the minimal polynomial of a sequence obtained from the componentwise linear transformation of a linear recurring sequence let s s1 s2 sm be a linear recurring sequence with terms in gf qn and t be a linear transformation of gf qn over gf q denote t s t s1 t s2 t sm in this paper we first present counter examples to show that the main result in a m youssef g gong on linear complexity of sequences over gf 2n theoret comput sci 352 2006 288 292 is not correct in general since lemma 3 in that paper is incorrect then we determine the minimal polynomial of t s if the canonical factorization of the minimal polynomial of s without multiple roots is known and thus present the solution to the main problem which was considered in the above paper but incorrectly solved additionally as a special case we determine the minimal polynomial of t s if the minimal polynomial of s is primitive finally we give an upper bound on the linear complexity of t s when t exhausts all possible linear transformations of gf qn over gf q this bound is tight in some cases

fast multipole method for scattering from an arbitrary pec target above or buried in a lossy half space the fast multipole method fmm was originally developed for perfect electric conductors pecs in free space through exploitation of the spectral properties of the free space green s function in the work reported here the fmm is modified for scattering from an arbitrary three dimensional 3 d pec target above or buried in a lossy half space the near terms in the fmm are handled via the original method of moments mom analysis wherein the half space green s function is evaluated efficiently and rigorously through application of the method of complex images the far fmm interactions which employ a clustering of expansion and testing functions utilize an approximation to the green s function dyadic via real image sources and far field reflection dyadics the half space fmm algorithm is validated through comparison with results computed via a rigorous mom analysis further a detailed comparison is performed on the memory and computational requirements of the mom and fmm algorithms for a target in the vicinity of a half space interface

on the limit products of a family of matrices in this paper we direct attention at bounded families of complex n n matrices in order to study their asymptotic behaviour we recall from linear algebra appl 322 2001 162 the concept of limit spectrum maximizing product and show that nondefective families always admit such limit products then we consider defective families in loc cite we proved that for finite families of 2 2 matrices defectivity is equivalent to the existence of defective such limit products this result led us to conjecture the validity of this property also for higher dimensions n 3 here instead by making use of the results obtained by bousch and mairesse j am math soc 15 2002 77 that disproved the well known finiteness conjecture we find some counterexamples to our conjecture in loc cite for all n 3 2002 elsevier science inc all rights reserved

a characterization of switched linear control systems with finite l_ 2 gain motivated by an open problem posed by j p hespanha we extend the notion of barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation we use these tools to prove that the finiteness of the l 2 gain is equivalent for a large set of switched linear control systems to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one

shift register synthesis modulo m the berlekamp algorithm takes a sequence of elements from a field and finds the shortest linear recurrence or linear feedback shift register that can generate the sequence we present an algorithm which generalizes berlekamp s to the case when the elements of the sequence are integers modulo m where m is an arbitrary but known integer details will be published elsewhere

convergence analysis of c2 hermite interpolatory subdivision schemes by explicit joint spectral radius formulas abstract subdivision schemes are popular iterative processes to build graphs of functions curves and surfaces we analyze the 2 point hermite c 2 subdivision scheme introduced by merrien in 26 for the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme in this paper by an explicit computation of the joint spectral radius of such pairs of matrices we determine necessary and sufficient conditions for the scheme to be c 2 convergent whenever it reproduces cubic polynomials in addition we present two one parameter families of convergent subdivision schemes belonging to the class in 26 possessing interesting properties from the shape control point of view

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

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

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

sets of nonnegative matrices without positive products abstract for an arbitrary irreducible set of nonnegative d d matrices we consider the following problem does there exist a strictly positive product with repetitions permitted of those matrices under some general assumptions we prove that if it does not exist then there is a partition of the set of basis vectors of r d on which all given matrices act as permutations moreover there always exists a unique maximal partition with the maximal number of parts possessing this property and the number of parts is expressed by eigenvalues of matrices this generalizes well known results of perron frobenius theory on primitivity of one matrix to families of matrices we present a polynomial algorithm to decide the existence of a positive product for a given finite set of matrices and to build the maximal partition similar results are obtained for scrambling products applications to the study of lyapunov exponents inhomogeneous markov chains etc are discussed

on the zero stability of variable stepsize multistep methods the spectral radius approach in this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients in particular we deal with the zero stability of the 3 step bdf method on grids with variable stepsize for the numerical solution of ivps for odes our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results which give a slight improvement to the best results already known from the literature the success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods

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

construction of highly stable two step w methods for ordinary differential equations we describe the construction of linearly implicit two step w methods of high stage order and desirable stability properties for the numerical solution of stiff differential systems the presented methods are demonstrated to be a and l stable when the stepsize is held constant they also preserve good stability properties in a variable stepsize environment under quite demanding conditions imposed on the stepsize pattern

on the asymptotic properties of a family of matrices in this paper we consider bounded families f of complex n n matrices after introducing the concept of asymptotic order we investigate how the norm of products of matrices behaves as the number of factors goes to infinity in the case of defective families f u sing the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so called generalized spectral radius with reference to the well known finiteness conjecture for finite families we also introduce the concepts of spectrum maximizing product and limit spectrum maximizing product showing that for finite families of 2 2matrices defectivity is equivalent to the existence of defective such limit products 2001 elsevier science inc all rights reserved

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

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

optimal control of discrete time bilinear systems with applications to switched linear stochastic systems abstract this paper aims at characterizing the most destabilizing switching law for discrete time switched systems governed by a set of bounded linear operators the switched system is embedded in a special class of discrete time bilinear control systems this allows us to apply the variational approach to the bilinear control system associated with a mayer type optimal control problem and a second order necessary optimality condition is derived optimal equivalence between the bilinear system and the switched system is analyzed which shows that any optimal control law can be equivalently expressed as a switching law this specific switching law is most unstable for the switched system and thus can be used to determine stability under arbitrary switching based on the second order moment of the state the proposed approach is applied to analyze uniform mean square stability of discrete time switched linear stochastic systems numerical simulations are presented to verify the usefulness of the theoretic results

extremal and barabanov semi norms of a semigroup generated by a bounded family of matrices abstract let s s i i i be an arbitrary family of complex n by n matrices where 1 n let s denote the joint spectral radius of s defined as s lim sup l sup i 1 i l i l s i 1 s i l 1 l which is independent of the norm used here a semi norm on c n is called extremal of s if it satisfies x 0 and x s i s x x x 1 x n c n and i i in this paper using an elementary analytical approach we show that if s is bounded in c n n then there always exists for s an extremal semi norm on c n if additionally s is compact in c n n this extremal semi norm has the barabanov type property i e to any x c n one can find an infinite sequence i n i with x s i 1 s i k s k x for each k 1 as a common starting point this directly implies the fundamental results barabanov s norm theorem berger wang s formula and elsner s reduction theorem

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

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

convergence rate of nonlinear switched systems abstract this paper is concerned with the convergence rate of the solutions of nonlinear switched systems we first consider a switched system which is asymptotically stable for a class of switching signals but not for all switching signals we show that solutions corresponding to that class of switching signals converge arbitrarily slowly to the origin then we consider analytic switched systems for which a common weak quadratic lyapunov function exists under two different sets of assumptions we provide explicit exponential convergence rates for switching signals with a fixed dwell time

minimum time control of boolean networks boolean networks bns are discrete time dynamical systems with boolean state variables bns have recently been attracting considerable interest as computational models for biological systems and in particular gene regulating networks boolean control networks bcns are boolean networks with boolean inputs we consider the problem of steering a bcn from a given state to a desired state in minimal time using the algebraic state space representation assr of bcns we derive several necessary conditions stated in the form of maximum principles mps for a control to be time optimal in the assr every state and input vector is a canonical vector using this special structure yields an explicit state feedback formula for all time optimal controls to demonstrate the theoretical results we develop a bcn model for the genetic switch controlling the lambda phage development upon infection of a bacteria our results suggest that this biological switch is designed in a way that guarantees minimal time respon

stability of positive linear switched systems on ordered banach spaces abstract we provide sufficient criteria for the stability of positive linear switched systems on ordered banach spaces the switched systems can be generated by finitely many bounded operators in infinite dimensional spaces with a general class of order inducing cones in the discrete time case we assume an appropriate interior point of the cone whereas in the continuous time case an appropriate interior point of the dual cone is sufficient for stability this is an extension of the concept of linear lyapunov functions for positive systems to the setting of infinite dimensional partially ordered spaces we illustrate our results with examples

the finite step realizability of the joint spectral radius of a pair of d d matrices one of which being rank one abstract we study the finite step realizability of the joint generalized spectral radius of a pair of real square matrices s 1 and s 2 one of which has rank 1 where 2 d let a denote the spectral radius of a square matrix a then we prove that there always exists a finite length word i 1 i m 1 2 m for some finite m 1 such that s i 1 s i m m sup n 1 max i 1 i n 1 2 n s i 1 s i n n in other words there holds the spectral finiteness property for s 1 s 2 explicit formula for computation of the joint spectral radius is derived this implies that the stability of the switched system induced by s 1 s 2 is algorithmically decidable in this case

stabilisation of discrete time bilinear systems by constant control inputs abstractin this paper the stabilisation problem of discrete time bilinear systems by using constant control inputs is considered it is shown that the spectrum of a matrix derived from the system

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

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

common polynomial lyapunov functions for linear switched systems in this paper we consider linear switched systems dot x t a u t x t x in r n u in u a u u in u compact and the problem of asymptotic stability for arbitrary switching functions uniform with respect to switching uas given a uas system it is always possible to build a common polynomial lyapunov function our main result is that the degree of that common polynomial lyapunov function is not uniformly bounded over all the uas systems this result answers a question raised by dayawansa and martin a generalization to a class of piecewise polynomial lyapunov functions is given

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

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

geometric and asymptotic properties associated with linear switched systems abstract consider a continuous time linear switched system on r n associated with a compact convex set of matrices when it is irreducible and its largest lyapunov exponent is zero there always exists a barabanov norm associated with the system this paper deals with two types of issues a properties of barabanov norms such as uniqueness up to homogeneity and strict convexity b asymptotic behavior of the extremal solutions of the linear switched system regarding issue a we provide partial answers and propose four related open problems as for issue b we establish when n 3 a poincare bendixson theorem under a regularity assumption on the set of matrices we then revisit a noteworthy result of n e barabanov describing the asymptotic behavior of linear switched system on r 3 associated with a pair of hurwitz matrices a a b c t after pointing out a gap in barabanov s proof we partially recover his result by alternative arguments

on the finiteness property for rational matrices in this chapter we analyze a recent conjecture stating that the finiteness property holds for pairs of binary matrices we show that the finiteness property holds for all pairs of binary matrices if and only if it holds for all sets of nonnegative rationalmatrices we provide a similar result for matrices with positive and negative entries we finally prove the conjecture for 2 2 matrices

brief paper explicit construction of a barabanov norm for a class of positive planar discrete time linear switched systems we consider the stability under arbitrary switching of a discrete time linear switched system a powerful approach for addressing this problem is based on studying the most unstable switching law musl if the solution of the switched system corresponding to the musl converges to the origin then the switched system is stable for any switching law the musl can be characterized using optimal control techniques this variational approach leads to a hamilton jacobi bellman equation describing the behavior of the switched system under the musl the solution of this equation is sometimes referred to as a barabanov norm of the switched system although the barabanov norm was studied extensively it seems that there are few examples where it was actually computed in closed form in this paper we consider a special class of positive planar discrete time linear switched systems and provide a closed form expression for a corresponding barabanov norm and a musl the unit circle in this norm is a parallelogram

stability of planar switched systems the linear single input case we study the stability of the origin for the dynamical system dot x t u t ax t 1 u t bx t where a and b are two 2 2 real matrices with eigenvalues having strictly negative real part x in mbox bf r 2 and u 0 infty to 0 1 is a completely random measurable function more precisely we find a coordinates invariant necessary and sufficient condition on a and b for the origin to be asymptotically stable for each function u the result is obtained without looking for a common lyapunov function but studying the locus in which the two vector fields ax and bx are collinear there are only three relevant parameters the first depends only on the eigenvalues of a the second depends only on the eigenvalues of b and the third contains the interrelation among the two systems and it is the cross ratio of the four eigenvectors of a and b in the projective line cp1 in the space of these parameters the shape and the convexity of the region in which there is stability are studied this bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation

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

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

fast multipole representation of green s function for an impedance half space in this paper exact and approximate multipole representations for fields of a horizontal or a vertical infinitesimal dipole above impedance surfaces are derived from the exact image formulation these multipole expansions are appropriate for applications in the method of moments through a formulation known as the fast multipole method unlike the existing approximate formulations these new multipole representations of the green s function for the half space impedance surface problem do not impose any restrictions on the location of source or observation points and are computationally very efficient

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

complex source beam moment method procedure for accelerating numerical integral equation solutions of radiation and scattering problems a new hybrid complex source beam moment method csb mom procedure is presented to accelerate the matrix vector product mvp computation in the iterative solution of moment method integral equation problems the scattering object in this csb mom algorithm is partitioned into groups in a manner similar to the fast multipole method fmm however unlike fmm the interactions between well separated groups are computed by using the complex source beams csbs which represent the radiation from the basis elements in the group the directional properties of csbs allow a fast evaluation for these interactions such that the overall mvp is computed very efficiently the direct solution time and the storage requirement of the csb mom method is numerically shown to be o n3 2 by optimally selecting the number of groups a low memory version of a new hybrid complex source beam moment method csb mom procedure is presented to accelerate the matrix vector product mvp computation in the iterative solution of moment method integral equation problems the scattering object in this csb mom algorithm is partitioned into groups in a manner similar to the fast multipole method fmm however unlike fmm the interactions between well separated groups are computed by using the complex source beams csbs which represent the radiation from the basis elements in the group the directional properties of csbs allow a fast evaluation for these interactions such that the overall mvp is computed very efficiently the direct solution time and the storage requirement of the csb mom method is numerically shown to be o n3 2 by optimally selecting the number of groups a low memory version of the csb mom method is also presented wherein separated group interactions are computed on the fly the csb mom method is also presented wherein separated group interactions are computed on the fly

synthesis and uniqueness of m sequences over gf q sup n as n phase sequences over gf q the construction of m sequences over gf q sup n from known m sequences over gf q is discussed an algorithm is given which generates m sequences over gf q sup n from m sequences over gf q where n phase shifts of known m sequences over gf q are determined by the iterative polynomials it is shown that n shift distinct m sequences over gf q sup n generated by the same m sequences over gf q are unique when the elements of gf q sup n are represented by n tuples of the elements from gf q

stability properties of a class of positive switched systems with rank one difference abstract given a single input continuous time positive system described by a pair a b with a a diagonal matrix we investigate under what conditions there exists a state feedback law u t c x t that makes the resulting controlled system positive and asymptotically stable by this meaning that a b c is metzler and hurwitz in the second part of this note we assume that the state space model switches among different state feedback laws c i i 1 2 p each of them ensuring the positivity and show that the asymptotic stability of this type of switched system is equivalent to the asymptotic stability of all its subsystems while its stabilizability is equivalent to the existence of an asymptotically stable subsystem

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

exact closed form expression of a sommerfeld integral for the impedance plane problem a sommerfeld integral for an impedance half plane is one of the classical problems in electromagnetic theory in this paper the integral is evaluated into two series representations which are expressed in terms of the exponential integral and the lommel function respectively it is shown that the series expansions absolutely converge for any parameters such as distance or surface impedance then based on the lommel function expansion an exact closed form expression of the integral is formulated this expression is written in terms of incomplete weber integrals which are directly related to incomplete cylindrical functions such as incomplete lipschitz hankel integrals additionally a complete asymptotic series of the integral is obtained based on the exponential integral expansion with conventional methods such as steepest descent method it is cumbersome to derive the divergent series the validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

relationships between m sequences over gf q and gf q sup m it is shown that m sequences over gf q sup m of length q sup nm 1 corresponding to primitive polynomials in gf q sup m x of degree n can be generated from known m sequences over gf q of length q sup nm 1 obtained from primitive polynomials in gf q x of degree mn a procedure for generating the m sequences over gf q sup 2 from m sequences over gf q was given which enables the generation of m sequences over gf p sup 2n in addition it was shown that all of the primitive polynomials in gf q sup m x can be obtained from a complete set of the primitive polynomials in gf q x

a note on stability conditions for planar switched systems this article is concerned with the stability problem for the planar linear switched system where the real matrices a 1 a 2 2 2 are hurwitz and u 0 0 1 is a measurable function we give coordinate invariant necessary and sufficient conditions on a 1 and a 2 under which the system is asymptotically stable for arbitrary switching functions u the new conditions unify those given in previous papers and are simpler to be verified since we reduced to study 4 cases instead of 20 most of the cases are analysed in terms of the function

regularity of non stationary subdivision a matrix approach in this paper we study scalar multivariate non stationary subdivision schemes with integer dilation matrix m and present a unifying general approach for checking their convergence and for determining their holder regularity latter in the case m mi m ge 2 m mi m 2 the combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius as an application we prove a recent conjecture by dyn et al on the holder regularity of the generalized daubechies wavelets we illustrate our results with several examples

overlap free words and spectra of matrices overlap free words are words over the binary alphabet a a b that do not contain factors of the form xvxvx where x a and v a we analyze the asymptotic growth of the number u n of overlap free words of length n as n we obtain explicit formulas for the minimal and maximal rates of growth of u n in terms of spectral characteristics the joint spectral subradius and the joint spectral radius of certain sets of matrices of dimension 20x20 using these descriptions we provide new estimates of the rates of growth that are within 0 4 and 0 03 of their exact values the best previously known bounds were within 11 and 3 respectively we then prove that the value of u n actually has the same rate of growth for almost all natural numbers n this average growth is distinct from the maximal and minimal rates and can also be expressed in terms of a spectral quantity the lyapunov exponent we use this expression to estimate it in order to obtain our estimates we introduce new algorithms to compute the spectral characteristics of sets of matrices these algorithms can be used in other contexts and are of independent interest

continuous dependence with respect to the input of trajectories of control affine systems we study the continuous dependence on the input of trajectories of control affine systems belonging to the class c0 m of all systems sigma of the form sigma dot x f 0 x sum i 1 m u i t f i x where f0 fm are continuous vector fields on some open subset of mathbb r n and the control functions belong to l 1 0 t mathbb r m we give a simple necessary and sufficient condition for a control sequence u j j 1 infty to cal t 0 converge to a control u infty i e to be such that for every system sigma in c0 the trajectories generated by the uj converge as j rightarrow infty to the trajectories generated by u infty we also characterize cal t k convergence the concept of control convergence that arises when we use instead of c0 m the class ck m of systems sigma where the fi are of class ck for k ge 1 in the scalar input case and we explain how the analogous characterization for the multi input case fails to be true unless one restricts oneself to the class c k comm m of systems for which the vector fields f0 fm commute as a preliminary we define a topology of trajectory convergence or t convergence on the set of all time varying vector fields omega times i ni x t mapsto f x t in mathbb r n where omega is an open subset of mathbb r n and i is an interval and we study some of its properties this enables us to make the definition of cal t k convergence precise for sequences and more generally for nets by saying that a net u alpha alpha in a in l 1 0 t mathbb r m cal t k converges to a limit u infty if for every system sigma in c0 m the time varying vector fields x t mapsto f 0 x sum i 1 m u i alpha t f i x cal t k converge to x t mapsto f 0 x sum i 1 m u i infty t f i x

the generalized spectral radius theorem an analytic geometric proof abstract let be a bounded set of complex matrices m a 1 a m a i the generalized spectral radius theorem states that where and are defined as follows lim sup m 1 m where m sup a a m with a the spectral radius lim sup m 1 m where m sup a a m with any matrix norm we give an elementary proof based on analytic and geometric tools which is in some ways simpler than the first proof by berger and wang

a note on marginal stability of switched systems in this note we present criteria for marginal stability and marginal instability of switched systems for switched nonlinear systems we prove that uniform stability is equivalent to the existence of a common weak lyapunov function cwlf that is generally not continuous for switched linear systems we present a unified treatment for marginal stability and marginal instability for both continuous time and discrete time switched systems in particular we prove that any marginally stable system admits a norm as a cwlf by exploiting the largest invariant set contained in a polyhedron several insightful algebraic characteristics are revealed for marginal stability and marginal instability

on linear complexity of sequences over gf 2 n in this paper we consider some aspects related to determining the linear complexity of sequences over gf 2n in particular we study the effect of changing the finite field basis on the minimal polynomials and thus on the linear complexity of sequences defined over gf 2n but given in their binary representation let a ai be a sequence over gf 2n then ai can be represented by ai j 0n 1 aij j aij gf 2 where is the root of the irreducible polynomial defining the field consider the sequence b bi whose elements are obtained from the same binary representation of a but assuming a different set of basis say 0 1 n 1 i e bi j 0r 1 aij j we study the relation between the minimal polynomial of a and b

joint spectral radius and holder regularity of wavelets abstract we give preliminary results on the holder exponent of wavelets of compact support in particular we give a nearly complete map of this exponent for the family of four coefficient multiresolution analyses and determine the smoothest one this will resolve two conjectures by colella and heil wavelets of compact support can be generated via infinite products of certain matrices the rate of growth of these products determines the regularity of the wavelet this rate can be determined via joint generalized or common spectral radius of the given set of matrices we outline a method for calculating this radius for a given set of matrices the method relies on guessing the particular finite optimal product which satisfies the finiteness conjecture and exhibits the fastest growth then we generate an optimal unit ball by taking the convex hull of the action of the semigroup of matrices scaled by their joint radius on the invariant ball of the scaled optimal product if this process terminates in a finite number of steps and the convex hull does not grow then the guessed optimal product is confirmed and the joint radius is determined

asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three asymptotic properties of extremal solutions of linear inclusions of order three with zero lyapunov exponent are investigated under certain conditions it is shown that all extremal solutions of such inclusions tend to the same up to a multiplicative factor solution which is central symmetric the structure of the convex set of extremal norm is studied a number of extremal points of this set are described

comprehensive lyapunov functions for linear switching systems abstract for a linear dynamical switching system we analyse maximal asymptotic growth of trajectories depending on the initial point both discrete and continuous time systems in r d are considered we prove the existence of a lyapunov norm in r d with the following property for every invariant linear subspace l r d of the system the restriction of the norm on l provides a tight upper bound for the growth of trajectories on l for this we introduce the concept of the spectral normal form of a family of matrices properties of the comprehensive lyapunov norms are analysed and methods of their construction are discussed

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

complete closed form expression of dyadic green s function and its far and near field approximations for an impedance half plane a closed form expression of the dyadic green s function is formulated for an impedance half plane which is written in terms of the incomplete cylindrical function of poisson form due to the branch cut of the logarithm function that is required to calculate the input argument of the incomplete cylindrical function the closed form representation consists of two formulations since the closed form expression contains a singularity at 0 the small argument expansion of the expression is also derived to rigorously characterize the behavior of the function at 0 the previously reported complete asymptotic expansion for the sommerfeld integral for an impedance half plane is not accurate for practically important cases such as near earth propagation and or when the surface is highly conductive hence in this paper a new complete asymptotic series of the sommerfeld integral are derived for the case that the existing asymptotic series is not accurate the two asymptotic series not only allow efficient numerical computation but also provide more accurate results for virtually all propagation scenarios based on the two asymptotic series the complete asymptotic series of the dyadic green s function is derived all derived formulations are numerically verified and their accuracies are investigated

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 stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem

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

matrix products with constraints on the sliding block relative frequencies of different factors abstract one of fundamental results of the theory of joint generalized spectral radius the berger wang theorem establishes equal ity between the joint and generalized spectral radii of a set of matrices generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with essential difficulties since known proofs of the berger wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products recently x dai 1 proved an analog of the berger wang theorem for the case when factors in matrix products are formed by some markov law we extend the concepts of joint and generalized spectral radii to products of matrices with constraints on the sliding block relative frequencies of occurrences of different factors and prove an analog of the berger wang theorem for this case