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

monotonic convergent iterative learning controller design based on interval model conversion this note presents a robust iterative learning controller design method for plants subject to interval model uncertainty in the a matrix of their state space description first order perturbation theory is used to find bounds on the eigenvalues and eigenvectors of the powers of a when a is an interval matrix these bounds are then used for calculation of the interval uncertainty of the markov matrix the bounds on the markov matrix are then used to design an iterative learning controller that ensures monotonic convergence for all systems in the interval plant

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

minimal surfaces in finite volume non compact hyperbolic 3 manifolds we prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3 manifold mathcal n we also obtain a least area incompressible properly embedded finite topology 2 sided surface we prove a properly embedded minimal surface of bounded curvature has finite topology this determines its asymptotic behavior some rigidity theorems are obtained

block lu factors of generalized companion matrix pencils we present formulas for computations involving companion matrix pencils as may arise in considering polynomial eigenvalue problems in particular we provide explicit companion matrix pencils for matrix polynomials expressed in a variety of polynomial bases including monomial orthogonal newton lagrange and bernstein bezier bases additionally we give a pair of explicit lu factors associated with each pencil and a prescription for block pivoting when required

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

parallel krylov solvers for the polynomial eigenvalue problem in slepc polynomial eigenvalue problems are often found in scientific computing applications when the coefficient matrices of the polynomial are large and sparse usually only a few eigenpairs are required and projection methods are the best choice we focus on krylov methods that operate on the companion linearization of the polynomial but exploit the block structure with the aim of being memory efficient in the representation of the krylov subspace basis the problem may appear in the form of a low degree polynomial quartic or quintic say expressed in the monomial basis or a high degree polynomial coming from interpolation of a nonlinear eigenproblem expressed in a nonmonomial basis we have implemented a parallel solver in slepc covering both cases that is able to compute exterior as well as interior eigenvalues via spectral transformation we discuss important issues such as scaling and restart and illustrate the robustness and performance of the solver with some numerical experiments

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

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

the singular set of mean curvature flow with generic singularities a mean curvature flow starting from a closed embedded hypersurface in mathbf r n 1 must develop singularities we show that if the flow has only generic singularities then the space time singular set is contained in finitely many compact embedded n 1 dimensional lipschitz submanifolds plus a set of dimension at most n 2 if the initial hypersurface is mean convex then all singularities are generic and the results apply in mathbf r 3 and mathbf r 4 we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time slice for 2 or 3 convex hypersurfaces in all dimensions the same arguments lead to the same conclusion the flow is completely smooth at almost all times and connected components of the singular set are contained in time slices a key technical point is a strong parabolic reifenberg property that we show in all dimensions and for all flows with only generic singularities we also show that the entire flow clears out very rapidly after a generic singularity these results are essentially optimal

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

generalized matrix diagonal stability and linear dynamical systems let a aij be a real square matrix and 1 p we present two analogous developments one for schur stability and the discrete time dynamical system x t 1 ax t and the other for hurwitz stability and the continuous time dynamical system x t ax t here is a description of the latter development for a we define and study hurwitz diagonal stability with respect to p norms abbreviated as hdsp hds2 is the usual concept of diagonal stability a is hdsp implies re 0 for every eigenvalue of a which means a is hurwitz stable abbreviated as hs when the off diagonal elements of a are nonnegative a is hs iff a is hdsp for all p for the dynamical system x t ax t we define diagonally invariant exponential stability relative to the p norm abbreviated as diesp meaning there exist time dependent sets which decrease exponentially and are invariant with respect to the system we show that diesp is a special type of exponential stability and the dynamical system has this property iff a is hdsp

computation of the monodromy matrix in floating point arithmetic with the wilkinson model in this study results have been obtained that compute the monodromy matrix in floating point arithmetic using the wilkinson model these results have been applied to the asymptotic stability of systems of linear difference equations with periodic coefficients also the effect of floating point arithmetic has been investigated on numerical examples

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

eigenvalue bounds of structures with uncertain but bounded parameters many analysis and design problems in engineering and science involve uncertainty to varying degrees this paper is concerned with the structural vibration problem involving uncertain material or geometric parameters specified as bounds on these parameters this produces interval stiffness and mass matrices and the problem is transformed into a generalized interval eigenvalue problem in interval mathematics however tighter bounds on the eigenvalues may be obtained by using the formulation of the structural dynamic problem often the stiffness and mass matrices can be formed as a non negative decomposition in the uncertain structural parameters in this case the eigenvalue bounds may be obtained from the parameter vertex solutions even more efficiently using interval extension from interval mathematics the generalized interval eigenvalue problem may be divided into two generalized eigenvalue problems for real symmetric matrix pairs the parameter vertex solution algorithm is compared with deif s solution the eigenvalue inclusion principle and the interval perturbation method in numerical examples

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

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

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

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 for schur and hurwitz diagonal stability of complex interval matrices abstract let c a j b be a complex interval matrix where a b r n n are real interval matrices and j 2 1 denote by d the set of positive definite diagonal n n matrices the schur respectively hurwitz diagonal stability of c abbreviated as sds respectively hds is defined in terms of the stein respectively lyapunov inequality associated with a generic matrix c c the inequality must have a common solution q d for all c c the paper develops two types of results for the sds respectively hds analysis which exploit different properties of c each type of results presents i a sufficient condition relying on the solvability of a matrix norm inequality sds analysis or a matrix measure inequality hds analysis ii a construction procedure for a concrete matrix that satisfies the inequality at i and concomitantly yields a common solution q d for some particular cases both necessary and sufficient conditions can be derived the schur and hurwitz approaches are developed in parallel for concision reasons as well as for emphasizing the similarities numerical examples illustrate the applicability of the theoretical results

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

robust stability test of a class of linear time invariant interval fractional order system using lyapunov inequality this paper provides a new analytical robust stability checking method of fractional order linear time invariant interval uncertain system this paper continues the authors previous work yangquan chen hyo sung ahn i podlubny robust stability check of fractional order linear time invariant systems with interval uncertainties in proceedings of the ieee conference on mechatronics and automation niagara falls canada july 2005 pp 210 215 where matrix perturbation theory was used for the new robust stability checking lyapunov inequality is utilized for finding the maximum eigenvalue of a hermitian matrix through numerical examples the usefulness and the effectiveness of the newly proposed method are verified

stability of interval matrices the real eigenvalue case c v hollot and a c bartlett 1987 showed that testing at most 2 to the n sup 2 certain matrices for stability is sufficient for verifying stability of an n n interval matrix with real eigenvalues it is proven that this upper bound can be reduced to 2 sup 2n 1 and a special case where testing only two matrices is needed is considered

heegaard surfaces and measured laminations i the waldhausen conjecture we give a proof of the so called generalized waldhausen conjecture which says that an orientable irreducible atoroidal 3 manifold has only finitely many heegaard splittings in each genus up to isotopy jaco and rubinstein have announced a proof of this conjecture using different methods

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

criteria for robust absolute stability of time varying nonlinear continuous time systems the present paper establishes results for the robust absolute stability of a class of nonlinear continuous time systems with time varying matrix uncertainties of polyhedral type and multiple time varying sector nonlinearities by using the variational method and the lyapunov second method criteria for robust absolute stability are obtained in different forms for the given class of systems specifically the parametric classes of lyapunov functions are determined which define the necessary and sufficient conditions of robust absolute stability the piecewise linear lyapunov functions of the infinity vector norm type are applied to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations several simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for special cases two examples are presented as applications of the present results to a particular second order system and to a specific class of systems with time varying interval matrices in the linear part

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

necessary and sufficient conditions for componentwise stability of interval matrix systems in the asymptotic stability as analysis of interval matrix systems some results are available that operate only as sufficient conditions based on a unique test matrix adequately built from the interval matrix our note reveals the complete role of this test matrix for fully characterizing the componentwise asymptotic stability cwas of interval matrix systems cwas is a special type of as which ensures the flow invariance of certain time dependent sets with respect to the state space trajectories hence the sufficient conditions for as get a new and deeper meaning by their reformulation as necessary and sufficient conditions with respect to the stronger property of cwas

lower bounds on the spectra of symmetric matrices with nonnegative entries if we normalize a symmetric n n matrix with nonnegative entriesso that its largest entry is 1 then its spectrum is bounded below by n2 the lower bound is achieved in all even dimensions for and only for adjacency matrices of complete bipartite graphs with equal parts

analysis of uncertain structural systems using interval analysis the imprecision or uncertainty present in many engineering analysis design problems can be modeled using probabilistic fuzzy or interval methods this work considers the modeling of uncertain structural systems using interval analysis by representing each uncertain input parameter as an interval number a static structural analysis problem can be expressed in the form of a system of linear interval equations in addition to the direct and gaussian elimination based solution approaches a combinatorial approach based on an exhaustive combination of the extreme values of the interval numbers and an inequality hased method are presented for finding the solution of interval equations the range or interval of the solution vector response parameters is found to increase with increasing size of the problem in all of the methods an interval truncation approach is proposed to limit the growth of intervals of response parameters so that realistic and accurate solutions can be obtained in the presence of large amounts of uncertainty numerical examples are presented to illustrate the computational aspects of the methods and also to indicate the importance of the truncation approach in practical problems the utility of interval methods in predicting the extreme values of the response parameters of structures is discussed

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

a parallel additive schwarz preconditioned jacobi davidson algorithm for polynomial eigenvalue problems in quantum dot simulation we develop a parallel jacobi davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation a jacobi davidson eigenvalue solver is implemented based on the portable extensible toolkit for scientific computation petsc the eigensolver thus inherits petsc s efficient and various parallel operations linear solvers preconditioning schemes and easy usages the parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by schrodinger s equations we find that the parallel restricted additive schwarz preconditioner in conjunction with a parallel krylov subspace method e g gmres can solve the correction equations the most costly step in the jacobi davidson algorithm very efficiently in parallel besides the overall performance is quite satisfactory we have observed near perfect superlinear speedup by using up to 320 processors the parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 intel 3 0ghz processors

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

simple criteria for stability of interval matrices in this paper the simple sufficient conditions are presented for the stability of a constant real matrix whose elements are not known precisely but we know the upper and lower bounds of intervals in which the elements of matrix a are confined these conditions are also applied in determining the stability and the degree of stability of a matrix

on stability of interval matrices new sufficient and sometimes necessary and sufficient conditions are obtained for schur and hurwitz stability of interval matrices by relying on the concept of connective stability and m matrices the necessity part is broadened to include interval matrices with mixed signs of the off diagonal elements provided the sign patterns follow that of the morishima matrix the obtained results are extended to cover convex combinations of interval matrices

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

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

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

diagonal stability of interval matrices and applications abstract let a i a r n n a a a be an interval matrix and 1 p we introduce the concept of schur and hurwitz diagonal stability relative to the holder p norm of a i abbreviated as sds p and hds p respectively this concept is formulated in terms of a matrix inequality using the p norm which must be satisfied by the same positive definite diagonal matrix for all a a i the inequality form is different for sds p and hds p the particular case of p 2 is equivalent to the condition of quadratic stability of a i the sds 2 inequality is equivalent to the stein inequality a a i a t pa p 0 and the hds 2 inequality is equivalent to the lyapunov inequality a a i a t p pa 0 in both cases p is a positive definite diagonal matrix and the notation 0 means negative definite the first part of the paper provides sds p and hds p criteria presents methods for finding the positive definite diagonal matrix requested by the definition of sds p and hds p analyzes the robustness of sds p and hds p and explores the connection with the schur and hurwitz stability of a i the second part shows that the sds p or hds p of a i is equivalent to the following properties of a discrete or continuous time dynamical interval system whose motion is described by a i the existence of a strong lyapunov function defined by the p norm and the existence of exponentially decreasing sets defined by the p norm that are invariant with respect to system s trajectories

interval tensors and their application in solving multi linear systems of equations abstract in this paper we introduce interval tensors and present some results about their eigenvalues positive definiteness and application in solving multi linear systems it is proved that the set of maximum z eigenvalues of a symmetric interval tensor is a compact interval also several bounds for eigenvalues of an interval tensor are proposed in addition necessary and sufficient conditions for having a positive definite interval tensor are presented and investigated furthermore solving tensor equations using interval methods is presented and the interval jacobi and gauss seidel algorithms are extended for interval multi linear systems finally some numerical experiments are carried out to illustrate the methods

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

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

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

systems with structured uncertainty relations between quadratic and robust stability the relation between the notions of robust stability and quadratic stability for uncertain systems with structured uncertainty due to both real and complex parameter variations is discussed examples are presented to demonstrate that for systems containing at least two uncertain blocks the notions of robust stability for complex parameter variations and quadratic stability for real parameter variations are not equivalent a byproduct of these examples is that for this class of systems quadratic stability for real perturbations need not imply quadratic stability for complex perturbations this is in stark contrast with the situation in the case of unstructured uncertainty for which it is known that quadratic stability for either real or complex perturbations is equivalent to robust stability for complex perturbations and thus equivalent to a small gain condition on the transfer matrix that the perturbation experiences

effective finiteness of irreducible heegaard splittings of non haken 3 manifolds the main result is a short effective proof of tao li s theorem that a closed non haken hyperbolic 3 manifold n has at most finitely many irreducible heegaard splittings along the way we show that n has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index 1 minimal surfaces this effective result together with the sequel with daniel ketover solves the classification problem for heegaard splittings of non haken hyperbolic 3 manifolds

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

network nestedness as generalized core periphery structures the concept of nestedness in particular for ecological and economical networks has been introduced as a structural characteristic of real interacting systems we suggest that the nestedness is in fact another way to express a mesoscale network property called the core periphery structure with real ecological mutualistic networks and synthetic model networks we reveal the strong correlation between the nestedness and core periphery ness likeness to the core periphery structure by defining the network level measures for nestedness and core periphery ness in the case of weighted and bipartite networks however at the same time via more sophisticated null model analysis we also discover that the degree the number of connected neighbors of a node distribution poses quite severe restrictions on the possible nestedness and core periphery parameter space therefore there must exist structurally interwoven properties in more fundamental levels of network formation behind this seemingly obvious relation between nestedness and core periphery structures

the space of embedded minimal surfaces of fixed genus in a 3 manifold iii planar domains this paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold in cm3 cm5 we describe the case where the surfaces are topologically disks on any fixed small scale although the focus of this paper general planar domains is more in line with cm6 we will prove a result here namely corollary iii 3 5 below which is needed in cm5 even for the case of disks roughly speaking there are two main themes in this paper the first is that stability leads to improved curvature estimates this allows us to find large graphical regions these graphical regions lead to two possibilities

mixed forward backward stability of the two level orthogonal arnoldi method for quadratic problems abstract we revisit the numerical stability of the two level orthogonal arnoldi toar method for computing an orthonormal basis of a second order krylov subspace associated with two given matrices we show that the computed basis is close to a basis for a second order krylov subspace associated with nearby matrices provided that the norms of the given matrices are not too large or too small thus we provide conditions that guarantee the numerical stability of the toar method in computing orthonormal bases of second order krylov subspaces we also study scaling the quadratic problem for improving the numerical stability of the toar procedure when the norms of the matrices are too large or too small

componentwise asymptotic stability of linear constant dynamical systems this note deals with a special type of asymptotic stability namely componentwise asymptotic stability with respect to the vector gamma t cwas of system s dot x ax bu t geq 0 where gamma t 0 componentwise inequality and gamma t rightarrow 0 as t rightarrow infty s is cwas if for each t 0 geq 0 and for each x t 0 leq gamma t 0 x t 0 with the components x i t 0 the free response of s satisfies x t leq gamma t for each t geq t 0 for gamma t underline underline delta alphae beta t t geq 0 with alpha 0 and beta 0 scalar the cweas e exponential may be defined s is cwas cweas if and only if dot gamma t geq bar a gamma t t geq 0 bar a alpha and bar a has the elements a ij and a ij i neq j these results may be used in order to evaluate in a more detailed manner the dynamical behavior of s as well as to stabilize s componentwise by a suitable linear state feedback

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

genus bounds for min max minimal surfaces we prove optimal genus bounds for minimal surfaces arising from the min max construction of simon smith this confirms a conjecture made by pitts rubinstein in 1986

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

bernstein bezoutian matrices several computational and structural properties of bezoutian matrices expressed with respect to the bernstein polynomial basis are shown the exploitation of such properties allows the design of fast algorithms for the solution of bernstein bezoutian linear systems without never making use of potentially ill conditioned reductions to the monomial basis in particular we devise an algorithm for the computation of the greatest common divisor gcd of two polynomials in bernstein form a series of numerical tests are reported and discussed which indicate that bernstein bezoutian matrices are much less sensitive to perturbations of the coefficients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the bernstein and the power basis

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

necessary and sufficient conditions for the hurwitz and schur stability of interval matrices establishes a set of new sufficient conditions for the hurwitz and schur stability of interval matrices the authors use these results to establish necessary and sufficient conditions for the hurwitz and schur stability of interval matrices the authors relate the above results to the existence of quadratic lyapunov functions for linear time invariant systems with interval valued coefficient matrices using the above results the authors develop an algorithm to determine the hurwitz and the schur stability properties of interval matrices the authors demonstrate the applicability of their results by means of two specific examples

componentwise stability of 1d and 2d linear discrete time systems this paper deals with a special type of asymptotic exponential stability namely componentwise asymptotic exponential stability for 1 d and 2 d linear discrete time systems the main motivation for these results is the need particularly felt in the evaluation in a more detailed manner of the dynamical behaviour of 1 d and 2 d linear discrete time systems necessary and sufficient conditions for componentwise asymptotic exponential stability are given

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

the space of embedded minimal surfaces of fixed genus in a 3 manifold i estimates off the axis for disks this paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed riemannian 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r3 with the flat metric this study is undertaken here and completed in cm6 these local results are then applied in cm7 where we describe the general structure of fixed genus surfaces in 3 manifolds there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r2 into r3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multi valued graph this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that such a disk

a multilevel jacobi davidson method for polynomial pde eigenvalue problems arising in plasma physics the simulation of drift instabilities in the plasma edge leads to cubic polynomial pde eigenvalue problems with parameter dependent coefficients the aim is to determine the wave number which leads to the maximum growth rate of the amplitude of the wave this requires the solution of a large number of pde eigenvalue problems since we are only interested in a smooth eigenfunction corresponding to the eigenvalue with largest imaginary part the jacobi davidson method can be applied unfortunately a naive implementation of this method is much too expensive for the large number of problems that have to be solved in this paper we will present a multilevel approach for the construction of an appropriate initial search space we will also discuss the efficient solution of the correction equation and we will show how optimal scaling helps to accelerate the convergence

coupling of eigenvalues of complex matrices at diabolic and exceptional points the paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters the cases of weak and strong coupling are distinguished and their geometric interpretation in two and three dimensional spaces is given general asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios two physical examples illustrate effectiveness and accuracy of the presented theory

on the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph we prove that if b is an essentially nonnegative symmetric matrix with minimum eigenvalue m b and if the graph of b is connected and bipartite with bipartition v1 v2 then there exists an eigenvector x belonging to m b such that xi 0 for i v1 and xi 0 for i v2 we then establish two facts concerning essentially nonnegative symmetric matrices with bipartite graph and point out how the above result can be used in conjunction with these facts to give a proof not requiring the perron frobenius theory of the following theorem of constantine 2 among all essentially nonnegative symmetric n by n matrices with minimum diagonal entry at least and maximum off diagonal entry at most m the matrix with smallest minimum eigenvalue is man in where an is the adjacency matrix of the complete bipartite graph k n2 n2

vector norms as lyapunov functions for linear systems a unified theory of quadratic and piecewise linear lyapunov functions for continuous and discrete time linear systems is presented the key to this work is the description of these lyapunov functions by vector norms the main results are sufficient and necessary conditions for a vector norm to be a lyapunov function as well as a method based on these conditions of constructing such lyapunov functions

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