countable abelian group actions and hyperfinite equivalence relations an equivalence relation e on a standard borel space is hyperfinite if e is the increasing union of countably many borel equivalence relations e n where all e n equivalence classs are finite in this article we establish the following theorem if a countable abelian group acts on a standard borel space in a borel manner then the orbit equivalence relation is hyperfinite the proof uses constructions and analysis of borel marker sets and regions in the space 2 mathbb z omega this technique is also applied to a problem of finding borel chromatic numbers for invariant borel subspaces of 2 mathbb z n

baire measurable paradoxical decompositions via matchings abstract we show that every locally finite bipartite borel graph satisfying a strengthening of hall s condition has a borel perfect matching on some comeager invariant borel set we apply this to show that if a group acting by borel automorphisms on a polish space has a paradoxical decomposition then it admits a paradoxical decomposition using pieces having the baire property this strengthens a theorem of dougherty and foreman who showed that there is a paradoxical decomposition of the unit ball in r 3 using baire measurable pieces we also obtain a baire category solution to the dynamical von neumann day problem if a is a nonamenable action of a group on a polish space x by borel automorphisms then there is a free baire measurable action of f 2 on x which is lipschitz with respect to a

stability and stabilization of continuous time switched linear systems this paper addresses two strategies for the stabilization of continuous time switched linear systems the first one is of open loop nature trajectory independent and is based on the determination of a minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function to be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition including chattering is expressed being nonconvex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

recovery of jumps and singularities of an unknown potential from limited data in dimension 1 we prove that in dimension 1 the jumps and singularities of an unknown potential appearing in the schrodinger equation can be recovered using the born approximation the result is based on an accurate determination of the first nonlinear term of the born series we shall also work in wider function space than the previous publications on this topic numerical examples illustrate the feasibility of this technique

measurable circle squaring laczkovich proved that if bounded subsets a and b of r k have the same non zero lebesgue measure and the box dimension of the boundary of each set is less than k then there is a partition of a into finitely many parts that can be translated to form a partition of b here we show that it can be additionally required that each part is both baire and lebesgue measurable as special cases this gives measurable and translation only versions of tarski s circle squaring and hilbert s third problem

adaptive control of piecewise linear systems the state tracking case nonlinear controlled systems at multiple operating points are modeled as piecewise linear systems where changes in operating points are modeled as switches between constituent linearized systems this note studies the adaptive state feedback for state tracking control problem for such systems piecewise linear reference model systems are used for generating desired state trajectories and their stability properties are studied adaptive state feedback control schemes are developed and their stability and tracking performance are analyzed and evaluated by simulation examples it is shown that exponential tracking performance can be achieved if the reference input is sufficiently rich and the switches are sufficiently slow

robust stability and stabilization of linear switched systems with dwell time sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty a lyapunov function in quadratic form which is non increasing at the switching instants is assigned to each subsystem during the dwell time this function varies piecewise linearly in time after switching occurs it becomes time invariant afterwards this function leads to asymptotic stability conditions for the nominal set of subsystems that can be readily extended to the case where these subsystems suffer from polytopic type parameter uncertainties the method proposed is then applied to stabilization via state feedback both for the nominal and the uncertain cases

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

on parameter convergence in adaptive control it is well known that the parameter error as well as the model plant mismatch error in a model reference adaptive scheme tends exponentially to zero iff a certain sufficient richness condition holds for signals inside the time varying plant control loop in this paper we give conditions on the reference signal the exogenous input to the adaptive loop namely that it have as many spectral lines as there are unknown parameters in order to guarantee parameter convergence

hybrid model reference adaptive control of piecewise affine systems this paper is concerned with the derivation of a model reference adaptive control mrac scheme for multimodal piecewise affine pwa and piecewise linear systems the control allows the plant to track asymptotically the states of a multimodal piecewise affine or smooth reference model the reference model can be characterized by a number and geometry of phase space regions that can be entirely different from those of the plant numerical simulations on a set of representative examples confirm the theoretical derivation and proof of stability

a nonconservative lmi condition for stability of switched systems with guaranteed dwell time ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems several methods have been proposed in the literature to address this problem but unfortunately they provide sufficient conditions only this technical note proposes the use of homogeneous polynomial lyapunov functions in the non restrictive case where all the subsystems are hurwitz showing that a sufficient condition can be provided in terms of an lmi feasibility test by exploiting a key representation of polynomials several properties are proved for this condition in particular that it is also necessary for a sufficiently large degree of these functions as a result the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well some examples illustrate the proposed approach

an inverse born approximation for the general nonlinear schrodinger operator on the line this work deals with the inverse scattering problems for the one dimensional schrodinger equation of the most general form with general nonlinearity f and with a real parameter k under some assumptions on f we prove that all singularities and jumps of f x u0 where u0 x k eikx can be recovered by the reflection coefficient with arbitrary large k

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

technical communique switched controllers and their applications in bilinear systems in this paper switched controllers are designed for a class of nonlinear processes the nonlinear processes together with the switched controllers constitute a type of switched nonlinear systems an invariant principle is presented for such switched nonlinear systems the invariant principle and the switched controllers are used to globally stabilize a class of bilinear systems that are not open loop neutrally stable

adaptive asymptotic tracking control of uncertain time driven switched linear systems this technical note establishes a novel result for adaptive asymptotic tracking control of uncertain switched linear systems the result exploits a recently proposed stability condition for switched systems in particular a time varying positive definite lyapunov function is used to develop a novel piecewise continuous model reference adaptive law and a dwell time switching law in contrast with previous research where asymptotic tracking was possible only in the presence of a common lyapunov function for the reference models in this work asymptotic tracking is shown in a more general setting additionally in the presence of persistence of excitation the controller parameter estimation errors will converge to zero asymptotically the main contribution of this work consists in establishing a symmetry between adaptive control of classical non switched linear systems and adaptive control of switched linear systems a practical example with an electro hydraulic system is adopted to illustrate the results

borel chromatic numbers we study in this paper graph coloring problems in the context of descriptive set theory we consider graphs g x r where the vertex set x is a standard borel space i e a complete separable metrizable space equipped with its algebra of borel sets and the edge relation r x 2 is definable i e borel analytic co analytic etc a borel n coloring of such a graph where 1 n n 0 is a borel map c x y with card y n such that xry c x c y if such a borel coloring exists we define the borel chromatic number of g in symbols x b g to be the smallest such n otherwise we say that g has uncountable borel chromatic number in symbols x b g n 0

partial recovery of potentials in generalized nonlinear schrodinger equations on the line the inverse born approximation method is applied to inverse scattering problems for two one dimensional nonlinear schrodinger equations we prove that all jumps and singularities of the unknown function are recovered from the born function sequence in particular one can locate any bounded interval on the line the result is based on the accurate determination of the first nonlinear term of the born sequence