeneous body without body forces the additional balance laws are identically satisfied if the standard force balance holds in approximate solutions for example obtained by finite elements the equilibrium is not satisfied exactly thus configurational forces occur on discretization nodes an implementation of the configurational force balance into the finite element scheme is presented the use of configurational forces is discussed with three main aspects it is demonstrated how configurational forces can be used to check and to improve the finite element solution examples from fracture mechanics and problems with material inhomogeneities are discussed copyright 2001 john wiley sons ltd

the expression of the generalized bott duffin inverse and its perturbation theory let l be a subspace of c n and p l be the orthogonal projection of c n onto l then for the nxn matrix a the generalized bott duffin b d inverse a l is given by a l p l ap l i p l in this paper we will give an expression of a l in the rest of the part the perturbation analysis of a l and the least squares solution of the generalized b d equation ax y b for x l and y l will be established under certain small perturbation of p lap l

new perturbation results on pseudo inverses of linear operators in banach spaces abstract let x and y be banach spaces let t x y be a bounded linear operator with closed range and let s x y be another bounded linear operator we study conditions on s t that guarantee the closeness of the range of s and obtain some new bounds on the pseudo inverse of s

on the continuity of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range we study the continuity problem of the generalized inverse of t and related least squares solutions to the operator equation tx y

on the perturbation of the least squares solutions in hilbert spaces abstract let h1 and h2 be hilbert spaces and let t h1 h2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

on perturbations for oblique projection generalized inverses of closed linear operators in banach spaces the main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces we provide a new stability characterization of oblique projection generalized inverses of closed linear operators under t bounded perturbations which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space

application of material forces to hyperelastostatic fracture mechanics ii computational setting the concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics to this end we consider the continuum mechanics of material forces as outlined in part i of this work p steinmann int j solid struct 37 7371 7391 which act contrary to the common physical forces on the material manifold or rather in the material space in the sequel it is proposed to discretize the corresponding quasi static balance of pseudo momentum by a standard galerkin finite element procedure as a result we obtain global discrete node point quantities the material node point forces which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical j integral

perturbation of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range in this paper we present some new results of the perturbation analysis on the generalized inverse of t and the related least squares solution to the operator equation tx y

the characterizations of the stable perturbation of a closed operator by a linear operator in banach spaces in this paper we investigate the invertibility of iy tt when t is a closed operator from x to y with a generalized inverse t and t is a linear operator whose domain contains d t and range is contained in d t the characterizations of the stable perturbation t t of t by t in banach spaces are obtained the results extend the main results in 7 8

invariances of frame sequences under perturbations this paper determines the exact relationships that hold among the major paley wiener perturbation theorems for frame sequences it is shown that major properties of a frame sequence such as excess deficit and rank remain invariant under paley wiener perturbations but need not be preserved by compact perturbations for localized frames which are frames with additional structure it is shown that the frame measure function is also preserved by paley wiener perturbations

perturbation for a pair of oblique projectors aa mn and bb mn abstract by using the method of generalized singular values decomposition we establish a canonical theorem for a pair of oblique projectors aa mn and bb mn give some new results for perturbations of aa mn and bb mn where a mn is the weighted moore penrose inverse of matrix a m and n are two hermitian positive definite matrices and b is a matrix obtained from a perturbation of a our work can be seen as the extended results for orthogonal projectors aa and bb some numerical examples are given to illustrate the results

perturbation analysis of generalized inverses of linear operators in banach spaces abstract let x 1 x 2 be two banach spaces and t x 1 x 2 be a bounded linear operator with a bounded generalized inverse t we study the perturbation problem of generalized inverse t and provide two new stability characterizations of generalized inverse we also give a continuity characterization of moore penrose inverse in hilbert spaces our results are new even in the case of matrices

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

perturbation of least squares problem in hilbert spaces let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

perturbation analysis of the least squares solution in hilbert spaces abstract let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we give a complete description of the perturbation analysis for the least squares solution to the operator equation t x y where x h 1 y h 2

on the error estimate for the projection of a point onto a linear manifold abstract suppose that the linear system a h x b is consistent the problem of finding the solution x to min a h x b 1 2 x p 2 2 is studied where a c n m b c m x p c n and a b p are given when a b and p are perturbed an error bound for the solution x is obtained roundoff error bounds for the singular value decomposition method the column pivoting method and the rank revealing method are obtained the results in this paper extend those of arioli laratta and menchi to the case when a is not of full rank and either m n or m n

theory and numerics for finite deformation fracture modelling using strong discontinuities a general finite element approach for the modelling of fracture is presented for the geometrically non linear case the kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements thus the deformation map is defined in terms of one continuous and one discontinuous portion considered as mutually independent giving rise to a weak formulation of the equilibrium consisting of two coupled equations in addition two different fracture criteria are considered firstly a principle stress criterion in terms of the material mandel stress in conjunction with a material cohesive zone law relating the cohesive mandel traction to a material displacement jump associated with the direct discontinuity secondly a criterion of griffith type is formulated in terms of the material crack driving force mcdf with the crack propagation direction determined by the direction of the force corresponding to the direction of maximum energy release apart from the material modelling the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria copyright 2005 john wiley sons ltd

on the semi continuity of generalized inverses in banach algebras abstract the main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a banach algebra with identity we first obtain a sufficient and necessary condition for the lower semi continuity of reflexive generalized inverses as set valued mappings based on this result we characterize the continuity of the moore penrose inverse in a c algebra and therefore derive some new and well known criteria in operator theory

a generalization of a classic theorem in the perturbation theory for linear operators let x and y be banach spaces and let t x y be a bounded linear operator with closed range in this paper we extend a classic perturbation result from the invertible case to the general case for the consistent operator equation tx y

perturbation analysis for the operator equationtx bin banach spaces letx1 x2be two banach spaces over the complex fieldcand lett x1 x2be a bounded linear operator with the generalized inverset lett t tbe a bounded linear operator with t t 1 suppose thatdimkert dimkert orr t kert 0 thenthas the generalized inverset i t t 1t with t t 1 t t this result generalizes theorem 3 9 ofna using this result we give some results of the perturbation analysis for the operator equationtx b

a framework for fracture modelling based on the material forces concept with xfem kinematics a theoretical and computational framework which covers both linear and non linear fracture behaviour is presented as a basis for the formulation we use the material forces concept due to the close relation between on one hand the eshelby energy momentum tensor and on the other hand material defects like cracks and material inhomogeneities by separating the discontinuous displacement from the continuous counterpart in line with the extended finite element method xfem we are able to formulate the weak equilibrium in two coupled problems representing the total deformation however in contrast to standard xfem where the direct motion discontinuity is used to model the crack we rather formulate an inverse motion discontinuity to model crack development the resulting formulation thus couples the continuous direct motion to the inverse discontinuous motion which may be used to simulate linear as well as non linear fracture in one and the same formulation in fact the linear fracture formulation can be retrieved from the non linear cohesive zone formulation simply by confining the cohesive zone to the crack tip these features are clarified in the two numerical examples which conclude the paper copyright 2005 john wiley sons ltd

perturbation and expression for inner inverses in banach spaces and its applications abstract it is well known that if t is invertible and t 1 is its inverse then t 1 i t t t 1 1 is the inverse of t for t t t 1 1 the main purpose of this paper is to study the following problem if t is an inner inverse of t and t t t 1 is t i t t t 1 an inner inverse of t if the answer is no when is t i t t t 1 an inner inverse of t in this paper we give a complete answer to these problems and provide some characterizations for t i t t t 1 to be an inner inverse of t in banach spaces utilizing these results we investigate similar problems for the generalized inverse 1 3 inverse 1 4 inverse 1 5 inverse 1 2 3 inverse 1 2 4 inverse moore penrose inverse group inverse and drazin inverse the results obtained in this paper extend and improve many recent results in this area

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

perturbation analysis for the reduced minimum modulus of bounded linear operator in banach spaces let x and y be two banach spaces over the complex field c and let t x y be a bounded linear operator with the generalized inverse t let t t dt be a bounded linear operator with t dt 1 2 let c t denote the reduced minimum modulus of t we first establish the upper semi continuity theorem for c t furthermore if dimkert dimkert or r t kert 0 then c t is continuous under the operator norm also we have a corollary in hilbert spaces which improves the result in linear algebra appl 262 1997 229 242 theorem 4 1

a new fast finite element method for dislocations based on interior discontinuities a new technique for the modelling of multiple dislocations based on introducing interior discontinuities is presented in contrast to existing methods the superposition of infinite domain solutions is avoided interior discontinuities are specified on the dislocation slip surfaces and the resulting boundary value problem is solved by a finite element method the accuracy of the proposed method is verified and its efficiency for multi dislocation problems is illustrated bounded core energies are incorporated into the method through regularization of the discontinuities at their edges though the method is applied to edge dislocations here its extension to other types of dislocations is straightforward copyright 2006 john wiley sons ltd

perturbation analysis for oblique projection generalized inverses of closed linear operators in banach spaces in this paper we study the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces by the method of the perturbation analysis of linear operators we obtain an explicit perturbation theorem and error estimates for the oblique projection generalized inverse of closed linear operators under the t bounded perturbation which extend the known results on the perturbation of the oblique projection generalized inverse of bounded linear operators in banach spaces

the perturbation of consistent least squares problems abstract suppose that the linear system ax b is consistent and a a a b b b are perturbed from a b respectively where a and are m n matrices b and b are m dimensional vectors let a f e a 2 b 2 e b 2 and a 2 a 2 be the condition number of a it is shown that if e 1 then the least squares solution x of a x b is close to a solution x of ax b even when a is not of full rank and rank rank a

some new perturbation theorems for generalized inverses of linear operators in banach spaces abstract et x y be banach spaces and let t b x y be a linear operator with closed range r t we consider the perturbation problem for the bounded outer inverse the bounded 2 3 inverse and the bounded generalized inverse of t with respect to the projectors some new perturbation theorems are given by the generalized neumman lemma and the concept of stable perturbations in banach spaces