a refinement of matrosov s theorem for differential inclusions this note presents a refinement of matrosov s theorem for a class of differential inclusions whose set valued map is defined as a closed convex hull of finitely many vector fields this class of systems may arise in the analysis of switched nonlinear systems when stability with arbitrary switching between the given vector fields is considered assuming uniform global stability of a compact set it is shown that uniform global attractivity of the set can be verified by tailoring matrosov functions to individual vector fields this refinement of matrosov s theorem is an extension of the existing matrosov results which may be easier to apply to certain differential inclusions than existing results as demonstrated by an example

a nested matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems a new infinitesimal sufficient condition is given for uniform global asymptotic stability ugas for time varying nonlinear systems it is used to show that a certain relaxed persistency of excitation condition called uniform spl delta persistency of excitation u spl delta pe is sufficient for uniform global asymptotic stability in certain situations u spl delta pe of the right hand side of a time varying differential equation is also shown to be necessary under a uniform lipschitz condition the infinitesimal sufficient condition for ugas involves the inner products of the flow field with the gradients of a finite number of possibly sign indefinite locally lipschitz lyapunov like functions these inner products are supposed to be bounded by functions that have a certain nested or triangular negative semidefinite structure this idea is reminiscent of a previous idea of matrosov who supplemented a lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the lyapunov function is zero for this reason we call the main result a nested matrosov theorem the utility of our results on stability analysis is illustrated through the well known case study of the nonholonomic integrator

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

a phase cell cluster expansion for a hierarchical o 3 4 model the formalism developed in a previous paper is applied to yield a phase cell cluster expansion for a hierarchical b model the field is expanded into modes with specific renormatization group scaling properties the present cluster expansion for a vacuum expectation value is formally the natural factorization of each term in the perturbation expansion into the contribution of modes connected to the variables in the expectation via interactions and that of the complementary set the expectation value is thus realized as a sum of contributions due to finite subsets of the modes we emphasize the following additional features 1 partitions of unity are not used 2 there are essentially no cut offs 3 the expansion is developed directly without an initial need to prove an ultraviolet stability bound the most difficult part of the traditional approach our main interest in the present phase cell cluster expansion is founded in the belief that it may be the right vehicle for proving the existence of a non trivial four dimensional field theory

construction of ym 4 with an infrared cutoff we provide the basis for a rigorous construction of the schwinger functions of the pure su 2 yang mills field theory in four dimensions in the trivial topological sector with a fixed infrared cutoff but no ultraviolet cutoff in a regularized axial gauge the construction exploits the positivity of the axial gauge at large field for small fields a different gauge more suited to perturbative computations is used this gauge and the corresponding propagator depends on large background fields of lower momenta the crucial point is to control in a non perturbative way the combined effect of the functional integrals over small field regions associated to a large background field and of the counterterms which restore the gauge invariance broken by the cutoff we prove that this combined effect is stabilizing if we use cutoffs of a certain type in momentum space we check the validity of the construction by showing that slavnov identities which express infinitesimal gauge invariance do hold non perturbatively

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

analyzing the stability of switched systems using common zeroing output systems this paper introduces the notion of common zeroing output systems czos to analyze the stability of switched systems the concept of czos allows one to verify weak zero state detectability it characterizes a common behavior of any individual subsystem when the output signal for each subsystem is approaching zero heuristically speaking it removes the effect of switching behavior and thus enables one to analyze stability properties in systems with complex switching signals with the help of czos the krasovskii lasalle theorem can be extended to switched nonlinear time varying systems with both arbitrary switching and more general restricted switching cases for switched nonlinear time invariant systems the needed detectability condition is further simplified leading to several new stability results particularly when a switched linear time invariant system is considered it is possible to generate a recursive method which combines a krasovskii lasalle result and a nested matrosov result to find a czos if it exists the power of the proposed czos is demonstrated by consensus problems in literature to obtain a stronger convergence result with weaker conditions

the short distance behavior of φ 4 3 we consider the 3 4 quantum field theory on a torus and study the short distance behavior we reproduce the standard result that the singularities can be removed by a simple mass renormalization for the resulting model we give anl p bound on the short distance regularity of the correlation functions to obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields

construction of the hierarchical φ4 trajectory we study the invariant unstable manifold of the trivial renormalization group fixed point tangent to the 4 vertex in the hierarchical approximation we parametrize it by a running 4 coupling with linear step function the manifold is studied as a fixed point of the renormalization group composed with a flow of the running coupling we present a rigorous construction of it beyond perturbation theory by means of a contraction mapping starting from a perturbative approximant of order seven we obtain a convergent representation in dimensions 2 d 28 9 with certain restrictions the perturbative approximant is logarithmically divergent in d 3 dimensions

nested matrosov function theorem for nonlinear delayed systems abstract in this paper the stability problem for time varying nonlinear delayed systems is investigated an infinite dimensional version of the nested matrosov function theorem is proposed to conclude uniform global asymptotic stability for time varying nonlinear delayed systems such a stability theorem requires a weak lyapunov functional and some auxiliary functionals and the upper bounds of the derivatives on these functionals must satisfy some nested conditions the proposed results can be used to analyze uniform asymptotic stability of time varying delayed systems where lasalle s invariance principle cannot be applied and barbalat s lemma is not sufficient to conclude the uniform attractivity property an example is given to illustrate the effectiveness of the proposed stability theorem

construction of the two dimensional sine gordon model for β 8π we present a rigorous renormalization group construction of the two dimensional massless and massive quantum sine gordon models in finite volume for the range 0 8 we prove analyticity in the coupling constant which implies the convergence of perturbation theory the field correlation functions and their generating functional are analyzed and shown to have the short distance asymptotics of the free field theory in the massive case the bounds are uniform in volume and we also obtain uniform estimates on the long distance decay of correlations

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

uniform asymptotic stability of switched nonlinear time varying systems and detectability of reduced limiting control systems this paper is concerned with the study of both local and global uniform asymptotic stability for switched nonlinear time varying nltv systems through the detectability of output maps with this aim the notion of reduced limiting control systems for switched nltv systems whose switchings verify time state dependent constraints and the concept of weak zero state detectability for those reduced limiting systems are introduced necessary and sufficient conditions for the global uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this detectability property these sufficient conditions in conjunction with the existence of multiple weak lyapunov functions yield a criterion for the global uniform asymptotic stability of families of trajectories of the switched system this criterion can be seen as an extension of the classical krasovskii lasalle theorem an interesting feature of the results is that no dwell time assumptions are made moreover they can be used for establishing the global uniform asymptotic stability of the switched nltv system under arbitrary switchings the effectiveness of the proposed results is illustrated by means of various interesting examples including the stability analysis of a semiquasi z source inverter

asymptotic stability in hybrid systems via nested matrosov functions a theorem on nested matrosov functions is extended to time varying hybrid systems it provides sufficient conditions for uniform global asymptotic stability of a compact set an application to parameter identification with state resets is made and illustrated on an example

a simplified design for strict lyapunov functions under matrosov conditions we construct strict lyapunov functions for broad classes of nonlinear systems satisfying matrosov type conditions our new constructions are simpler than the designs available in the literature we illustrate the practical interest of our designs using a globally asymptotically stable biological model

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

a matrosov theorem for adversarial markov decision processes matrosov s relaxation of lyapunov conditions for uniform global asymptotic stability in time varying systems is extended to stochastic set valued discrete time systems nested matrosov functions are used to give conditions for stability that complement invariance principles for time invariant systems unlike invariance principles matrosov functions also can be applied to general time varying systems

the renormalization group according to balaban ii large fields this is an expository account of balaban s approach to the renormalization group the method is illustrated with a treatment of the ultraviolet problem for the scalar 4 model on a toroidal lattice in dimension d 3 in this paper we control the large field contribution to the partition function

stability and persistent excitation in signal sets persistent excitation pe conditions have been widely used to analyze stability properties of various parameter identification algorithms and to establish uniform global asymptotic stability ugas for a large class of nonlinear time varying systems in order to generalize such conditions to a more general setting a new pe condition is proposed with three basic ingredients a signal set to represent a family of time functions e g trajectories a pseudo distance measure to describe the convergence and some binary relations e g state to output relations closely related to detectability this pe condition is a necessary condition to guarantee ugas under uniform global stability and an integral inequality it becomes a sufficient condition of ugas a novel concept m pair which aims at simplifying the checking of the pe condition is introduced by using m pair it is possible to simplify the structure of the referred signal set in the spirit of the classic krasovskii lasalle theorem and to extend the dimension of the reference signal set similar to the matrosov theorem thus the framework of m pair not only unifies these well known results but also generates more flexibility in checking the pe conditions when applied to nonlinear switched systems three new tools to verify the pe condition are obtained finally an example illustrates that a nonlinear time varying switched system with arbitrary switching can be shown to be ugas without using a common lyapunov function

matrosov theorem for parameterized families of discrete time systems a version of matrosov s theorem for parameterized discrete time time varying systems is presented the theorem is a discrete time version of the continuous time result in loria et al 2002 d persistency of excitation a necessary and sufficient condition for uniform attractivity 2002 submitted for publication our result facilitates controller design for sampled data nonlinear systems via their approximate discrete time models an application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete time plant models is presented

quantum diffusion of the random schrodinger evolution in the scaling limit we consider random schrodinger equations on r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and psi t the solution with initial data psi 0 the space and time variables scale as x sim lambda 2 varkappa 2 text and t sim lambda 2 varkappa text with 0 varkappa varkappa 0 left d right we prove that in the limit 0 the expectation of the wigner distribution of psi t converges weakly to the solution of a heat equation in the space variable x for arbitrary l 2 initial data

stochastic quantization reflection positivity and quantum fields we investigate stochastic quantization as a mathematical tool for quantum field theory we test the method for the free scalar field we find that the usual method of stochastic quantization is incompatible with establishing a hilbert space interpretation for transition probabilities in quantum theory in particular we prove that for any finite stochastic time the standard probability measure violates reflection positivity as a consequence if one desires to use stochastic quantization in constructive quantum field theory one needs to find a more robust procedure than the standard one

a renormalizable field theory the massive gross neveu model in two dimensions the euclidean massive gross neveu model in two dimensions is just renormalizable and asymptotically free thanks to the pauli principle bare perturbation theory with an ultra violet cut off and the correct ansatz for the bare mass is convergent in a disk whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant therefore the theory can be fully constructed in a perturbative way it satisfies the o s axioms and is the borel sum of the renormalized perturbation expansion of the model