arithmetic representations of fundamental groups i let x be a normal algebraic variety over a finitely generated field k of characteristic zero and let ell be a prime say that a continuous ell adic representation rho of pi 1 acute mathrm e text t x bar k is arithmetic if there exists a finite extension k of k and a representation tilde rho of pi 1 acute mathrm e text t x k with rho a subquotient of tilde rho pi 1 x bar k we show that there exists an integer n n x ell such that every nontrivial semisimple arithmetic representation of pi 1 acute mathrm e text t x bar k is nontrivial mod ell n as a corollary we prove that any nontrivial ell adic representation of pi 1 acute mathrm e text t x bar k which arises from geometry is nontrivial mod ell n

bmo spaces associated with semigroups of operators we study bmo spaces associated with semigroup of operators on noncommutative function spaces i e von neumann algebras and apply the results to boundedness of fourier multipliers on non abelian discrete groups we prove an interpolation theorem for bmo spaces and prove the boundedness of a class of fourier multipliers on noncommutative lp spaces for all 1 p with optimal constants in p

john nirenberg inequality and atomic decomposition for noncommutative martingales in this paper we study the john nirenberg inequality for bmo and the atomic decomposition for h1 of noncommutative martingales we first establish a crude version of the column resp row john nirenberg inequality for all 0 p by an extreme point property of lp space for 0 p 1 we then obtain a fine version of this inequality the latter corresponds exactly to the classical john nirenberg inequality and enables us to obtain an exponential integrability inequality like in the classical case these results extend and improve junge and musat s john nirenberg inequality by duality we obtain the corresponding q atomic decomposition for different hardy spaces h1 for all 1 q which extends the 2 atomic decomposition previously obtained by bekjan et al finally we give a negative answer to a question posed by junge and musat about bmo

an operator valued t1 theory for symmetric czos abstract we provide a natural bmo criterion for the l 2 boundedness of calderon zygmund operators with operator valued kernels satisfying a symmetric property our arguments involve both classical and quantum probability theory in the appendix we give a proof of the l 2 boundedness of the commutators r j b whenever b belongs to the bourgain s vector valued bmo space where r j is the j th riesz transform a common ingredient is the operator valued haar multiplier studied by blasco and pott

a combinatorial approach to nonlinear functional expansions an introduction with an example in this paper we present a new approach to causal functionals we introduce combinatorial interpretations of the solutions of nonlinear differential equations with forcing terms this theory parallels the algebraic approach with formal power series in noncommutative variables developed by fliess lamnabhi and lamnabhi lagarrigue this theory makes use of certain combinatorial objects called weighted increasing trees weighted paths and histories we can deduce very efficient algorithms for the computation of the corresponding volterra kernels we present an introduction to our combinatorial theory an example with a nonlinear circuit gives the flavor of our approach the complete proofs and general theory will be discussed in a further paper

on the radius of convergence of interconnected analytic nonlinear input output systems a complete analysis is presented of the radii of convergence of the parallel product cascade and feedback interconnections of analytic nonlinear input output systems represented as fliess operators such operators are described by convergent functional series which are indexed by words over a noncommutative alphabet their generating series are therefore specified in terms of noncommutative formal power series given growth conditions for the coefficients of the generating series for the subsystems the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent in the process of deriving the radius of convergence for the feedback connection it is shown definitively that local convergence is preserved under feedback this had been an open problem in the literature until recently

faa di bruno hopf algebra of the output feedback group for multivariable fliess operators abstract given two nonlinear input output systems written in terms of chen fliess functional expansions i e fliess operators it is known that the feedback interconnected system is always well defined and in the same class an explicit formula for the generating series of a single input single output closed loop system was provided by the first two authors in earlier work via hopf algebra methods this paper is a sequel it has four main innovations first the full multivariable extension of the theory is presented next a major simplification of the basic setup is introduced using a new type of grading that has recently appeared in the literature this grading also facilitates a fully recursive algorithm to compute the antipode of the hopf algebra of the output feedback group and thus the corresponding feedback product can be computed much more efficiently the final innovation is an improved convergence analysis of the antipode operation namely the radius of convergence of the antipode is computed

pseudo localization of singular integrals and noncommutative calderon zygmund theory abstract the weak type 1 1 boundedness of singular integrals acting on matrix valued functions has remained open since the 1980s mainly because the methods provided by the vector valued theory are not strong enough in fact we can also consider the action of generalized calderon zygmund operators on functions taking values in any other von neumann algebra our main tools for its solution are two first the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases l 2 sense outside of the support of the function on which it acts this gives rise to a pseudo localization principle which is of independent interest even in the classical theory second we construct a noncommutative form of calderon zygmund decomposition by means of the recent theory of noncommutative martingales this is a corner stone in the theory as application we obtain the sharp asymptotic behavior of the constants for the strong l p inequalities also unknown up to now our methods settle some basics for a systematic study of a noncommutative calderon zygmund theory

an operator valued tb theorem we establish conditions similar to the tb theorem of david journe and semmes which guarantee the boundedness of an integral transformation t with l x valued kernel on lxp rn where 1 p and x is a banach space with the unconditionality property of martingale differences umd

computable convergence bounds of series expansions for infinite dimensional linear analytic systems and application this paper deals with the convergence of series expansions of trajectories for semi linear infinite dimensional systems which are analytic in state and affine in input a special case of such expansions corresponds to volterra series which are extensively used for the analysis the simulation and the control of weakly nonlinear finite dimensional systems the main results of this paper give computable bounds for both the convergence radius and the truncation error of the series these results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality they are available for distributed and boundary control systems as an illustration these results are applied to an epidemic population dynamic model in this example it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues the relevance of the method is illustrated by simulations

invertibility of switched nonlinear systems this article addresses the invertibility problem for switched nonlinear systems affine in controls the problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state we extend the concept of switch singular pairs introduced recently to nonlinear systems and develop a formula for checking if the given state and output form a switch singular pair a necessary and sufficient condition for the invertibility of switched nonlinear systems is given which requires the invertibility of individual subsystems and the nonexistence of switch singular pairs when all the subsystems are invertible we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs detailed examples are included to illustrate these newly developed concepts

interconnection of nonlinear causal systems for nonlinear systems it is not a mathematically obvious fact that the simplest feedback system yields equations which are solvable even when the gain in the loop is

on two definitions of observation spaces this paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls

bounds for the hilbert transform with matrix a2 weights we establish a new estimate for the hilbert transform in l 2 space endowed with a matrix weight

lie brackets and local controllability a sufficient condition for scalar input systems we develop a general formalism based on exponential lie series which can be used to study control variations we apply the formalism to prove a result conjectured by h hermes that gives a sufficient condition for small time local controllability for single input systems

algebraic differential equations and rational control systems an equivalence is shown between realizability of input output i o operators by rational control systems and high order algebraic differential equations for i o pairs this generalizes to nonlinear systems the equivalence between autoregressive representations and finite dimensional linear realizability

a combinatorial hopf algebra for nonlinear output feedback control systems abstract in this work a combinatorial description is provided of a faa di bruno type hopf algebra which naturally appears in the context of fliess operators in nonlinear feedback control theory it is a connected graded commutative and non cocommutative hopf algebra defined on rooted circle trees a cancellation free forest formula for its antipode is given

commutators in the two weight setting let r be the vector of riesz transforms on mathbb r n and let mu lambda in a p be two weights on mathbb r n 1 p infty the two weight norm inequality for the commutator b r l p mu rightarrow l p lambda is shown to be equivalent to the function b being in a bmo space adapted to mu and lambda this is a common extension of a result of coifman rochberg weiss in the case of both lambda and mu being lebesgue measure and bloom in the case of dimension one

a quantitative condition for the well posedness of interconnected dynamical systems a sufficient condition for the well posedness of interconnected dynamical systems is reported the condition is stated in terms of the uniform instantaneous gains of subsystems and interconnections and includes vidyasagar s condition 8 as a special case the result is useful for testing the validity of the dynamical model of a large scale system which is obtained by combining the models of subsystems

embeddings of vector valued bergman spaces abstract we show that a dyadic version of the carleson embedding theorem for the bergman space extends to vector valued functions and operator valued measures this is in contrast to a result by nazarov treil volberg in the context of the hardy space we also discuss some embeddings for analytic vector valued functions

non causal fliess operators and their shuffle algebra fliess operators as a class of non linear operators have been well studied in several respects they have a well developed realization theory and convenient representations in terms of directed infinite products of exponential lie series their interconnection as subsystems has been studied as has their relationship to rational systems they find applications in such diverse areas as discretization methods for controls systems optimal control neural network analysis and the numerical solution of stochastic differential equations one issue concerning fliess operators however that has received little attention is their possible generalization to the non causal case examples of such operators appear implicitly in the literature addressing hilbert adjoints of causal non linear operators and system inversion for the purpose of output tracking but a general systematic treatment of the subject has not appeared in this paper a non causal extension of a fliess operator is developed with the primary focus being on local convergence continuity the associated shuffle algebra and computing adjoint operators

an extended linearization approach to nonlinear system inversion the inversion problem for nonlinear systems is considered from both the internal viewpoint state equation representations and the external viewpoint volterra series representations in order to circumvent difficulties in the known exact inversion theory a method for obtaining what will be called a partial inverse system is proposed for a given nonlinear system a cascade system typically containing differentiators is found such that the overall nonlinear system has a family of constant operating points with inputs and outputs identical furthermore the linearization of the overall system about each of these constant operating points is the identity system

profinite braid groups galois representations and complex multiplications in this paper we consider two closely correlated subjects one is a pro i analogue of the braid group and the other is a construction of the universal 1 adic power series for complex multiplications of fermat type or equivalently for jacobi sums both arise from and constitute a first step in the study of the canonical representation of the absolute galois group gq gal q q in the outer automorphism group of the profinite fundamental group of pq o 1 oo

wavelets and the angle between past and future the main result the riesz projectionp or equivalently hilbert transformt is bounded in the weighted spacel2 w wherewis a matrix valued weight if and only ifsupi 1 i iw 1 2 1 i iw 1 1 2 wheresupremumis taken over all intervalsi motivation for this problem comes from stationary processes riesz projection is bounded means the angle between past and future of a stationary process with spectral measurewis nonzero in the scalar case the result is the well known hunt muckenhoupt wheeden theorem the main step in our proof is to show that a vector haar system forms an unconditional basis inl2 w as a byproduct of our approach we get some new results about bases of wavelets in weighted spaces in both scalar and vector valued cases

on galois representations arising from towers of coverings of p1 0 1 we propose a new way to describe universally thel adic galois representations associated to each almost pro l tower of etale coverings ofp1 0 1 this generalizes our universal power series for jacobi sums cf i which arises from the tower of fermat curves of degreeln n and contains the case of the tower of modular curves of level 2mln m fixed n as another important special case as a fundamental tool we shall establish and use an almost pro l version of the theorems of blanchfield and of lyndon in fox free differential calculus

generating series for interconnected analytic nonlinear systems given two analytic nonlinear input output systems represented as fliess operators four system interconnections are considered in a unified setting the parallel connection product connection cascade connection and feedback connection in each case the corresponding generating series is produced and conditions for the convergence of the corresponding fliess operator are given in the process an existing notion of a composition product for formal power series has its set of known properties significantly expanded in addition the notion of a feedback product for formal power series is shown to be well defined in a broad context and its basic properties are characterized

a faa di bruno hopf algebra for a group of fliess operators with applications to feedback a faa di bruno type hopf algebra is developed for a group of integral operators known as fliess operators where operator composition is the group product such operators are normally written in terms of generating series over a noncommutative alphabet using a general series expansion for the antipode an explicit formula for the generating series of the compositional inverse operator is derived the result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product that is the generating series for the fliess operator representation of the closed loop system written in terms of the generating series of the fliess operator component systems this formula is employed to provide a proof that local convergence is preserved under feedback

technical communique extending the domain of definition of functional series for nonlinear systems functional series are the basis of much of existing nonlinear system theory the present note extends the well known local series beyond its radius of convergence to describe the entire system trajectory for as long as it exists first steps towards a generalization to systems defined on analytic manifolds are made

characterizations of a 2 matrix power weights abstract in the scalar setting the power functions x for 1 1 are the canonical examples of a 2 weights in this paper we study two types of power functions in the matrix setting with the goal of obtaining canonical examples of a 2 matrix weights we first study type 1 matrix power functions which are n n matrix functions whose entries are of the form a x our main result characterizes when these power functions are a 2 matrix weights and has two extensions to type 1 power functions of several variables we also study type 2 matrix power functions which are n n matrix functions whose eigenvalues are of the form a x we find necessary conditions for these to be a 2 matrix weights and give an example showing that even nice functions of this form can fail to be a 2 matrix weights

paraproducts and hankel operators of schatten class via p john nirenberg theorem we give an interpolation free proof of the known fact that a dyadic paraproduct is of schatten von neumann class sp if and only if its symbol is in the dyadic besov space bpd our main tools are a product formula for paraproducts and a p john nirenberg theorem due to rochberg and semmes we use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols using an averaging technique by petermichl we retrieve peller s characterizations of scalar and vector hankel operators of schatten von neumann class sp for 1 p we then employ vector techniques to characterise little hankel operators of schatten von neumann class answering a question by bonami and peloso furthermore using a bilinear version of our product formula we obtain characterizations for boundedness compactness and schatten class membership of products of dyadic paraproducts

a unified approach to generating series for mixed cascades of analytic nonlinear input output systems the primary goal is to describe in a unified fashion the generating series for the cascade connection of any two analytic nonlinear input output systems in particular it will be shown that a single general definition of a composition product can be formulated in terms of formal power series to describe any possible cascade connection of analytic integral operators fliess operators and analytic functions provided that the composite system is well defined and resides in one of these two classes in each case the radius of convergence of the interconnection is computed

failure of the matrix weighted bilinear carleson embedding theorem abstract we prove failure of the natural formulation of a matrix weighted bilinear carleson embedding theorem featuring a matrix valued carleson sequence as well as products of norms for the embedding we show that assuming an a 2 weight is also not sufficient indeed a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding we prove the optimal dependence of the embedding on this quantity we show that any improvement of a recent matrix weighted bilinear embedding featuring a scalar carleson sequence and inner products instead of norms must fail in particular replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products any formulation using norms even in the presence of a scalar carleson sequence must fail as a positive result we prove the so called matrix weighted redundancy condition in full generality

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

general siso takagi sugeno fuzzy systems with linear rule consequent are universal approximators takagi sugeno ts fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice virtually all the ts fuzzy systems use linear rule consequent at present there exist no results qualitative or quantitative to answer the fundamentally important question that is especially critical to ts fuzzy systems as fuzzy controllers and models are ts fuzzy systems with linear rule consequent universal approximators if the answer is yes then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration in this paper we provide answers to these questions for a general class of single input single output siso fuzzy systems that use any type of continuous input fuzzy sets ts fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case we first constructively prove that this general class of siso ts fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove by utilizing the weierstrass approximation theorem that the general ts fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision furthermore we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound an illustrative numerical example is provided

a weak type inequality for non commutative martingales and applications we prove a weak type 1 1 inequality for square functions of non commutative martingales that are simultaneously bounded in l 2 and l 1 more precisely the following non commutative analogue of a classical result of burkholder holds there exists an absolute constant k 0 such that if m is a semi finite von neumann algebra and mn 1 1 is an increasing filtration of von neumann subalgebras of m then for any given martingale x xn 1 1 that is bounded in l 2 m l 1 m adapted to mn 1 1 there exist two martingale difference sequences a an 1 1 and b bn 1 1 with dxn an bn for every n 1 1 x n 1 aan 1 2

a t1 theorem for integral transformations with operator valued kernel we consider generalized calderon zygmund operators whose kernel k x y takes values in l x continuous linear operators on the banach space x and satis es variants of the classical standard estimates involving rboundedness which has recently become a crucial notion in connection with operator valued singular integrals boundedness criteria in the spirit of the t1 theorem of david and journe are proved for such operators on the bochner spaces lp rn x where 1 p and x is a umd space for some results x is also required to have pisier s property in the special case t1 t 1 0 we obtain an essentially complete analogue of the scalar valued theorem we also provide su cient conditions for the general case but they are stronger in general than the necessary t1 t 1 bmo type conditions although they reduce to them in the classical situation a counterexample is given to show that the natural necessary conditions are not su cient in general

discrete time approximations of fliess operators a convenient way to represent a nonlinear input output system in control theory is via a chen fliess functional expansion or fliess operator the general goal of this paper is to describe how to approximate fliess operators with iterated sums and to provide accurate error estimates for two different scenarios one where the series coefficients are growing at a local convergence rate and the other where they are growing at a global convergence rate in each case it is shown that the error estimates are asymptotically achievable for certain worst case inputs the paper then focuses on the special case where the operators are rational i e they have rational generating series and thus are realizable in terms of bilinear ordinary differential state equations in particular it is shown that a discretization of the state equation via a kind of euler approximation coincides exactly with the discrete time fliess operator approximator of the continuous time rational operator

bilinear system interconnections and generating series of weighted petri nets this paper has three objectives the first objective is to provide a simpler proof concerning a sufficient condition due to ferfera under which bilinearity is preserved for cascade interconnections the next objective is to provide a specific counterexample to show that this condition does not apply to the feedback connection the final objective is to show that the well known correspondence between rational series and formal power series recognized by weighted finite state automata can be generalized to produce a correspondence between the generating series of cascaded and feedback connected bilinear systems and a class of weighted petri nets

fliess operators on lp spaces convergence and continuity fliess operators as input output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory in the classical analysis of these operators certain growth conditions on the coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed upperbound in some emerging applications however it is more natural to consider other classes of inputs in this paper lp function spaces are considered in particular it is shown that the classic growth conditions also provide su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 t where t is bounded and p 1 in addition stronger global growth conditions are given that apply even for the case where t is unbounded when the coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

weighted interpolation in paley wiener spaces and finite time controllability this paper considers the solution of weighted interpolation problems in model subspaces of the hardy space h 2 that are canonically isometric to paley wiener spaces of analytic functions a new necessary and sufficient condition is given on the set of interpolation points which guarantees that a solution in h 2 can be transferred to a solution in a model space the techniques used rely on the reproducing kernel thesis for hankel operators which is given here with an explicit constant one of the applications of this work is to the finite time controllability of diagonal systems specified by a c 0 semigroup c 2010 elsevier inc all rights reserved

left inversion of analytic nonlinear siso systems via formal power series methods given a single input single output siso system f and a function y in the range of f the left inversion problem is to determine an input u such that y f u the goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where f is an analytic mapping in the sense that it has a convergent chen fliess functional expansion and y is a real analytic function in particular it will be shown that given a certain condition on the generating series c of f a corresponding unique analytic u can always be determined via operations on formal power series the condition on c turns out to be equivalent to having a well defined relative degree when f has an input affine analytic state space realization with finite dimension but the method is applicable even when f does not have such a realization the technique is demonstrated on four examples including a continuous stirred chemical reactor

a study of the matrix carleson embedding theorem with applications to sparse operators abstract in this paper we study the dyadic carleson embedding theorem in the matrix weighted setting we provide two new proofs of this theorem which highlight connections between the matrix carleson embedding theorem and both maximal functions and h 1 bmo duality along the way we establish boundedness results about maximal functions associated to matrix a 2 weights and duality results concerning h 1 and bmo sequence spaces in the matrix setting as an application we then use this carleson embedding theorem to show that if s is a sparse operator then the operator norm of s on l 2 w satisfies s l 2 w l 2 w w a 2 3 2 for every matrix a 2 weight w

finite tripod variants of i om on ihara s question oda matsumoto conjecture in this note we introduce and prove a wide generalization and sharpening of ihara s question oda matsumoto conjecture for short i om that leads to a quite concrete topological combinatorial description of absolute galois groups in particular of mathrm gal mathbb q mathrm aut overline mathbb q as envisioned by grothendieck in his esquisse d un programme

p johnson homomorphisms and pro p groups abstract we propose an approach to study non abelian iwasawa theory using the idea of johnson homomorphisms in low dimensional topology we introduce arithmetic analogues of johnson homomorphisms maps called the p johnson homomorphisms maps associated to the zassenhaus filtration of a pro p galois group over a z p extension of a number field we give their cohomological interpretation in terms of massey products in galois cohomology