a property of left linear rewrite systems preserving recognizability abstract let r be a left linear rewrite system over sign r and let g sign r where g 1 sign r and 0 sign r we show that r effectively preserves recognizability if and only if r effectively preserves recognizability

sensitivity analysis of discrete markov chains via matrix calculus abstract markov chains are often used as mathematical models of natural phenomena with transition probabilities defined in terms of parameters that are of interest in the scientific question at hand sensitivity analysis is an important way to quantify the effects of changes in these parameters on the behavior of the chain many properties of markov chains can be written as simple matrix expressions and hence matrix calculus is a powerful approach to sensitivity analysis using matrix calculus we derive the sensitivity and elasticity of a variety of properties of absorbing and ergodic finite state chains for absorbing chains we present the sensitivities of the moments of the number of visits to each transient state the moments of the time to absorption the mean number of states visited before absorption the quasistationary distribution and the probabilities of absorption in each of several absorbing states for ergodic chains we present the sensitivity of the stationary distribution the mean first passage time matrix the fundamental matrix and the kemeny constant we include two examples of application of the results to demographic and ecological problems

bottom up tree pushdown automata classification and connection with rewrite systems we define different types of bottom up tree pushdown automata and study their connections with rewrite systems along this line of research we complete and generalize the results of gallier book and salomaa we define the notion of a tail reduction free trf rewrite system using the decidability of ground reducibility we prove the decidability of the trf property monadic rewrite systems of book gallier and salomaa become a natural particular case of trf rewrite systems we associate a deterministic bottom up tree pushdown automaton with any left linear trf rewrite system finally we generalize monadic rewrite systems by introducing the notion of a semi monadic rewrite system and show that like a monadic rewrite system it preserves recognizability

markov chain sensitivity measured by mean first passage times abstract the purpose of this article is to present results concerning the sensitivity of the stationary probabilities for an n state time homogeneous irreducible markov chain in terms of the mean first passage times in the chain

differential equations for roaming pseudospectra paths to extremal points and boundary tracking when studying the varepsilon pseudospectrum of a matrix one is often interested in computing the extremal points having maximum real part or modulus this is a crucial step for example when computing the distance to instability of a stable system using the property that the pseudospectrum is determined via perturbations by rank 1 matrices we derive differential equations on the manifold of normalized rank 1 matrices whose solutions tend to the critical rank 1 perturbations associated with the extremal points of locally maximum real part and modulus this approach also allows us to track the boundary contour of the pseudospectrum in a neighborhood of the extremal points the technique we propose is related to an idea recently developed by guglielmi and overton who derived discrete dynamical systems instead of the continuous ones we present the method turns out to be fast in comparison with those previously proposed in the literature and appears to be promising in dealing with large sparse problems

structured maximal perturbations for hamiltonian eigenvalue problems we discuss the effect of structure preserving perturbations on complex or real hamiltonian eigenproblems and characterize the structured worst case effect perturbations we derive explicit expressions for the maximal hamiltonian perturbations

low rank differential equations for hamiltonian matrix nearness problems for a hamiltonian matrix with purely imaginary eigenvalues we aim to determine the nearest hamiltonian matrix such that some or all eigenvalues leave the imaginary axis conversely for a hamiltonian matrix with all eigenvalues lying off the imaginary axis we look for a nearest hamiltonian matrix that has a pair of imaginary eigenvalues the hamiltonian matrices can be allowed to be complex or restricted to be real such hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems they are closely related to the problem of determining extremal points of hamiltonian pseudospectra we obtain a characterization of optimal perturbations which turn out to be of low rank and are attractive stationary points of low rank differential equations that we derive we use a two level approach where in the inner level we determine extremal points of the hamiltonian varepsilon pseudospectrum for a given varepsilon by following the low rank differential equations into a stationary point and on the outer level we optimize for varepsilon this permits us to give fast algorithms exhibiting quadratic convergence for solving the considered hamiltonian matrix nearness problems

linear generalized semi monadic rewrite systems effectively preserve recognizability abstract we introduce the notion of the generalized semi monadic rewrite system which is a generalization of well known rewrite systems the ground rewrite system the monadic rewrite system and the semi monadic rewrite system we show that linear generalized semi monadic rewrite systems effectively preserve recognizability we show that a tree language l is recognizable if and only if there exists a rewrite system r such that r r 1 is a linear generalized semi monadic rewrite system and that l is the union of finitely many r classes we show several decidability and undecidability results on rewrite systems effectively preserving recognizability and on generalized semi monadic rewrite systems for example we show that for a rewrite system r effectively preserving recognizability it is decidable if r is locally confluent moreover we show that preserving recognizability and effectively preserving recognizability are modular properties of linear collapse free rewrite systems finally as a consequence of our results on trees we get that restricted right left overlapping string rewrite systems effectively preserve recognizability

structured pseudospectra in structural engineering this paper presents a new method for computing the pseudospectra of a matrix that respects a prescribed sparsity structure the pseudospectrum is defined as the set of points in the complex plane to which an eigenvalue of the matrix can be shifted by a perturbation of a certain size a canonical form for sparsity preserving perturbations is given and a computable formula for the corresponding structured pseudospectra is derived this formula relates the computation of structured pseudospectra to the computation of the structured singular value ssv of an associated matrix although the computation of the ssv in general is an np hard problem algorithms for its approximation are available and demonstrate good performance when applied to the computation of structured pseudospectra of medium sized or highly sparse matrices the method is applied to a wing vibration problem where it is compared with the matrix polynomial approach and to the stability analysis of truss structures new measures for the vulnerability of a truss structure are proposed which are related to the distance to singularity of the associated stiffness matrix copyright 2005 john wiley sons ltd

eigenvalue patterned condition numbers toeplitz and hankel cases we continue the study started in noschese and pasquini eigenvalue condition numbers zero structured versus traditional j comput appl math 185 2006 174 189 concerning the sensitivity of simple eigenvalues of a matrix a to perturbations in a that belong to a chosen subspace of matrices in noschese and pasquini eigenvalue condition numbers zero structured versus traditional j comput appl math 185 2006 174 189 the zero structured perturbations have been considered here we focus on patterned perturbations and the cases of the toeplitz and of the hankel matrices are investigated in detail useful expressions of the absolute patterned condition number of the eigenvalue l and of the analogue of the matrix yx h which leads to the traditional condition number of l are given matlab codes are defined to compare traditional zero structured and patterned condition numbers a report on significant numerical tests is included

fractional pseudospectra and their localizations abstract motivated by the growing successful use of fractional differential equations in the modeling of different important phenomena in this paper we derive tools for practical analysis of the robust asymptotic stability of a incommensurate fractional order linear system first the concept of fractional pseudospectra is introduced second driven by the simplicity and usefulness of spectral localizations in the analysis of various matrix properties we introduce adequate localization techniques using the ideas that come from diagonally dominant matrices in order to localize the fractional pseudospectra in such way many theoretical and practical applications of pseudospectra robust stability transient behavior nonnormal dynamics etc in fractional order differential systems can be linked to the specificity of the matrix entries allowing one to understand certain phenomena in practice better third we consider the fractional distance to instability in l l 1 and l 2 norms and determine efficient lower bounds finally this novel approach is implemented on the realistic model of empirical food web to link the stability that incorporates hereditary dynamics of living organisms with the empirical data and their uncertainty limitations

descendants of a recognizable tree language for prefix constrained linear monadic term rewriting with position cutting strategy abstract for a tree language l a finite set z of regular path languages and a set s of z prefix constrained linear monadic term rewriting rules over the position cutting descendant of l for s is the set s l of trees reachable from a tree in l by rewriting in s by position cutting strategy if l is recognizable then s l is recognizable as well moreover if s is finite then we can construct a tree automaton recognizing s l for a recognizable tree language l and a finite set z of regular path languages we study the set d z l of position cutting descendants of l for all sets of z prefix constrained linear monadic term rewriting rules over we show that d z l is finite and that if l is given by a tree automaton a and each element of z is given by an automaton then we can construct a set r 1 r k of z prefix constrained linear monadic term rewriting systems over such that d z l r 1 l r k l

monadic thue systems abstract certain infinite thue systems over a finite alphabet are studied in particular systems s e such that for each a e the set u u a s is a context freelanguage the syntactic structure of sets of ancestors and sets of descendants is considered as well as that of unions of congruence classes taken over infinite context free languages or regular sets the common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable even for finite systems the word problem for confluent systems of this type is shown to be tractable the question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

entrywise perturbation theory and error analysis for markov chains grassmann taksar and heyman introduced a variant of gaussian climination for computing the steady state vector of a markov chain in this paper we prove that their algorithm is stable and that the problem itself is well conditioned in the sense of entrywise relative error thus the algorithm computes each entry of the steady state vector with low relative error even the small steady state probabilities are computed accurately the key to our analysis is to focus on entrywise relative error in both the data and the computed solution rather than making the standard assessments of error based on norms our conclusions do not depend on any condition numbers for the problem

on the first and second order derivatives of the perron vector abstract in a previous work 5 the authors developed formulas for the second order partial derivatives of the perron root as a function of the matrix entries at an essentially nonnegative and irreducible matrix these formulas which involve the group generalized inverse of an associated m matrix were used to investigate the concavity and convexity of the perron root as a function of the entries the authors now combine the above results together with an approach taken in an earlier joint paper 6 of the second author with l elsner and c johnson and they develop formulas for the second order derivatives of an appropriately normalized perron vector with respect to the matrix entries which again are given in terms the group generalized inverse of an associated m matrix convexity properties of the perron vector as a function of the entries of the matrix are then examined in addition formulas for the first derivative of the perron vector resulting from different normalizations of this eigenvector are also given a by product of one of these formulas yields that the group generalized inverse of a singular and irreducible m matrix can be diagonally scaled to a matrix which is entrywise column diagonally dominant

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

some undecidability results concerning the property of preserving regularity a finite string rewriting system r preserves regularity if and only if it preserves regularity where is the alphabet containing exactly those letters that have occurrences in the rules of r this proves a conjecture of gyenizse and vagvolgyi 1997 in addition some undecidability results are presented that generalize results of gilleron and tison 1995 from term rewriting systems to string rewriting systems it follows that the property of being regularity preserving is undecidable for term rewriting systems thus answering another question of gyenizse and vagvolgyi 1997 finally it is shown that it is undecidable in general whether a finite lengthreducing and confluent string rewriting system yields a regular set of normal forms for each regular language

perturbation results for nearly uncoupled markov chains with applications to iterative methods the standard perturbation theory for linear equations states that nearly uncoupled markov chains numcs are very sensitive to small changes in the elements indeed some algorithms such as standard gaussian elimination will obtain poor results for such problems a structured perturbation theory is given that shows that numcs usually lead to well conditioned problems it is shown that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve these structured error bounds a variant of gaussian elimination due to grassman taksar and heyman was recently shown by o cinneide to achieve such bounds

density of smooth functions between two manifolds in sobolev spaces abstract we give some results concerning density or non density of smooth functions between two compact manifolds m n and n k in sobolev spaces w 1 p m n n k in particular we study the case n s k

perforated strict pseudospectra of demmel s matrices abstract regarding the pseudospectra of matrices it is easy to derive some conclusions from computer made graphics however those conclusions might be false without further analysis indeed it happened when discussing the presence of a local maximum at the origin for the least singular value of the characteristic matrix of the n n real toeplitz matrix whose first row and first column are 1 b b 2 b n 1 and 1 0 0 0 t respectively which was first taken into account by demmel in this paper we determine the value of a local maximum for the last singular value of the characteristic matrix of such toeplitz matrices in the cases n 3 and n 5 we also specify the point where this local maximum is attained at

derivatives and perturbations of eigenvectors for a matrix bf a z whose entries are complex valued functions of a complex variable z results are presented concerning derivatives of an eigenvector bf x z of bf a z associated with a simple eigenvalue lambda z when bf x z is restricted to satisfy a constraint of the form sigma bf x z 1 where sigma is a rather arbitrary scaling function the differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix finally an application is given which concerns the analysis of a finite markov chain subject to perturbations in the transition probabilities

spectral value sets a graphical tool for robustness analysis in this note we present a method to determine and visualize the set of all complex numbers to which at least one eigenvalue of a matrix a can be shifted by real or complex perturbations of the form a a a d e where d and e are fixed matrices and the unknown disturbance matrix satisfies 0 both real and complex perturbations are studied the graphical method may be considered as an extension of the classical root locus method to multivariable systems and multiparameter perturbations the results are illustrated by various examples

bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications abstract we consider a class of symmetric tridiagonal matrices which may be viewed as perturbations of toeplitz matrices the toeplitz structure is destroyed since two elements on each off diagonal are perturbed based on a careful analysis we derive sharp bounds for the extremal eigenvalues of this class of matrices in terms of the original data of the given matrix in this way we also obtain a lower bound for the smallest singular value of certain matrices some numerical results indicate that our bounds are extremely good

a closure property of regular languages note given a semigroup homomorphism x p y where p y is the powerset of y we show that the inverse image k 1 of a regular set k y is regular too this theorem has several consequences generalizing previously known results

a counterexample for two conjectures about stability one measure of the stability of a matrix a is the distance to the nearest unstable matrix b recently van loan presented an algorithm to compute b which depended on a conjecture about the location of its eigenvalues we provide a counterexample to this conjecture which shows that the algorithm may overestimate the distance to b by an arbitrary amount the same counterexample invalidates another conjecture and algorithm of the author

structured pseudospectra and random eigenvalues problems in vibrating systems this paper introduces the concept of pseudospectra as a generalized tool for uncertainty quantification and propagation in structural dynamics different types of pseudospectra of matrices and matrix polynomials are explained particular emphasis is given to structured pseudospectra for matrix polynomials which offer a deterministic way of dealing with uncertainties for structural dynamic systems the pseudospectra analysis is compared with the results from monte carlo simulations of uncertain discrete systems two illustrative example problems one with probabilistic uncertainty with various types of statistical distributions and the other with interval type of uncertainty are studied in detail excellent agreement is found between the pseudospectra results and monte carlo simulation results

some decision problems about controlled rewriting systems abstract a description of deterministic context free languages by some kind of finitely generated right congruences the sets of relations generating such right congruences are called controlled rewriting systems or c systems has been given previously in the first part of this paper we study several decision problems on c systems namely the confluence problem the equivalence problem the refinement problem the class equivalence problem and the class inclusion problem in the second part of the paper we deduce from the main theorem given previously a characterisation of dcfls by means of finitely generated congruences the sets of relations generating such congruences are now ordinary finite semi thue systems we study three decision problems on finite semi thue systems namely the class equivalence problem the word problem for the syntactic congruence of one class and the partial confluence problem all the problems are investigated in several classes of c systems or semi thue systems for every class we give an answer which may be yes the problem is decidable no the problem is undecidable or eq which means that the problem is recursively equivalent to the equivalence problem for dpda

the undecidability of the semi unification problem abstract the semi unification problem sup is a natural generalization of both first order unification and matching the problem arises in various branches of computer science and logic although several special cases of sup are known to be decidable the problem in general has been open for several years we show that sup in general is undecidable by reducing what we call the boundedness problem of turing machines to sup the undecidability of this boundedness problem is established by a technique developed in the mid 1960s to prove related results about turing machines

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

group inverses of m matrices associated with nonnegative matrices having few eigenvalues abstract we continue here earlier investigations of the structure of group generalized inverses a of singular and irreducible m matrices a ri m where m is an n n nonnegative and irreducible matrix whose spectral radius is r and where m is assumed to belong to one of several special classes of nonnegative matrices we now focus on cases where m has only a few distinct eigenvalues we are particularly interested in instances of such matrices m which cause all or almost all of the off diagonal entries of the corresponding a to be nonpositive we then apply our results to the study of the sign pattern of the off diagonal entries of the group generalized inverse of a ri m where m comes from one of the following families of nonnegative matrices 1 the magic square of order n 4k generated by matlab 2 the doubly regular tournament matrices the latter type is an example of a special type of a nonnegative matrix m for which the group inverse of the associated m matrix satisfies a a t for some 0

componentwise pseudospectrum of a matrix abstract given a nonnegative weight matrix e defining componentwise structured perturbations e e we introduce a componentwise pseudospectrum of a square matrix a as contour sets of the function c a i 1 e where m is the spectral radius of a square matrix m and discuss how to compute it

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

relative error bounds for the lu decomposition via the gth algorithm recently the author showed that the grassmann taksar heyman gth algorithm computes the steady state distribution of a finite state markov chain with low relative error here it is shown that the lu decomposition computed in the course of the gth algorithm also has low relative error the proof requires a refinement of the methods used in the earlier paper

term rewriting restricted to ground terms we study rewriting of ground terms for an arbitrary term rewrite system r over a ranked alphabet we restrict the rewriting relation r to ground terms we introduce the relation r g r t t we show that r g r t t we show that for a given term rewrite system r and a given ground term rewrite system s over a ranked alphabet it is decidable if r g s we show that for a given left linear right ground term rewrite system r over a ranked alphabet it is decidable if there is a ground term rewrite system s over such that r g s

on structured componentwise condition numbers for hamiltonian eigenvalue problems abstract in this paper we introduce the structured componentwise perturbation analysis to hamiltonian eigenvalue problems the explicit expressions for the relative structured componentwise condition number for hamiltonian eigenvalue problems are derived also we will compare the relative structured componentwise condition number with the relative unstructured componentwise condition number and the relative structured normwise condition number respectively we show the superiority of the relative structured componentwise condition number through numerical examples

eigenvalues pseudospectrum and structured perturbations we investigate the behavior of eigenvalues under structured perturbations we show that for many common structures such as complex symmetric toeplitz symmetric toeplitz circulant and others the structured condition number is equal to the unstructured condition number for normwise perturbations and prove similar results for real perturbations an exception are complex skewsymmetric matrices we also investigate componentwise complex and real perturbations here hermitian and skew hermitian matrices are exceptional for real perturbations furthermore we characterize the structured complex and real pseudospectrum for a number of structures and show that often there is little or no significant difference to the usual unstructured pseudospectrum

perron frobenius theory over real closed fields and fractional power series expansions abstract some of the main results of the perron frobenius theory of square nonnegative matrices over the reals are extended to matrices with elements in a real closed field we use the results to prove the existence of a fractional power series expansion for the perron frobenius eigenvalue and normalized eigenvector of real square nonnegative irreducible matrices which are obtained by perturbing a possibly reducible nonnegative matrix further we identify a system of equations and inequalities whose solution yields the coefficients of these expansions for irreducible matrices our analysis assures that any solution of this system yields a fractional power series with a positive radius of convergence

global analysis of a p ginzburg landau energy with radial structure abstract this paper is concerned with the asymptotic behavior of a p ginzburg landau functional with radial structure as parameter goes to zero in the case of p 2 by analyzing the functional globally we show that the singularity of p ginzburg landau energy concentrates on the origin by the fact the singularity can be balanced by some infinitesimal weight we prove that an energy with a proper weight is globally bounded

spectral simplex method we develop an iterative optimization method for finding the maximal and minimal spectral radius of a matrix over a compact set of nonnegative matrices we consider matrix sets with product structure i e all rows are chosen independently from given compact sets row uncertainty sets if all the uncertainty sets are finite or polyhedral the algorithm finds the matrix with maximal minimal spectral radius within a few iterations it is proved that the algorithm avoids cycling and terminates within finite time the proofs are based on spectral properties of rank one corrections of nonnegative matrices the practical efficiency is demonstrated in numerical examples and statistics in dimensions up to 500 some generalizations to non polyhedral uncertainty sets including euclidean balls are derived finally we consider applications to spectral graph theory mathematical economics dynamical systems and difference equations

computing unstructured and structured polynomial pseudospectrum approximations abstract in many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial the sensitivity commonly is described by condition numbers or pseudospectra however the determination of pseudospectra of matrix polynomials is very demanding computationally this paper describes a new approach to computing approximations of pseudospectra of matrix polynomials by using rank one or projected rank one perturbations these perturbations are inspired by wilkinson s analysis of eigenvalue sensitivity this approach allows the approximation of both structured and unstructured pseudospectra computed examples show the method to perform much better than a method based on random rank one perturbations both for the approximation of structured and unstructured i e standard polynomial pseudospectra

robust stability of linear systems described by higher order dynamic equations the stability radius of higher order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices is studied different perturbation norms are considered the aim is to derive robustness criteria that are expressed directly in terms of the original data previous results on robust stability of hurwitz and schur polynomials are extended to monic matrix polynomials for disturbances acting via a uniform input matrix computable formulas are obtained whereas for perturbations with multiple input matrices structured singular values are involved

structured eigenvalue condition numbers and linearizations for matrix polynomials this work is concerned with eigenvalue problems for structured matrix polynomials including complex symmetric hermitian even odd palindromic and anti palindromic matrix polynomials most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size recently linearizations have been classified for which the pencil reflects the structure of the original polynomial a question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations for all structures under consideration we show that this cannot happen if the matrix polynomial is well scaled there is always a structured linearization for which the structured eigenvalue condition number does not differ much this implies for example that a structure preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial

pseudo spectra theory of tensors and tensor polynomial eigenvalue problems abstract this paper is devoted to the extension of the pseudo spectra theory from matrices to tensors based on the definition of an eigenpair of real symmetric tensors and results on the pseudo spectrum of square matrices we first introduce the pseudo spectrum of a complex tensor and investigate its fundamental properties such as its computational interpretations and the link with the stability of its related homogeneous dynamical system we then extend the pseudo spectrum to the setting of tensor polynomial eigenvalue problems we further derive basic structure of the pseudo spectrum for tensor polynomial eigenvalue problems including the symmetry boundedness and number of connected components under suitable mild assumptions finally we discuss the implication of the pseudo spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial

an ode based method for computing the distance of coprime polynomials to common divisibility the problem of computing the distance of two real coprime polynomials to the set of polynomials with a nontrivial greatest common divisor gcd appears in computer algebra signal processing and control theory it has been studied in the literature under the names approximate common divisor varepsilon gcd and distance to uncontrollability existing solution methods use different types of local optimization methods and require a user defined initial approximation in this paper we propose a new method that allows us to include constraints on the coefficients of the polynomials moreover the method proposed in the paper is more robust to the initial approximation than the newton type optimization methods available in the literature our approach consists of two steps 1 reformulate the problem as a problem of determining the structured distance to singularity of an associated sylvester matrix and 2 integrate a system of odes which describes the gradient associated to the functional to be minim

m matrix generalized inverses of m matrices abstract let p be an n n nonnegative irreducible and stochastic matrix and consider the associated singular and irreducible m matrix q i p we show that without further restrictive spectral or combinatorial assumptions on p it is reasonable to expect that the group inverse q of q will be an m matrix only when p lies in a small wedge about a rank 1 nonnegative matrix a wedge whose existence is itself dependent on p we go on to consider the question of when q has a nonzero generalized inverse not necessarily the group inverse which is an m matrix we give examples of all sorts of classes of p for which such is the case but point out that the general problem remains unsolved

on nonsingularity verification of uncertain matrices over a quadratically constrained set nonsingularity of a matrix is investigated in this technical note that depends affinely on several uncertainty matrices whose real and imaginary parts are constrained by some quadratic forms based on some equivalent descriptions for these uncertainty matrices a sufficient condition is derived this condition is given in a linear matrix inequality form and can in principle be efficiently validated the obtained result includes some well known results in systems and control theory as a special case such as the lyapunov equation based stability criterion for linear time invariant systems the d g scaling based upper bound for the mixed structured singular value a sufficient condition for the stability of spatially interconnected systems etc a computationally verifiable condition is also provided that indicates situations under which the aforementioned condition becomes necessary

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

acute perturbation of the group inverse abstract for any n n complex matrix a let a g be the group inverse of a when a is singular a matrix b a e is said to be an acute perturbation of a if e is small and the spectral radius b g b a g a 1 the acute perturbation coincides with the stable perturbation of the group inverse if the matrix b satisfies condition r b n a 0 n b r a 0 which was introduced by castro gonzalez et al 2008 8 furthermore several examples are provided to illustrate the acute perturbation of the group inverse

interconnected systems with uncertain couplings explicit formulae for mu values spectral value sets and stability radii in this paper we study the variation of the spectrum of block diagonal systems under perturbations of compatible block structure with fixed zero blocks at arbitrarily prescribed locations gershgorin type perturbations we derive explicit and computable formulae for the associated mu values the results are then applied to characterize spectral value sets and stability radii for such perturbed systems by specializing our results to the scalar diagonal case the classical eigenvalue inclusion theorems of gershgorin brauer and brualdi are obtained as corollaries moreover it follows that the inclusion regions of brauer and brualdi are optimal for the corresponding perturbation structures

computation of the structured singular value via moment lmi relaxations the structured singular value ssv provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties unfortunately computing the ssv is an np hard problem and the polynomial time algorithms available in the literature are only able to provide except for some special cases upper and lower bounds on the exact value of the ssv in this work we present a new algorithm to compute an upper bound on the ssv in case of mixed real complex uncertainties the underlying idea of the developed approach is to formulate the ssv computation as a nonconvex polynomial optimization problem which is relaxed into a sequence of convex optimization problems through moment based relaxation techniques two heuristics to compute a lower bound on the ssv are also discussed the analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the robust control toolbox in matlab and it provides in most of the cases coincident lower and upper bounds on the structured singular value

on the computation of structured singular values and pseudospectra structured singular values and pseudospectra play an important role in assessing the properties of a linear system under structured perturbations this paper discusses computational aspects of structured pseudospectra for structures that admit an eigenvalue minimization characterization including the classes of real skew symmetric hermitian and hamiltonian perturbations for all these structures we develop algorithms that require o n2 operations per grid point combining the schur decomposition with a lanczos method these algorithms form the basis of a graphical matlab interface for plotting structured pseudospectra 2009 elsevier b v all rights reserved

beam tracking for interference alignment in slowly fading mimo interference channels a perturbations approach under a linear framework in this paper the beam design for signal space interference alignment in slowly fading multiuser multiple input multiple output mimo interference channels is considered based on a linear formulation for interference alignment a predictive beam tracking algorithm is proposed using matrix perturbation theory the proposed algorithm based on a mixture of iteration and update computes interference aligning beamforming vectors at the current time by updating the previous beam vectors based on the channel difference between the two time steps during the predictively updating phase and yields significant reduction in computational complexity compared with existing methods recalculating beams at each time step the tracking performance of the algorithm is analyzed in terms of mean square error and sum rate loss between the predictively updating approach and the recalculating approach and the impact of imperfect channel knowledge is also investigated under the state space channel model numerical results show that the proposed algorithm has almost the same performance as non predictive methods in sum rate thus the proposed algorithm provides a very efficient way to realize interference alignment in a realistic slowly fading mimo channel environment

a note on structured pseudospectra of block matrices in this note we consider the question of equivalence of pseudospectra and structured pseudospectra of block matrices the structures we study are all so called double structures that is the blocks of the given matrix are of the same structure as the block matrix the approach is based on that of non block matrices which are also briefly studied and the use of distance to singularity we also list some open problems and conjectures

first order structured perturbation theory for multiple zero eigenvalues of skew adjoint matrices abstract given a matrix belonging to some class of structured matrices we consider the question of comparing the sensitivity of its eigenvalues under two different kinds of perturbations either unstructured i e arbitrary or structured i e those belonging to the same class of matrices as the unperturbed one in a previous paper kressner et al 2009 13 the authors compared the structured and unstructured condition numbers of possibly multiple eigenvalues for several different matrix and pencil structures only one case was left out of the analysis namely the one where the asymptotic order of perturbed eigenvalues under structured perturbations is different from the asymptotic order under unstructured ones this is precisely the case we consider in the present paper given a matrix which is skew adjoint with respect to a symmetric scalar product and has a zero eigenvalue with a certain jordan structure first order expansions are obtained for the perturbed eigenvalues under structured perturbation as well as bounds on the structured condition number similar results are obtained for structured perturbations of symmetric skew symmetric and palindromic matrix pencils

w1 p convergence of ginzburg landau type minimizers abstract this paper is concerned about the w 1 p convergence for a minimizer u of a ginzburg landau type functional this proof is completed by establishing an important estimation the l loc p estimation of u

a note on structured pseudospectra in this note we study the notion of structured pseudospectra we prove that for toeplitz circulant hankel and symmetric structures the structured pseudospectrum equals the unstructured pseudospectrum we show that this is false for hermitian and skew hermitian structures we generalize the result to pseudospectra of matrix polynomials indeed we prove that the structured pseudospectrum equals the unstructured pseudospectrum for matrix polynomials with toeplitz circulant hankel and symmetric structures we conclude by giving a formula for structured pseudospectra of real matrix polynomials the particular type of perturbations used for these pseudospectra arise in control theory

some undecidability results for non monadic church rosser thue systems abstract in this paper the computational power of finite church rosser thue systems that are non monadic is explored in particular it is shown that there exists a finite non monadic church rosser thue system t such that the set of linear sentences that are true under the interpretation given by t is undecidable this contrasts with the situation for finite church rosser thue systems that are monadic in addition some problems are presented that are decidable for finite monadic church rosser thue systems but that are undecidable in general for finite non monadic church rosser thue systems finally it is shown how degrees of complexity can be encoded in finite non monadic church rosser thue systems

an iterative method for computing robustness of polynomial stability we propose a method for computing the distance of a stable polynomial to the set of unstable ones both in the hurwitz and in the schur case the method is based on the reformulation of the problem as the structured distance to instability of a companion matrix associated to a polynomial we first introduce the structured e pseudospectrum of a companion matrix and write a system of ordinary differential equations which maximize the real part or the absolute value of elements of the structured e pseudospectrum and then exploit the knowledge of the derivative of the maximizers with respect to e to devise a quadratically convergent iteration furthermore we use a variant of the same odes to compute the boundary of structured pseudospectra and compare them to unstructured ones an extension to constrained perturbations is also considered

nonnegative alternating circulants leading to m matrix group inverses abstract let c be the set of all an n n nonnegative irreducible alternating circulant matrices we characterize a subset c of c such that if b c and the perron root of b is r then the group inverse ri b of the singular and irreducible m matrixri b is also an m matrix this is equivalent to the fact that for each such b the perron root at b is a concave function in each of the off diagonal entries the characterization for the case when n is odd presents more difficulties than for the case when n is even so the two cases are treated separately

reductions in tree replacement systems abstract certain kinds of church rosser tree replacement systems reduction systems are studied most results presented here generalize results proved for string rewriting systems by book 1982 the concept of a tree rewriting system with a norm is introduced and studied a new tree reducing machine which reduces every input tree to an irreducible tree is defined as a consequence when the norm under consideration is the size of the tree the number of nodes the word problem for finite church rosser reduction systems is decidable in linear time in the size of the input tree church rosser monadic reduction systems generalizing the monadic thue systems of book 1982 and book et al 1982 are also studied it is shown that every congruence class and every finite union of congruence classes defined by such a system is accepted by a deterministic bottom up tree pushdown automation this new form of a tree automation is briefly studied

losing recognizability we consider the ranked alphabet s consisting of a binary symbol we give a rewrite system r over s such that r effectively preserves recognizability on any ranked alphabet obtained by adding finitely many nullary symbols to s however r does not preserve recognizability on the ranked alphabet obtained by adding one unary and one nullary symbol to s

backward error condition numbers and pseudospectra for the multiparameter eigenvalue problem we define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem mep the pseudospectrum for the mep is defined and characterized we show that the distance from a right definite mep to the closest non right definite mep is related to the smallest unbounded pseudospectrum some numerical results are given

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

an iterative method for computing the pseudospectral abscissa for a class of nonlinear eigenvalue problems we consider the following class of nonlinear eigenvalue problems left sum i 1 m a i p i lambda right v 0 where a 1 ldots a m are given n times n matrices and the functions p 1 ldots p m are assumed to be entire this does not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations our aim is to compute the epsilon pseudospectral abscissa i e the supremum of the real parts of the points in the epsilon pseudospectrum which is the complex set obtained by joining all solutions of the eigenvalue problem under perturbations delta a i i 1 m of norm at most epsilon of the matrices a i i 1 m under mild assumptions guaranteeing the existence of a globally rightmost point of the epsilon pseudospectrum we prove that it is sufficient to restrict the analysis to rank one perturbations of the form delta a i beta i u v where u in mathbb c n and v in mathbb c n wit

erratum addendum differential equations for roaming pseudospectra paths to extremal points and boundary tracking we correct an error that incurred in our paper published in siam j numer anal 49 2011 pp 1194 1209 and give new algorithms for tracking the boundary of the pseudospectrum which are suitable for large sparse matrices