unconditional schauder decompositions and stopping times in the lebesgue bochner spaces we extend troitsky s study of martingales in banach lattices to include stopping times results from the theory of unconditional schauder decompositions and multipliers are used to derive an optional stopping theorem for unbounded stopping times we also apply these techniques to convergent nets of stopped processes as well as to unconditional schauder decompositions in vector valued l p spaces 1 p

natural frequencies of structures with interval parameters abstract this paper deals with the evaluation of the lower and upper bounds of the natural frequencies of structures with uncertain but bounded parameters the solution of the generalized interval eigenvalue problem is pursued by taking into account the actual variability and dependencies of uncertain structural parameters affecting the mass and stiffness matrices to this aim interval uncertainties are handled by applying the improved interval analysis via extra unitary interval eui recently introduced by the first two authors by associating an eui to each uncertain but bounded parameter the cases of mass and stiffness matrices affected by fully disjoint completely or partially coincident uncertainties are considered then based on sensitivity analysis it is shown that the bounds of the interval eigenvalues can be evaluated as solution of two appropriate deterministic eigenvalue problems without requiring any combinatorial procedure if the eigenvalues are monotonic functions of the uncertain parameters then the exact bounds are obtained the accuracy of the proposed method is demonstrated by numerical results concerning truss and beam structures with material and or geometrical uncertainties

new sufficient conditions for robust stability analysis of interval matrices abstract this letter presents new sufficient conditions for robust hurwitz stability of interval matrices the proposed conditions are based on two approaches i finding a common lyapunov matrix for the interval family and ii converting the robust stability problem into a robust non singularity problem using kronecker operations the main contribution of the letter is to derive accurate and computationally simple optimal estimates of the robustness margin and spectral bound of general interval matrices the evaluation of the condition relies on the solutions of linear matrix inequalities lmis and eigenvalue problems both of which are solved very efficiently the improvements gained by using the proposed conditions are demonstrated through application to previous examples in the literature

interval analysis of vibrating systems interval calculus is a tool to evaluate a mathematical expression for ranges of values of its parameters the basic mathematical operations are defined in the interval algebra vibrating systems having system parameters initial conditions or forcing functions defined as intervals rather than single valued quantities have been modelled using interval calculus the eigenvalue problem for determining natural frequencies and vibrating modes of a system in the case of system matrix elements given as a range of values cannot be solved by exhaustion due to the prohibitively large number of solutions of the point number eigenvalue problem eigenvalue solution with interval evaluation of the commonly used numerical techniques is not feasible because if they are applied directly the solution intervals diverge an optimization technique was used to obtain the minimum radius intervals of the solution for the eigenvalue sensitivity problem to assure monotonicity and absolute inclusion necessary for convergence to the exact interval a converging interval halving sequence was developed for finite width interval matrices for numerical tests of the method a monte carlo solution was developed the results showed that interval analysis can predict the range of the eigenvalues with sufficient accuracy the response of a linear system to general excitation for interval matrices of the system parameters cannot be found with interval evaluation of the commonly used numerical techniques because if they are applied directly the solution intervals diverge interval modal analysis and the interval solution of the eigenvalue problem were developed an application is presented for the dynamic response of a rotor with interval bearing properties

quadratic variation of martingales in riesz spaces national research foundation of south africa grant specific unique reference number uid 85672 and by gnampa of italy u 2012 000574 20 07 2012 and u 2012 000388 09 05 2012

the generalized eigenvalue problem for tridiagonal symmetric interval matrices an algorithm to compute the eigenintervals of the generalized eigenvalue problem for tridiagonal symmetric matrices a b with interval entries is developed the algorithm is based on the sturm sequences and requires twice as much computational effort as for point matrices the algorithm is applied to the problem of computing the natural frequencies of a vibrating beam where the manufacturing tolerance is taken into consideration

sufficient regularity conditions for complex interval matrices and approximations of eigenvalues sets abstract in this paper two approaches are described to establish verifiable sufficient regularity conditions of complex interval matrices in the first approach a complex interval matrix is mapped to a real block interval matrix and then its sufficient regularity conditions are obtained in the second approach a necessary condition for the singularity of a complex interval matrix is derived and used to get its sufficient regularity conditions as an application the above derived sufficient regularity conditions are used to investigate the location of the outer approximations of individual eigenvalue sets of complex interval matrices two algorithms are proposed and results obtained are compared with those obtained by earlier methods and monte carlo simulations the advantages of these algorithms are that they can detect gaps in between the approximations of the whole eigenvalue sets the second algorithm is very effective compared to the first algorithm from the computational time point of view several numerical examples and statistical experiments are worked out to validate and demonstrate the efficacy of our work

characterizing and approximating eigenvalue sets of symmetric interval matrices we consider the eigenvalue problem for the case where the input matrix is symmetric and its entries are perturbed with perturbations belonging to some given intervals we present a characterization of some of the exact boundary points which allows us to introduce an inner approximation algorithm that in many case estimates exact bounds to our knowledge this is the first algorithm that is able to guarantee exactness we illustrate our approach by several examples and numerical experiments

computing bounds to real eigenvalues of real interval matrices in this study a new method with algorithms for computing bounds to real eigenvalues of real interval matrices is developed the algorithms are based on the properties of continuous functions the method can provide the tightest eigenvalue bounds and improve some former research results numerical examples illustrate the applicability and effectiveness of the new method copyright 2007 john wiley sons ltd

a filtering method for the interval eigenvalue problem we consider the general problem of computing intervals that contain the real eigenvalues of interval matrices given an outer approximation superset of the real eigenvalue set of an interval matrix we propose a filtering method that iteratively improves the approximation even though our method is based on a sufficient regularity condition it is very efficient in practice and our experimental results suggest that it improves in general significantly the initial outer approximation the proposed method works for general as well as for symmetric interval matrices

a unified framework for deterministic and probabilistic mathscr d stability analysis of uncertain polynomial matrices in control theory we are often interested in robust mathscr d stability analysis which aims at verifying if all the eigenvalues of an uncertain matrix lie in a given region mathscr d although many algorithms have been developed to provide conditions for an uncertain matrix to be robustly mathscr d stable the problem of computing the probability of an uncertain matrix to be mathscr d stable is still unexplored the goal of this paper is to fill this gap in two directions first the only constraint on the stability region mathscr d that we impose is that its complement is a semialgebraic set this comprises many important cases in robust control theory second the mathscr d stability analysis problem is formulated in a probabilistic framework by assuming that only few probabilistic information is available on the uncertain parameters such as support and some moments we will show how to compute the minimum probability that the matrix is mathscr d stable by using convex relaxations based on the theory of moments

ito s rule and levy s theorem in vector lattices abstract the change of variable formula or ito s rule is studied in a dedekind complete vector lattice e with weak order unit e using the functional calculus we prove that for a holder continuous semimartingale x t x a m t b t t j and a twice continuously differentiable function f the formula 0 1 f x t f x a 0 t f x s d m s 0 t f x s d b s 1 2 0 t f x s d m s 0 s t j holds the first integral in the formula is an ito integral with reference to the local martingale m and the second and third integrals are dobrakov type integrals of a vector valued function with reference to a vector valued measure using the formula we prove levy s characterization of brownian motion as being a continuous martingale with compensator te the proof of this result yields a concrete description of abstract brownian motion defined in vector lattices

the interval eigenvalue problem using constraint interval analysis with an application to linear differential equations abstract we study the interval eigenvalue problem and its application to stability theory of interval linear differential equations the interval eigenvalue problem is studied from a constraint interval representation affording advantages not found in previous approaches

the extreme eigenvalues and stability of real symmetric interval matrices a novel khoritonov like algorithm for computing the minimal and maximal eigenvalues of n n dimensional symmetric interval matrices is presented it is proved that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2 sup n 1 symmetric vertex matrices whereas its minimal eigenvalue coincides with the minimal of another special set of 2 sup n 1 symmetric vertex matrices as immediate corollaries of this algorithm weak necessary and sufficient conditions for testing the hurwitz and schur stability of symmetric interval matrices where one has to test the stability of 2 sup n 1 and 2 sup n symmetric vertex matrices respectively are obtained

unconditional martingale difference sequences in banach spaces abstract for a banach space y the question of whether l p y has an unconditional basis if 1 p and y has unconditional basis stood unsolved for a long time and was answered in the negative by aldous in this work we prove a weaker positive result related to this question we show that if y j is a basis of y and d i is a martingale difference sequence spanning l p then the sequence d i y j is a basis of l p y for 1 p moreover if 1 p and y j is unconditional then d i y j is strictly dominated by an unconditional tensor product basis in addition for 1 p we show that if d i l p is a martingale difference sequence then there exists a constant k 0 so that i j n i j y j d i l p y k i j n i j y j d i l p holds for every sequence y j y and every choice of finitely supported scalars i j

an algorithm for checking stability of symmetric interval matrices a branch and bound algorithm for checking hurwitz and schur stability of symmetric interval matrices is proposed the algorithm in a finite number of steps either verifies stability or finds a symmetric matrix which is not stable it can also be used for checking positive definiteness of asymmetric interval matrices

robust control for linear systems with structured state space uncertainty abstract a recent sufficient condition for the stability robustness of linear systems with time varying norm bounded state space uncertainty is extended to include the structure of the uncertainty our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed loop eigenvector matrix therefore this robustness result is especially well suited to the design of control systems using eigenstructure assignment when the uncertainty is time invariant our norm is also an upper bound on the incremental eigenvalue perturbations we also consider the use of perron weightings to reduce conservatism and the extension of the results to discrete time systems

exact maximum singular value calculation of an interval matrix in this note we present a method for calculating the maximum singular value of an interval matrix first we provide an algorithm for calculating the maximum singular value of a square interval matrix then based on this algorithm we extend the result to non square interval matrix case and to the case of computing the minimum singular value through numerical examples the validity of the suggested methods is illustrated particularly we compare the newly proposed method with an existing method to show that the new method finds the correct bound of the maximum singular value with no exception

bounds on eigenvalues of real and complex interval matrices we present a cheap and tight formula for bounding real and imaginary parts of eigenvalues of real or complex interval matrices it outperforms the classical formulae not only for the complex case but also for the real case in particular it generalizes and improves the results by rohn 1998 5 and hertz 2009 19 the main idea behind is to reduce the problem to enclosing eigenvalues of symmetric interval matrices for which diverse methods can be utilized the result helps in analysing stability of uncertain dynamical systems since the formula gives sufficient conditions for testing schur and hurwitz stability of interval matrices it may also serve as a starting point for some iteration methods

stability of a hurwitz polynomial under coefficient perturbations necessary and sufficient conditions necessary and sufficient conditions for a perturbed polynomial to remain hurwitz are given the conditions allow considerable freedom in allocating different weights to various coefficients to reflect different levels of uncertainty in the coefficients the new conditions are an extension of a previous result of the author in which sufficient conditions for the same problem were obtained

sufficient conditions on stability of interval matrices connections and new results the stability properties of interval matrices are discussed sufficient conditions are given for these properties based upon gershgorin s theorem and its extension and they are shown to improve previous results the derivation helps clarify relations between certain previously available results and shows they can be derived using gershgorin s theorem furthermore the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular m matrices

necessary and sufficient conditions for the output feedback regularization of descriptor systems the problem of output feedback regularization of descriptor systems is studied necessary and sufficient conditions are given to regulate the descriptor systems by derivative and proportional output feedback into a closed loop regular system which is of index at most one

hybrid uncertain natural frequency analysis for structures with random and interval fields abstract this paper presents a robust non deterministic free vibration analysis for engineering structures involving hybrid yet spatially dependent uncertain system parameters distinguished from the conventional hybrid uncertain eigenvalue problem the concept of interval field is enclosed with random field model such that both the stochastic and non stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non deterministic free vibration analysis in order to determine the probabilistic characteristics i e means and standard deviations of the extremities of structural natural frequencies an extended unified interval stochastic sampling x uiss method is implemented for the purpose of effective hybrid uncertain free vibration analysis by meticulously blending sharpness promised interval eigenvalue analysis with stochastic sampling techniques the stochastic profiles i e probability density functions pdfs and the cumulative distribution functions cdfs of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods the applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures

fuzzy finite element method for frequency response function analysis of uncertain structures a concept is presented for incorporating fuzzy uncertainties in dynamic e nite element analyses of uncertain structures the objective is twofold the e rst goal is to clarify and extend the classical fuzzy e nite element ffe method as it was introduced for static analyses the shortcomings of the classical approach are described and an extension to a generalized approach is proposed this generalized approach is proven to be a more realistic and therefore more reliable concept for taking uncertainty into account the second goal is to illustrate the applicability of the method for dynamic analyses the classical and the generalized approach are compared using an eigenvalue analysis of a simple numerical example the ffe method is also applied to the calculation of the total envelope frequency response function frf using the modal superposition principle this method requires safe approximations of the individual mode envelope frequency response functions for this purpose a number of safe approximate optimization strategies are introduced the numerical example shows that useful results are obtained using this ffe approach for frf calculations

positive definiteness of hermitian interval matrices abstract we present a new necessary and sufficient criterion to check the positive definiteness of hermitian interval matrices it is shown that an n n hermitian interval matrix is positive definite if and only if its 4 n 1 n 1 specially chosen hermitian vertex matrices are positive definite

technical communique maximal perturbation bounds for robust α stability of matrix second order systems with one parameter perturbations in this paper the robust a stability problem of matrix second order systems with perturbations in the form of a one parameter family of matrices is investigated all the system matrices including the second order differential coefficient matrices are assumed to have such perturbations based on the kronecker product a necessary and sufficient condition for the robust a stability problem is presented by transforming such a problem into checking the nonsingularity of a class of uncertain matrices then a closed form for the maximal perturbation bounds for preserving the a stability is given finally illustrative examples are given to show that our results are effective and less conservative than the results obtained by other researchers

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

regularity radius and real eigenvalue range abstract let a p be a given n n interval parametric matrix the set of all n n parametric matrices when the m dimensional parameter vector p varies within a given interval vector p whose entries depend affine linearly on p also let l denote the set of all real eigenvalues of the bundle a p b p in this paper first the concept of regularity radius r a p of the interval parametric matrix a p is introduced it is then shown that there exists certain relationship between the problem of establishing if r belongs to l or not and the numerical value of the regularity radius of the interval parametric matrix a p b p the results presented may be useful in designing a method for determining or assessing the set l

an evolution strategy method for computing eigenvalue bounds of interval matrices in this paper a modified method by means of an evolution strategy es is introduced for computing eigenvalue bounds of interval matrices then a sufficient condition theorem of the evolutionary strategy is presented which guarantees the convergence in probability of the method the numerical examples show the method can effectively yield accurate bounds for interval eigenvalues

stability analysis of dynamic interval systems abstract the stability analysis of interval matrices with certain stability margins is considered by means of similarity transformation and gershgorin s theorem new criteria are developed and illustrated by examples we also demonstrate a counter example to a recent result by argoun 1986

the ito integral for martingales in vector lattices abstract in the setting of a dedekind complete riesz space on which a conditional expectation is defined we study the ito integral this integral obtains the integration of a stochastic process of a continuous parameter relative to a martingale a considerable part of the paper is devoted to aspects of integration of a vector valued function with respect to a vector measure here we use the dobrakov integral that is in our case a generalization of the well known bartle integral we discuss natural and predictable processes to prove a general version of the doob meyer decomposition of a submartingale this decomposition provides an essential tool used in the definition of the stochastic integral we derive the properties of the stochastic integral that are useful in developing the theory of stochastic processes in riesz spaces

a banach lattice approach to convergent integrably bounded set valued martingales and their positive parts denote by cf x the set of all nonempty convex closed subsets of a separable banach space x let be a complete probability space and denote by l1 cf x the complete metric space of equivalence classes of a e equal integrably bounded cf x valued functions for any preassigned filtration i we describe the space of convergent integrably bounded cf x valued martingales in terms of the closure of i 1 l1 i cf x in l1 cf x in particular we provide a formula to calculate the join of two such martingales and the positive part of such a martingale our object is achieved by considering the more general setting of a near vector lattice s d endowed with a riesz metric d by means of radstrom s embedding theorem for such spaces a link is established between the space of convergent martingales in s and the space of convergent martingales in the radstrom completion r s of s this link provides information about the former space of martingales via known properties of measure free martingales in riesz normed vector lattices applicable to r s we also apply our general results to the spaces of convergent ck x valued martingales where ck x denotes the set of all nonempty convex compact subsets of x

peeling off a nonconvex cover of an actual convex problem hidden convexity convexity is without a doubt one of the most desirable features in optimization many optimization problems that are nonconvex in their original settings may become convex after performing certain equivalent transformations this paper studies the conditions for such hidden convexity more specifically some transformation independent sufficient conditions have been derived for identifying hidden convexity the derived sufficient conditions are readily verifiable for quadratic optimization problems the global minimizer of a hidden convex programming problem can be identified using a local search algorithm

novel estimations for the eigenvalue bounds of complex interval matrices our work proposes two methods that estimate the eigenvalue bounds left right for real and imaginary parts of complex interval matrices the first method expresses each bound as an algebraic sum of weighted matrix measures where the measure corresponds to the spectral norm and the weighting matrix is diagonal and positive definite with unknown entries the optimization with respect to the entries of the weighting matrix yields the best value of the bound the computational approach is ensured as the minimization of a linear objective function subject to bilinear matrix inequality constraints and interval constraints the bounds are proved to be better than those provided by other estimation techniques the second method constructs four real matrices so that each of them can be exploited as a comparison matrix for the complex interval matrix and allows the estimation of one of the eigenvalue bounds the two methods we propose rely on different mathematical backgrounds and between the resulting bounds no firm inequality can be stated therefore they are equally useful in applications as reflected by the numerical case studies presented in the paper

an interval algorithm for uncertain dynamic stability analysis abstract analysis on the stability of dynamic systems is a very important field in structural elasticity theory for possible engineering situations where the system parameters are uncertain but bounded the propagation from the initial uncertainties of parameters to the consequent uncertainty of dynamic stability of structure is elaborately investigated by interval analysis for the first time the interval dynamic stability issue is studied through the column structure as a generalized interval eigenvalue problem and an analytic theorem is rigorously derived and numerically verified for solving the generalized interval eigenvalue problem moreover to avoid the exaggeration on actual bounds of solution an effective approach is established to limit the dependency phenomenon in the interval dynamic stability analysis the interval relationships of the excitation parameter and the critical load frequency with five types of system parameters are further bridged according to the interval arithmetic to propose a three stage interval scheme for evaluating the effects of interval system parameters on the dynamic stability of structures numerical studies demonstrate that the uncertainty of the constant part of load influences the boundaries of principal instability region much more than that of the variable part of load within all structural parameters the uncertainty of column length has the most effect on the boundaries which could make the critical frequency of load be magnified about ten times the uncertainty of parameter in particular if all the system parameters are interval with an identical uncertainty degree the uncertainty of parameter would be propagated in the system of dynamic stability and enlarged as high as 20 times the uncertainty of parameter consequently the impacts of parameter uncertainties on the dynamic stability of structures are fairly significant

robustness of discrete systems under structured uncertainties we give a sufficient condition for the robust stability of the discrete system x k 1 f e x k where the perturbation e is bounded e u a necessary and sufficient condition is derived in the case where f is a positive or a negative matrix and u f the stability of interval matrices is also considered numerical examples are given for illustration

the kolmogorov centsov theorem and brownian motion in vector lattices abstract the well known kolmogorov centsov theorem is proved in a dedekind complete vector lattice riesz space with weak order unit on which a strictly positive conditional expectation is defined it gives conditions that guarantee the holder continuity of a stochastic process in the space we discuss the notion of independence of projections and elements in the vector lattice and use this together with the kolmogorov centsov theorem to give an abstract definition of brownian motion in a vector lattice this definition captures the fact that the increments in a brownian motion are normally distributed and that the paths are continuous

on the invariance of the sign pattern of matrix eigenvectors under perturbation abstract we study how much perturbation a in a real matrix a is allowed for the i th real eigenvector not to change sign

stability of interval parameter matrices the stability of interval matrices has been studied by many authors in this paper the stability of interval parameter matrices is introduced and some criteria for this stability are given our results are effective for testing the stability of interval matrices and are more general than previous results furthermore criteria for the stability of time varying matrices are obtained

comparison of statistical and interval analysis for the energy flow uncertainties in structural vibrating systems in this paper the energy flow confidence between two structural multimodal systems coupled by a joint with uncertain parameters is computed by two different methods the first one assumes that the joint parameters perturbed randomly the statistical moments of the energy flow are calculated by an analytical procedure the second one uses interval analysis the joint parameters are considered interval variables and the interval of the energy flow is determined the properties of the statistical and interval solution are investigated and compared

a characterization of convex cones of matrices with constant regular inertia abstract let a be a convex cone of n n matrices in this paper we present a necessary and sufficient condition for a to contain matrices with a constant regular inertia based on a version of the lyapunov equation the condition involves only the normalized extreme points of a this extends a previous paper by the authors where a robust stability criterion for a was obtained

modal analysis of structures with uncertain but bounded parameters via interval analysis in this paper the modal interval analysis method to estimate modal parameters frequency response function frf and mode shapes of structures with uncertain but bounded is presented although the system parameters or properties are uncertain in many engineering problems but their probable range of values i e upper and lower bounds can be provided from practical experience and engineering knowledge moreover to avoid the resonance of a structure and to consider the dynamic response of an uncertain one and also for reliability and stability analysis we often need the bounds of the ranges of structural characteristic parameters such as natural frequency and normal mode by using modal analysis and interval calculus we investigate the method of computing upper and lower bounds of paramenters such as natural frequencies modal shapes and frfs theoretically it is possible to analyze the uncertain but bounded of a structure by using modal analysis and interval calculus they can be estimated by modal interval analysis on the basis of the estimated intervals the engineering structure parameters can be applied into engineering design a numerical example is presented for a tower structure and the results illustrate that the proposed method is effective a comparison of the modal interval method with the results of monte carlo simulation serves to validate the solutions and to identify the bounded ranges of parameters

a dynamic evolution scheme for structures with interval uncertainties by using bidirectional sequential kriging method abstract this paper proposes a bidirectional sequential kriging bsk method for nonlinear interval uncertainty quantification of dynamic systems different from existing surrogate based methods which build surrogate model at preselected samples the proposed method constructs surrogate models sequentially small amount of initial samples are generated by latin hypercube sampling lhs and a crude kriging surrogate model ksm is constructed then two index functions named lower confidence bound lcb and upper confidence bound ucb respectively are defined for sample collocation the index functions guide the search for the optima of structural response after iterations a sophisticated surrogate model will be obtained and the interval bounds of dynamic response can be calculated quickly with some auxiliary algorithms such as genetic algorithm ga numerical examples and engineering application are studied to verify the effectiveness of bsk the results show that compared with existing methods the proposed method can achieve higher accuracy with good efficiency in nonlinear interval analysis

solution theorems for the standard eigenvalue problem of structures with uncertain but bounded parameters generalized eigenvalue problems from the modal analysis are often converted to the standard eigenvalue problems in this paper it evaluates the upper and lower bounds on the eigenvalues of the standard eigenvalue problem of structures subject to severely deficient information about the structural parameters here we focus on non probabilistic interval analysis models of uncertainty which are adapted to the case of severe lack of information on uncertainty non probabilistic interval analysis method in which uncertainties are defined by interval numbers appears as an alternative to the classical probabilistic models for the standard eigenvalue problem of structures with uncertain but bounded parameters the vertex solution theorem the positive semi definite solution theorem and the parameter decomposition solution theorem for the standard eigenvalue problem are presented and compared with deif s solution theorem in numerical examples it is shown that for the upper and lower bounds on the eigenvalues of the standard eigenvalue problem with uncertain but bounded parameters the presented vertex solution theorem is unconditional and the positive semi definite solution theorem and the parameter decomposition solution theorem have less limitary conditions compared with deif s solution theorem the effectiveness of the vertex solution theorem the positive semi definite solution theorem and the parameter decomposition solution theorem are illustrated by numerical examples

comments on assessing the stability of linear time invariant continuous interval dynamic systems the first part of this note identifies and analyzes two weak points of the commented paper i example 1 reports an incorrect result for which we provide a complete explanation and correction ii the practical use of theorem 1 reveals major drawbacks such as a severe limitation in the applicability due to the restrictive conditions requested by the hypothesis and a relatively low accuracy of the results i e rough values for the right outer bounds of the eigenvalue ranges of interval matrices the second part of the note presents three methods that circumvent the use of theorem 1 and provide the following information about the eigenvalue range of an interval matrix a right outer bound relying on an inequality proved by j rohn see references the right end point by solving a constrained global maximization problem a right outer bound for arbitrary interval matrices and the right end point for some classes of interval matrices by calculating the spectral abscissa of a constant matrix that majorizes the considered interval matrix for illustration by numerical examples throughout the note we use three interval matrices related to example 1 from the commented paper

necessary and sufficient conditions for stability of symmetric interval matrices it is shown that a family of symmetric interval matrices is schur if and only if the vertex matrices are schur this result is also applicable to hurwitz stability and aperiodic properties of linear systems

an algorithm for addressing the real interval eigenvalue problem in this paper we present an algorithm for approximating the range of the real eigenvalues of interval matrices such matrices could be used to model real life problems where data sets suffer from bounded variations such as uncertainties e g tolerances on parameters measurement errors or to study problems for given states the algorithm that we propose is a subdivision algorithm that exploits sophisticated techniques from interval analysis the quality of the computed approximation and the running time of the algorithm depend on a given input accuracy we also present an efficient c implementation and illustrate its efficiency on various data sets in most of the cases we manage to compute efficiently the exact boundary points limited by floating point representation

an interval kalman filtering with minimal conservatism abstract the interval kalman filtering ikf can handle parametric interval uncertainties of the system matrices and it computes the lower and upper boundaries of the estimated states in this paper we propose an alternative form of interval kalman filtering to reduce the conservatism inherent to the existing interval kalman filtering first we address why the existing interval kalman filtering scheme induces conservatism in the boundary estimation then to remove the conservatism we derive noise covariance matrices taking into account of interval uncertainties as well as process and measurement noises following the typical derivation process of the standard kalman filtering a new recursive form of interval kalman filtering is derived through numerical simulations the superiority of the new algorithm over existing ikf is illustrated

computation of bounds for eigenvalues of structures with interval parameters in this paper the computation of eigenvalue bounds for generalized interval eigenvalue problem is considered two algorithms based on the properties of continuous functions are developed for evaluating upper and lower eigenvalue bounds of structures with interval parameters the method can provide the tightest bounds within a given precision numerical examples illustrate the effectiveness of the proposed method

assessing the stability of linear time invariant continuous interval dynamic systems it is known that stability analysis of linear time invariant dynamic systems under parameter uncertainties can be equated to estimating the range of the eigenvalues of matrices whose elements are intervals in this note first the problem of finding tight outer bounds on the eigenvalue ranges is considered a method for computing such bounds is suggested which consists essentially of setting up and solving a system of n mildly nonlinear algebraic equations n being the size of the interval matrix investigated the main result of the note however is a method for determining the right end point of the exact eigenvalue ranges the latter makes use of the outer bounds it is applicable if certain computationally verifiable monotonicity conditions are fulfilled the methods suggested can be applied for robust stability analysis of both continuous and discrete time systems numerical examples illustrating the applicability of the new methods are also provided

real eigenvalue bounds of standard and generalized real interval eigenvalue problems abstract in this paper we study the real eigenvalue bounds of standard and generalized real interval eigenvalue problems based on some known sufficient conditions for the regularity of interval matrices a new algorithm for computing real eigenvalue bounds for real interval matrices is proposed it can easily be extended for applying to generalized real interval eigenvalue problems one advantage of the method is that the computation procedure takes less time than former methods we have proposed therefore it can be applied for solving larger interval eigenvalue problems finally some numerical examples are presented which show the effectiveness of the method

eigenvalues of tridiagonal symmetric interval matrices an algorithm for the determination of the eigenvalues of tridiagonal symmetric interval matrices is presented the intervals of the eigenvalues are not overestimated but exactly calculated the algorithm is an efficient one because it only needs twice as many operations as the sturm algorithm which is used for real matrices

convergence property of interval matrices and interval polynomials in association with robust control system design and analysis the hurwitz property of interval matrices and interval polynomials has recently been actively investigated however its discrete counterpart the convergence property has seemingly not been much discussed in this paper this property is studied in comparison with the hurwitz counterpart some conditions under which interval matrices or interval polynomials are convergent are derived

necessary and sufficient conditions for stability of a class of interval matrices in this paper some necessary and sufficient conditions for stability complete instability and composite stability type of a class of interval matrices are given

a description of norm convergent martingales on vector valued lp spaces abstract norm convergent martingales on tensor products of banach spaces are considered in a measure free setting as a consequence we obtain the following characterization for convergent martingales on vector valued l p spaces let be a probability space x a banach space and n an increasing sequence of sub algebras of in order for f n n n 1 to be a convergent martingale in l p x 1 p it is necessary and sufficient that for each i n there exists a convergent martingale x i n n n 1 in l p and y i x such that for each n n we have f n s i 1 x i n s y i for all s where i 1 lim n x i n l p and lim i y i 0

comments on a necessary and sufficient condition for the stability of interval matrices in a recent paper by bialas a new proof of a known result concerning interval polynomials was given this result is summarized here and a much stronger result is mentioned bialas main theorem on the stability of interval matrices is shown to be false

semidefinite programming for dynamic steady state analysis of structures under uncertain harmonic loads this paper presents semidefinite programming approaches for computing confidential bounds for the dynamic steady state responses of a damped structure subjected to uncertain driving loads we assume that amplitudes of harmonic driving loads obey a non probabilistic uncertainty model the semidefinite programming problems are formulated for finding confidential bounds for various characteristic amounts of dynamic steady state response including the modulus and phase angle of the complex amplitude of the displacement and stress numerical examples demonstrate that sufficiently tight bounds can be obtained by solving the presented semidefinite programming problems

an algorithm to determine linear independence of a set of interval vectors this paper presents an algorithm to determine the linear dependence and independence of a set of interval vectors the algorithm is developed on the basis of a necessary and sufficient condition for linear independence of an arbitrary interval vector relative to other linearly independent interval vectors the necessary and sufficient condition can be checked by a linear programming method then the developed necessary and sufficient condition will be employed to check the linear independence of a set of interval vectors through numerical examples the validity of the developed algorithm is illustrated

on sufficient conditions for the stability of interval matrices using gershgorin s diagonal dominance theorem new sufficient conditions for the stability of interval matrices are obtained the new conditions remove a restriction in heinen s 1983 analysis and are applicable to a wider class of problems than previously possible

efficient isolation of polynomial s real roots this paper revisits an algorithm isolating the real roots of a univariate polynomial using descartes rule of signs it follows work of vincent uspensky collins and akritas johnson krandick our first contribution is a generic algorithm which enables one to describe all the known algorithms based on descartes rule of sign and the bisection strategy in a unified framework using that framework a new algorithm is presented which is optimal in terms of memory usage and as fast as both collins and akritas algorithm and krandick s variant independently of the input polynomial from this new algorithm we derive an adaptive semi numerical version using multi precision interval arithmetic we finally show that these critical optimizations have important consequences since our new algorithm still works with huge polynomials including orthogonal polynomials of degree 1000 and more which were out of reach of previous methods

operator martingale decompositions and the radon nikodým property in banach spaces abstract we consider submartingales and uniform amarts of maps acting between a banach lattice and a banach lattice or a banach space in this measure free setting of martingale theory it is known that a banach space y has the radon nikod m property if and only if every uniformly norm bounded martingale defined on the chaney schaefer l tensor product e l y where e is a suitable banach lattice is norm convergent we present applications of this result firstly an analogues characterization for banach lattices y with the radon nikod m property is given in terms of a suitable set of submartingales supermartingales on e l y secondly we derive a riesz decomposition for uniform amarts of maps acting between a banach lattice and a banach space this result is used to characterize banach spaces with the radon nikod m property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent in the case 1 p our results yield l p y space analogues of some of the well known results on uniform amarts in l 1 y spaces

eigenvalue bounds for symmetric matrices with entries in one interval in this paper we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval based on some matrix computation theories a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results furthermore the ideas are spread to general symmetric interval matrices finally numerical examples are presented which show the validity of the proposed methods

analysis of mechanical systems using interval computations applied to finite element methods this paper addresses the problem of mechanical systems in which parameters are uncertain and bounded interval calculation is used to find an envelope of transfer functions for mechanical systems modelled with finite elements within this context a new formulation has been developed for finite element problems involving bounded parameters to avoid the problems of overestimation an iterative algorithm is introduced which leads to a conservative solution for linear mechanical problems a method to ensure the convergence of this algorithm is also proposed this new algorithm has been tested on simple mechanical systems and leads to a conservative envelope of the transfer functions