a variation embedding theorem and applications fractional sobolev spaces also known as besov or slobodetski spaces arise in many areas of analysis stochastic analysis in particular we prove an embedding into certain q variation spaces applications include a new route to a regularity result by kusuoka for stochastic differential equations integration against besov paths a regularity criterion for rough paths and a new regularity result for cameron martin paths associated to fractional brownian motion

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

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

p variation of strong markov processes let t t 0 t be a strong markov process with values in a complete separable metric space x and with transition probability function ps t x dy 0 s t t x x for any h 0 t and a 0 consider the function h a sup ps t x y x y a x x 0 s t s h t it is shown that a certain growth condition on h a as a 0 and h stays fixed implies the almost sure boundedness of the p variation of t where p depends on the rate of growth

general rough integration levy rough paths and a levy kintchine type formula we consider rough paths with jumps in particular the analogue of lyons extension theorem and rough integration are established in a jump setting offering a pathwise view on stochastic integration against cadlag processes a class of levy rough paths is introduced and characterized by a sub ellipticity condition on the left invariant diffusion vector fields and a certain integrability property of the carnot caratheodory norm with respect to the levy measure on the group using hunt s framework of lie group valued levy processes examples of levy rough paths include a standard multi dimensional levy process enhanced with a stochastic area as constructed by d williams the pure area poisson process and brownian motion in a magnetic field an explicit formula for the expected signature is given

euler estimates for rough differential equations abstract we consider controlled ordinary differential equations and give new estimates for higher order euler schemes our proofs are inspired by recent work of a m davie who considers first and second order schemes in order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group such euler estimates have powerful applications by a simple limit argument they apply to rough path differential equations rdes in the sense of t lyons and hence also to stochastic differential equations driven by brownian motion or other random rough paths with sufficient integrability in the context of the latter we obtain strong remainder estimates in stochastic taylor expansions a la azencott ben arous castell and platen although our findings appear novel even in the case of driving brownian motion our main insight is the genuine rough path nature of quantitative remainder estimates in stochastic taylor expansions there are several other applications of which we discuss in detail l q convergence in lyons universal limit theorem and moment control of rde solutions

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

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

rank one characterization of joint spectral radius of finite matrix family abstract in this paper we study the joint generalized spectral radius of a finite set of matrices in terms of its rank one approximation by singular value decomposition in the first part of the paper we show that any finite set of matrices with at most one element s rank being greater than one satisfies the finiteness property under the framework of invariant extremal norm formula for the computation of joint generalized spectral radius for this class of matrix family is derived based on that in the second part we further study the joint generalized spectral radius of finite sets of general matrices through constructing rank one approximations in terms of singular value decomposition and some new characterizations of joint generalized spectral radius are obtained several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach

angular momentum theory umbral calculus and combinatorics abstract the theory of the unitary irreducible representations of the unitary group su 2 is reviewed with the objective of demonstrating the fundamental role that the umbral calculus and combinatorics play for the physicist the study of the group su 2 is synonymous with the theory of angular momentum in quantum theory and kronecker products of irreducible unitary representations comprise the mathematical apparatus for building composite systems from simpler constituents the clebsch gordon coefficients which are essential to the binary theory of composite systems are shown to have an umbral calculus origin macmahon s master theorem is shown to be the basic result for generating the mathematical quantities needed for bringing comprehension to the properties of composite systems

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

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

criteria for the stability of the finiteness property and for the uniqueness of barabanov norms a set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product the extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research in this article we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set but also for all sufficiently nearby sets of equal cardinality we also prove a theorem giving conditions under which the barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar and show that in certain cases these conditions are also persistent under small perturbations

optimal norms and the computation of joint spectral radius of matrices the notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix the finiteness conjecture fc claims that among the infinite products made from the elements of a given finite set of matrices there is a certain periodic product made from the repetition of the optimal product whose rate of growth is maximal fc has been disproved in this paper it is conjectured that fc is almost always true and an algorithm is presented to verify the optimality of a given product the algorithm uses optimal norms as a special subset of extremal norms several conjectures related to optimal norms and non decomposable sets of matrices are presented the algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi resolution analyses and wavelets the results of related numerical experiments are presented

differential equations driven by rough paths with jumps abstract we develop the rough path counterpart of ito stochastic integration and differential equations driven by general semimartingales this significantly enlarges the classes of ito forward stochastic differential equations treatable with pathwise methods a number of applications are discussed

the reconstruction theorem in besov spaces abstract the theory of regularity structures 9 sets up an abstract framework of modelled distributions generalising the usual holder functions and allowing one to give a meaning to several ill posed stochastic pdes a key result in that theory is the so called reconstruction theorem it defines a continuous linear operator that maps spaces of modelled distributions into the usual space of distributions in the present paper we extend the scope of this theorem to analogues to the whole class of besov spaces b p q with non integer regularity indices we then show that these spaces behave very much like their classical counterparts by obtaining the corresponding embedding theorems and schauder type estimates

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

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

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

finiteness property of pairs of 2 2 sign matrices via real extremal polytope norms this paper deals with the joint spectral radius of a finite set of matrices we s ay that a set of matrices has the finiteness property if the maximal rate of growth in the multiplicative semigroup it generates is given by the powers of a finite product here we address the problem of establishing the finiteness properties of p airs of 2 2 sign matrices such problem is related to the conjecture that pairs of sign ma trices fulfil the finiteness property for any dimension this would imply by a recent result by j ungers and blondel that finite sets of rational matrices fulfil the finiteness property wh ich would be very important property in terms of the computation of the joint spectral radius the finiteness property of n n sign matrices remains open but the knowledge of its validity in the 2 2 case gives the hope to solve it by using some induction argument on the dimension as a main tool of our proof we make use of a procedure to compute a so called real polytope extremal norm for the set in particular we present an algorithm which under some suitable assumptions is able to check if a certain product in the multiplicative semig roup is spectrum maximizing for pairs of sign matrices we develop the computations exactly and hence are able to prove analytically the finiteness property on the other hand the algorithm c an be used in a floating point arithmetic and provide a general tool for approximating the join t spectral radius of a set of matrices

corrigendum addendum to sets of matrices all infinite products of which converge abstract this corrigendum addendum supplies corrected statements and proofs of some results in our paper appearing in linear algebra appl 161 1992 227 263 these results concern special kinds of bounded semigroups of matrices it also reports on progress on the topics of this paper made in the last eight years

criteria of stability for continuous time switched systems by using liao type exponents for a given continuous time switched system its stability is determined by i the structure of the system ii the admissible switching set and iii the duration time in each subsystem both ii and iii are stability characterizations of switching dynamics for continuous time switched systems in this paper we introduce a new type of characteristic exponent the liao type exponent which is computable to capture the stability feature of continuous time switched systems we obtain two criteria of asymptotic exponential stability for linear and nonlinear cases by estimating the corresponding liao type exponents respectively the proposed approach is constructive and the obtained results allow existence of switching constraints several examples are provided to illustrate theoretical outcomes of the paper

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

convergence en loi des suites d integrales stochastiques sur l espace mathbb d 1 de skorokhod for sequences of stochastic integrals k n x n functional limit theorems are presented these results hold under simple natural conditions

ramification of rough paths the stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended chen s multiplicative property involves the durr connes kreimer coproduct on rooted trees this turns out to be the natural setting for a non geometric theory of rough paths

rough differential equations driven by signals in besov spaces rough differential equations are solved for signals in general besov spaces unifying in particular the known results in holder and p variation topology to this end the paracontrolled distribution approach which has been introduced by gubinelli imkeller and perkowski 24 to analyze singular stochastic pdes is extended from holder to besov spaces as an application we solve stochastic differential equations driven by random functions in besov spaces and gaussian processes in a pathwise sense

characterization of joint spectral radius via trace the joint spectral radius for a bounded collection of the square matrices with complex entries and of the same size is characterized by the trace of matrices this characterization allows us to give some estimates concerning the computation of the joint spectral radius

an algorithm for finding extremal polytope norms of matrix families in this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered we present an algorithm which under some suitable assumptions is able to check if a certain product in the multiplicative semigroup is spectrum maximizing the algorithm proceeds by attempting to construct a suitable extremal norm for the family namely a complex polytope norm as examples for testing our technique we first consider the set of two 2 dimensional matrices recently analyzed by blondel nesterov and theys to disprove the finiteness conjecture and then a set of 3 dimensional matrices arising in the zero stability analysis of the 4 step bdf formula for ordinary differential 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 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

an efficient lower bound for the generalized spectral radius of a set of matrices the generalized spectral radius gsr is a fundamental concept in studying the regularity of compactly supported wavelets here we describe an efficient method for estimating a lower bound for the gsr let mq be the set of all q q matrices with complex entries suppose a0 am 1 is a collection of m matrices in mq let ln be the set of all products of length n of the elements of define n maxa ln a in where a is the spectral radius of a the generalized spectral radius of is then lim supn n the standard method for estimating from below and at level n is to calculate the spectral radii of all mn products in ln and select the largest here we use three elementary theorems from linear algebra combinatorics and number theory to show that the same result can be obtained with no more than mnn matrix calculations

on the accuracy of the ellipsoid norm approximation of the joint spectral radius the joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set this quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate we introduce in this paper an approximation hat rho that is based on ellipsoid norms that can be computed by convex optimization and that is such that the joint spectral radius belongs to the interval hat rho sqrt n hat rho where n is the dimension of the matrices we also provide a simple approximation for the special case where the entries of the matrices are non negative in this case the approximation is proved to be within a factor at most m m is the number of matrices of the exact value

almost sure stability of discrete time switched linear systems a topological point of view in this paper we study the stability of discrete time switched linear systems via symbolic topology formulation and the multiplicative ergodic theorem a sufficient and necessary condition for mu a almost sure stability is derived where mu a is the parry measure of the topological markov chain with a prescribed transition 0 1 matrix a the obtained mu a almost sure stability is invariant under small perturbations of the system the topological description of stable processes of switched linear systems in terms of hausdorff dimension is given and it is shown that our approach captures the maximal set of stable processes for linear switched systems the obtained results cover the stochastic markov jump linear systems where the measure is the natural markov measure defined by the transition probability matrix two examples are provided to illustrate the theoretical outcomes of the paper

computing the joint spectral radius this paper presents algorithms for finding an arbitrarily small interval that contains the joint spectral radius of a finite set of matrices it also presents a numerical criterion for verifying in certain cases that the joint spectral radius is the maximum of the spectral radii of the given matrices error bounds are derived for the case where calculations are done with finite precision and the matrices are not known exactly the algorithms are implemented and applied to estimate holder exponents of the orthonormal wavelets n constructed by daubechies for 3 n 8

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

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

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

random walks and levy processes as rough paths we consider random walks and levy processes in a homogeneous group g for all p 0 we completely characterise almost all g valued levy processes whose sample paths have finite p variation and give sufficient conditions under which a sequence of g valued random walks converges in law to a levy process in p variation topology in the case that g is the free nilpotent lie group over mathbb r d so that processes of finite p variation are identified with rough paths we demonstrate applications of our results to weak convergence of stochastic flows and provide a levy khintchine formula for the characteristic function of the signature of a levy process at the heart of our analysis is a criterion for tightness of p variation for a collection of cadlag strong markov processes

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

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