multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

linear switching systems with slow growth of trajectories abstract we prove the existence of positive linear switching systems continuous time whose trajectories grow to infinity but slower than a given increasing function this implies that unlike the situation with linear ode the maximal growth of trajectories of linear systems may be arbitrarily slow for systems generated by a finite set of matrices this phenomenon is proved to be impossible in dimension 2 while in all bigger dimensions the sublinear growth may occur the corresponding examples are provided and several open problems are formulated

on the asymptotic regularity of a family of matrices abstract in this paper we consider bounded families f of complex n n matrices we give sufficient conditions under which the sequence k f 1 k k 1 where k f is the supremum of the spectral radii of all possible products of k matrices chosen in f is convergent to its supremum f the so called generalized spectral radius of f we also illustrate a possible practical application

a hormander type multiplier theorem for multilinear operators abstract in this paper we prove a hormander type multiplier theorem for multilinear operators as a corollary we can weaken the regularity assumption for multilinear fourier multipliers to assure the boundedness

parametric and nonparametric identification of linear systems in the presence of nonlinear distortions a frequency domain approach this paper studies the asymptotic behavior of nonparametric and parametric frequency domain identification methods to model linear dynamic systems in the presence of nonlinear distortions under some general conditions for random multisine excitations in the first part a related linear dynamic system rlds approximation to the nonlinear system nls is defined and it is shown that the differences between the nls and the rlds can be modeled as stochastic variables with known properties in the second part a parametric model for the rlds is identified convergence in probability of this model to the rlds is proven a function of dependency is defined to detect and separate the presence of unmodeled dynamics and nonlinear distortions and to bound the bias error on the transfer function estimate

matrix semigroups with constant spectral radius abstract multiplicative matrix semigroups with constant spectral radius c s r are studied and applied to several problems of algebra combinatorics functional equations and dynamical systems we show that all such semigroups are characterized by means of irreducible ones each irreducible c s r semigroup defines walks on euclidean sphere all its nonsingular elements are similar in the same basis to orthogonal we classify all nonnegative c s r semigroups and arbitrary low dimensional semigroups for higher dimensions we describe five classes and leave an open problem on completeness of that list the problem of algorithmic recognition of c s r property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones

on sets of eigenvalues of matrices with prescribed row sums and prescribed graph abstract motivated by a work of boros brualdi crama and hoffman we consider the sets of i possible perron roots of nonnegative matrices with prescribed row sums and associated graph and ii possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries to characterize the set of perron roots or possible eigenvalues of matrices in these classes we introduce following an idea of al pin elsner and van den driessche the concept of row uniform matrix which is a nonnegative matrix where all nonzero entries in every row are equal furthermore we completely characterize the sets of possible perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study extending known results to the reducible case we derive new sharp bounds on the set of eigenvalues or perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries in the last section of the paper a new constructive proof of the camion hoffman theorem is given

a gap result for the norms of semigroups of matrices let be a matrix norm on md c and let a be a finite set of matrices in md c we define mn a to be the maximum norm of a product of n elements of a we show that there is a gap in the possible growth of mn a showing that mn a grows either at least exponentially or is bounded by a polynomial in n of degree at most d 1 moreover we show that the growth is bounded by a polynomial if and only if every element of the semigroup generated by a has all of its eigenvalues on or inside the unit circle

on the complexity of computing the capacity of codes that avoid forbidden difference patterns some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed the maximal number of n bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n the coefficient of the exponent is the capacity of the forbidden patterns in this paper new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy the computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given subsequently a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be np hard when the sets of forbidden patterns are defined over an extended set of symbols finally the existence of extremal norms is proved for any set of matrices arising in the capacity computation based on this result a second capacity approximating algorithm is proposed the usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately

on the spectral radius of 0 1 matrices we determine the maximum spectral radius for 0 1 matrices with k2 andk2 1 1 s respectively and for symmetric 0 1 matrices with zero trace and e k21 s graphs with e edges in all cases equality is characterized

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

the greedy strategy for optimizing the perron eigenvalue we address the problems of minimizing and of maximizing the spectral radius over a compact family of non negative matrices those problems being hard in general can be efficiently solved for some special families we consider the so called product families where each matrix is composed of rows chosen independently from given sets a recently introduced greedy method works very fast however it is applicable mostly for strictly positive matrices for sparse matrices it often diverges and gives a wrong answer we present the selective greedy method that works equally well for all non negative product families including sparse ones for this method we prove a quadratic rate of convergence and demonstrate its efficiency in numerical examples the numerical examples are realised for two cases finite uncertainty sets and polyhedral uncertainty sets given by systems of linear inequalities in dimensions up to 2000 the matrices with minimal maximal spectral radii in product families are found within a few iterations applications to dynamical systems and to the graph theory are considered

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

maximal graphs and graphs with maximal spectral radius abstract a maximal graph is a connected graph with degree sequence not majorized by the degree sequence of any other graph relationships between three different sequences of integers that describe maximal graphs are given recursive and explicit forms of the characteristic polynomial of the adjacency matrix of a maximal graph are derived a correspondence between maximal graphs and connected graphs with stepwise adjacency matrices shows that among all connected graphs with n vertices and e edges the graph with maximal spectral radius is a maximal graph such maximal graphs are identified for certain values of n and e

on codes with local joint constraints we study the largest number of sequences with the property that any two sequences do not contain specified pairs of patterns we show that this number increases exponentially with the length of the sequences and that the exponent or capacity is the logarithm of the joint spectral radius of an appropriately defined set of matrices we illustrate a new heuristic for computing the joint spectral radius and use it to compute the capacity for several simple collections the problem of computing the achievable rate region of a collection of codes is introduced and it is shown that the region may be computed via a similar analysis

on the marginal instability of linear switched systems abstract stability properties for continuous time linear switched systems are at first determined by the largest lyapunov exponent associated with the system which is the analogue of the joint spectral radius for the discrete time case the purpose of this paper is to provide a characterization of marginally unstable systems i e systems for which the lyapunov exponent is equal to zero and there exists an unbounded trajectory and to analyze the asymptotic behavior of their trajectories our main contribution consists in pointing out a resonance phenomenon associated with marginal instability in the course of our study we derive an upper bound of the state at time t which is polynomial in t and whose degree is computed from the resonance structure of the system we also derive analogous results for discrete time linear switched systems

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

technical communique estimation of the risk for an unstable behaviour of feedback systems in the presence of nonlinear distortions a criterion to verify experimentally the stability of a nonlinear system captured in a feedback loop is proposed the basic idea is to split the output power in coherent linearly related to the input and noncoherent power the remaining power a nonlinear power gain measuring the sensitivity of the noncoherent output power to input variations is introduced using the small gain theorem it is possible to check the local stability of the feedback for the actual class of excitation signals the risk is estimated that the feedback system would become unstable for another realization of the experiment using extreme value statistics

the hormander multiplier theorem ii the bilinear local l 2 case we use wavelets of tensor product type to obtain the boundedness of bilinear multiplier operators on mathbb r n times mathbb r n associated with hormander multipliers on mathbb r 2n with minimal smoothness we focus on the local l 2 case and we obtain boundedness under the minimal smoothness assumption of n 2 derivatives we also provide counterexamples to obtain necessary conditions for all sets of indices

a maximum principle for the stability analysis of positive bilinear control systems with applications to positive linear switched systems we consider a continuous time bilinear control system bcs with metzler matrices each entry in the transition matrix of such a system is nonnegative making the positive orthant an invariant set of the dynamics motivated by the stability analysis of positive linear switched systems plsss we define a control as optimal if for a fixed final time it maximizes the spectral radius of the transition matrix our main result is a first order necessary condition for optimality in the form of a maximum principle mp the proof of this mp combines the standard needle variation with a basic result from the perron frobenius theory of nonnegative matrices we describe several applications of this mp to the stability analysis of plsss under arbitrary switching

on infinite words determined by l systems abstract a deterministic l system generates an infinite word if each word in its derivation sequence is a prefix of the next yielding as a limit we generalize this notion to arbitrary l systems via the concept of prefix languages a prefix language is a language l such that for all x y l x is a prefix of y or y is a prefix of x every infinite prefix language determines a single infinite word where c is a class of l systems e g 0l dt0l we denote by c the class of infinite words determined by the prefix languages in c this allows us to speak of infinite 0l words infinite dt0l words etc we categorize the infinite words determined by a variety of l systems showing that the whole hierarchy collapses to just three distinct classes of infinite words pd0l d0l and cd0l our results are obtained with the help of a pumping lemma which we prove for t0l systems

hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non negative matrices abstract recently blondel nesterov and protasov proved 1 2 that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non negative matrices with independent row column uncertainty we show that this result can be obtained as a simple consequence of the so called hourglass alternative used in 3 by the author and his companions to analyze the minimax relations between the spectral radii of matrix products axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property which includes the sets of non negative matrices with independent row uncertainty this class of matrices supplemented by the zero and identity matrices forms a semiring with the minkowski operations of addition and multiplication of matrix sets which gives means to construct new sets of non negative matrices possessing the finiteness property for the generalized and the lower spectral radii

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

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

the maximal eigenvalue of 0 1 matrices with prescribed number of ones abstract we determine the maximum spectral radius for 0 1 matrices with m 2 l ones for l 2 m 2 m 3 for all m and for a fixed l and m m l similar results are obtained for symmetric 0 1 matrix with zero diagonal in all cases equality is characterized

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

the hormander multiplier theorem for multilinear operators in this paper we provide a version of the mikhlin hormander multiplier theorem for multilinear operators in the case where the target space is l p for pe 1 this extends a recent result of tomita 15 who proved an analogous result for p 1

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

on asymptotic properties of matrix semigroups with an invariant cone recently several research efforts showed that the analysis of joint spectral characteristics of sets of matrices can be simplified when these matrices share an invariant cone we prove new results in this direction we prove that the joint spectral subradius is continuous in the neighborhood of sets of matrices that leave an embedded pair of cones invariant we show that both the averaged maximal spectral radius as well as the maximal trace where the maximum is taken on all the products of the same length t converge towards the joint spectral radius when t increases provided that the matrices share an invariant cone and additionally one of them is primitive

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

l 2 times l 2 rightarrow l 1 l2 l2 l1 boundedness criteria we obtain a sharp l 2 times l 2 rightarrow l 1 boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function in terms of the l q integrability of this function precisely we show that boundedness holds if and only if q 4 we discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions our second result is an optimal l 2 times l 2 rightarrow l 1 boundedness criterion for bilinear operators associated with multipliers with l infty derivatives this result provides the main tool in the proof of the first theorem and is also manifested in terms of the l q integrability of the multiplier the optimal range is q 4 which in the absence of plancherel s identity on l 1 should be compared to q infty in the classical l 2 rightarrow l 2 boundedness for linear multiplier operators