asymptotic solution of the wang uhlenbeck recurrence time problem a langevin particle is initiated at the origin with positive velocity its trajectory is terminated when it returns to the origin in 1945 wang and uhlenbeck posed the problem of finding the joint probability density function pdf of the recurrence time and velocity naming it the recurrence time problem we show that the short time asymptotics of the recurrence pdf is similar to that of the integrated brownian motion solved in 1963 by mckean we recover the long time t ensuremath 3 2 decay of the first arrival pdf of diffusion by solving asymptotically an appropriate variant of mckean s integral equation

a feynman kac formula for anticommuting brownian motion motivated by application to quantum physics anticommuting analogues of wiener measure and brownian motion are constructed the corresponding ito integrals are defined and the existence and uniqueness of solutions to a class of stochastic differential equations is established this machinery is used to provide a feynman kac formula for a class of hamiltonians several specific examples are considered

bouncing ball with a finite restitution chattering locking and chaos we study the dynamic evolution of a bouncing ball on a vibrating platform such as the membrane of a loudspeaker as a function of its coefficient of restitution ensuremath alpha and demonstrate primarily that the presence of chaos in this system is by no means inevitable and its development is by no means obvious indeed we show that generic trajectories starting under experimental conditions terminate in a region of chattering where the memory of earlier dynamics is lost before repeating themselves in a periodic way as a consequence except possibly for the strictly elastic case ensuremath alpha 1 the evolution to chaos via a period doubling route will not be observed our arguments are corroborated by numerical studies concerning especially the divergence of the mean period of generic trajectories as the elastic limit is approached ensuremath alpha ensuremath rightarrow 1

new algorithm for observer error linearization with a diffeomorphism on the outputs in this paper we give the necessary and sufficient conditions that guarantee the existence of a diffeomorphism which allows one to transform a multi output nonlinear dynamical system into a normal observable canonical form in particular we propose an algorithm that permits one to derive such diffeomorphism the main feature of the canonical form is that it is obtained by allowing a diffeomorphism on the outputs and it also allows one to design an observer with linear error dynamics we first consider multi output nonlinear dynamical system without inputs we then extend the results obtained to multi output nonlinear dynamical system with inputs

a non commutative martingale representation theorem for non fock quantum brownian motion abstract a non commutative theory of stochastic integration is constructed in which the integrators are the components of the quantum brownian motion with non unit variance unlike the unit variance fock case there is a kunita watanabe type representation theorem for processes which are martingales with respect to the generated filtration

distribution spaces and a new construction of stochastic processes associated with the grassmann algebra we associate with the grassmann algebra a topological algebra of distributions which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments as well as their derivatives

observer design for mimo non uniformly observable systems the design of high gain observers is usually based on normal forms of observability if the system is observable for every input uniform observability the gain of the observer does not required a solution to differential equation for multiple input multiple output mimo non uniformly observable systems we give here a sufficient condition that the input must satisfy in order to design an observer unlike uniformly observable systems the observer gain of non uniformly observable systems is derived from a lyapunov differential equation

stochastic calculus in superspace ii differential forms supermanifolds and the atiyah singer index theorem for pt i see ibid vol 25 p 447 68 1992 starting with vector bundles over manifolds supermanifolds are constructed whose function algebras correspond to twisted differential forms stochastic calculus for bosonic and fermionic brownian paths is used to provide a geometric construction of brownian paths on these supermanifolds a feynman kac formula for the heat kernel of the laplace beltrami operator is then derived this is used to provide a simple rigorous version of the supersymmetric proofs of the atiyah singer index theorem

fermionic path integration and grassmann brownian motion a grassmann probability theory with anticommuting random variables and stochastic processes is developed using an extension of berezin integration to infinite dimensional spaces a kolmogorov type consistency condition allows integration on spaces of paths in anticommuting space one particular stochastic process grassmann brownian motion is described and the associated measure used to give a path integral formula for the kernel of the evolution operator in fermionic quantum mechanics the fourier mode expansion of grassmann brownian motion is derived

the convex hull for a random acceleration process in two dimensions we compute exactly the mean perimeter l t and the mean area a t of the convex hull of a random acceleration process of duration t in two dimensions we use an exact mapping that relates via cauchy s formulae the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process x t y t to the computation of the extreme value statistics of the associated one dimensional component process x t the latter can be computed exactly for the one dimensional random acceleration process even though the process is non markovian physically our results are relevant in describing the average shape of a semi flexible ideal polymer chain in two dimensions

finite time observer design for nonlinear impulsive systems with impact perturbation this paper investigates the observer design problem of nonlinear impulsive systems with impact perturbation by using the concept of normal form this paper proposes a full order finite time observer which guarantees the finite time convergence independent of the impact perturbation reduced order observer is additionally developed for the proposed normal form an illustrative example is given in order to illustrate the capability of the proposed method

supersymmetry and the atiyah singer index theorem using a recently introduced index for supersymmetric theories we present a simple derivation of the atiyah singer index theorem for classical complexes and itsg index generalization using elementary properties of quantum mechanical supersymmetric systems

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

reduced order dynamic observer error linearisation a natural extension of observer error linearisation this paper introduces the concept of reduced order dynamic observer error linearisation rdoel for multi output systems which is the problem of transforming a nonlinear system into a nonlinear observer canonical form with the aid of auxiliary dynamics the proposed rdoel framework is not only a modified version of dynamic observer error linearisation doel but also a natural extension of observer error linearisation oel we provide three necessary conditions for the rdoel problem and then derive a necessary and sufficient condition described in terms of lie algebras of vector fields furthermore from the result we also give a geometric necessary and sufficient condition for the oel problem which has not yet been completely established in the case where a diffeomorphism on the output of general form is considered

stochastic calculus in superspace i supersymmetric hamiltonians various analytic results which combine fermionic brownian motion with stochastic integration are described and it is shown that a wide class of stochastic differential equations in superspace have solutions such solutions are then used to derive a feynman kac formula for a supersymmetric system in terms of the supercharge whose square is the hamiltonian of the system this is achieved by introducing superpaths parametrized by a commuting and an anticommuting time variable

berezin integrals and poisson processes we show that the calculation of berezin integrals over anticommuting variables can be reduced to the evaluation of expectations of functionals of poisson processes via an appropriate feynman kac formula in this way the tools of ordinary analysis can be applied to berezin integrals and as an example we prove a simple upper bound possible applications of our results are briefly mentioned

dynamics of inelastic collapse we consider a particle randomly accelerated by gaussian white noise on the half line xg0 the collisions of the particle with the wall at x 0 are inelastic the velocities just before and after reflection are related by v f ensuremath mathrm rv i where r is the coefficient of restitution cornell swift and bray have shown that for rl r c e ensuremath ensuremath pi sqrt 3 0 163 there is inelastic collapse i e boundary collisions localize the particle at x 0 the probability that the particle is not yet absorbed at the boundary after a time t decays for long times as t ensuremath ensuremath theta r the exponent ensuremath theta r is calculated exactly we also consider the case of partial survival boundary conditions i e elastic reflection with probability p and absorption with probability 1 ensuremath p and derive the analogous persistence exponent ensuremath varphi p the exact exponents satisfy the swift bray conjecture ensuremath theta r ensuremath varphi r 2 ensuremath theta r

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

observer linearization by output dependent time scale transformations we study the problem of observer linearization for single output dynamical systems in the presence of output dependent time scaling changes an alternative algorithm for the solution of the observer linearization problem is introduced the algorithm employs an exterior calculus approach that provides a simple procedure for the solution of the observer linearization problem by means of an output dependent time scale transformation

path integrals in n 2 supersymmetric quantum mechanics by use of path integrals in n 2 supersymmetric quantum mechanics a formula for the kernel of the supersymmetric time evolution operator is found structurally it is similar to the feynman kac formula used in the path integrals of quantum mechanics crucial to the development of this formula is a generalization of the wiener measure based upon the free supercharges rather than the free hamiltonian

a new reduced order observer normal form for nonlinear discrete time systems this paper presents a new observability normal form for discrete time nonlinear systems this form enables us to design a reduced order observer necessary and sufficient geometrical conditions for the existence of a coordinate change to transform a discrete time nonlinear system into such normal form are given an illustrative example is given to show the effectiveness of our approach

on the area under a continuous time brownian motion till its first passage time the area swept out under a one dimensional brownian motion till its first passage time is analysed using a fokker planck technique we obtain an exact expression for the area distribution for the zero drift case and provide various asymptotic results for the nonzero drift case emphasizing the critical nature of the behaviour in the limit of vanishing drift the results offer important insights into the asymptotic behaviour of a number of discrete models we also provide a succinct derivation for the distribution of the maximum displacement observed during a first passage

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

extended high gain observer design for a class of mimo non uniformly observable systems abstract this paper deals with the design of an extended high gain observer ehgo for a large class of multiple input multiple output mimo non uniformly observable systems indeed the involved system nonlinearities does not assume the usual triangular structure as it is often the case when a high gain approach is adopted this structure constraint relaxation is achieved by an appropriate use of the characteristic indices associated to the state components there are two fundamental features that are worth to be pointed out firstly the gain of the proposed ehgo is provided by an appropriate riccati ordinary differential equation ode secondly the convergence of the observation error is guaranteed under some persistent excitation condition which can be checked on line simulation results are given to illustrate the performance of the proposed ehgo

persistence of a continuous stochastic process with discrete time sampling we introduce the concept of discrete time persistence which deals with zero crossings of a continuous stochastic process x t measured at discrete times t n delta t for a gaussian markov process with relaxation rate mu we show that the persistence no crossing probability decays as rho a n for large n where a exp mu delta t and we compute rho a to high precision we also define the concept of alternating persistence which corresponds to a 1 corresponding to motion in an unstable potential mu 0 there is a nonzero probability of having no zero crossings in infinite time and we show how to calculate it

on constructing nonlinear observers two algorithmic approaches to constructing nonlinear observers currently exist here one of these is improved by finding an explicit solution to the partial differential equations describing the change of state coordinates thereby avoiding expensive bracket computations this simplifies the algorithm and has implications for system identification

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

linearization by generalized input output injection abstract the problem under interest is the linearization of nonlinear mimo systems by generalized input output injection in order to design observers with linear error dynamics the method is based on the study of the structure of the input output differential equations thus the problem is solved as a realization problem in this note one considers the linearization under two kinds of inout output injection in the first case the transformation depends on the output and time derivatives of the input whereas in the second case derivatives of both the input and the output are considered necessary and sufficient conditions are obtained which generalize the ones on standard input output injection linearization

nonlinear observer design by observer error linearization this paper studies the observer design problem by the observer error linearization approach for nonlinear systems with and without inputs necessary and sufficient conditions for the existence of the linearization transformation are derived for nonlinear systems without inputs the conditions are shown to be corrections to an existing result a computation procedure and a different set of necessary and sufficient conditions based on the computation procedure are presented

probability distribution functions for a single particle vibrating in one dimension experimental study and theoretical analysis abstract we consider the form of the rebound velocity 0 particle velocity and height h probability density functions pdfs for the one dimensional motion of a single particle on a sinusoidally oscillating base the motion is considered in the limit of high excitation vibration frequency collision rate experimentally we find that these pdfs are well approximated by p 0 0 0 exp 0 2 a gaussian p exp 2 and a boltzmann type function p h h exp 2 gh where is a constant and g is the acceleration due to gravity we develop an analytical model which accurately predicts the general form for the rebound velocity pdf the other two pdfs are then analytically shown to follow as a consequence scaling laws for the particle granular temperature with peak base velocity and particle base restitution coefficient determined from previous work can also be predicted from the pdf a fine scale spiky structure in the rebound velocity pdf is found using numerical simulations to be a consequence of resonance phenomena between the particle and vibrating base good agreement between scaling laws from the theory and simulation is found but insufficient data is obtainable to derive accuracy exponents experimentally

extended output depending normal form this paper concerns the design of a nonlinear observer through a transformation of a nonlinear system into an observer form that supports a high gain observer sufficient geometrical condition is deduced to guarantee the existence of change of coordinates allowing the transformation of a nonlinear system into the proposed normal form the result is highlighted by studying the susceptible exposed infected and recovered seir model

high gain observer for a class of non triangular systems this paper presents a high gain observer for a class of mimo nonlinear systems involving some uncertainties the latter is particularly composed of cascade subsystems where each subsystem is associated with a subset of the output variables and assumes a triangular dependence on its own state variables and may depend on the state variables of all other subsystems the main contribution consists in extending the available results to allow more interconnections between the subsystems of fundamental interest it is shown that the underlying observation error exponentially converges to zero in the absence of uncertainties moreover the observation error can be made as small as desired by properly specifying the observer design parameter in the case where uncertainties are considered

local observer for infinitesimally observable nonlinear systems in this article we present a local observer based on infinitesimal observability along trajectories the gain of this observer derives from a dynamical lyapunov equation the inputs for which the observer converges are those which render the system observable at each point of the limit of the unknown trajectory infinitesimal observability at infinity along trajectories the performance of the proposed observer is illustrated with an application to dc machine

partial observer normal form for nonlinear system in this paper we investigate the estimation problem for a class of partially observable nonlinear systems for the proposed partial observer normal form ponf necessary and sufficient conditions are deduced to guarantee the existence of a change of coordinates which can transform the studied system into the proposed ponf examples are provided to illustrate the effectiveness of the proposed results

verifiable conditions for multioutput observer error linearizability the verifiable necessary and sufficient conditions for the multioutput observer error linearization problem have not been found over the last three decades in this technical note we give the unverifiable necessary and sufficient conditions which are more specific than the ones in the literature using these conditions a verifiable sufficient condition also necessary with additional restriction on the problem is derived since our proofs are constructive a desired state transformation can also be found in the theorem also our algorithmic conditions can be easily implemented by a matlab programming

inelastic collapse of a ball bouncing on a randomly vibrating platform a theoretical study is undertaken of the dynamics of a ball which is bouncing inelastically on a randomly vibrating platform of interest are the distributions of the number of flights n f and the total time ensuremath tau c until the ball has effectively collapsed i e coalesced with the platform in the strictly elastic case both distributions have power law tails characterized by exponents that are universal i e independent of the detail of the platform noise distribution however in the inelastic case both distributions have exponential tails p n f ensuremath sim mathrm exp ensuremath ensuremath theta 1 n f and p ensuremath tau c ensuremath sim mathrm exp ensuremath ensuremath theta 2 ensuremath tau c the decay exponents ensuremath theta 1 and ensuremath theta 2 depend continuously on the coefficient of restitution and are nonuniversal however as one approaches the elastic limit they vanish in a manner which turns out to be universal an explicit expression for ensuremath theta 1 is provided for a particular case of the platform noise distribution

observer with linear error dynamics for nonlinear multi output systems based upon nonlinear state coordinate transformations and a condensed dual brunovsk form for multi output systems a method for designing nonlinear state observers with linear error dynamics is presented necessary and sufficient conditions are provided for identifying the class of nonlinear systems for which state observers with linear error dynamics can be obtained

graded manifolds supermanifolds and infinite dimensional grassmann algebras a new approach to superdifferentiable functions of grassmann variables is developed which avoids ambiguities in odd derivatives this is used to give an improved definition of supermanifold over a finite dimensional grassmann algebra a natural embedding of super manifolds over grassmann algebras with increasing number l of generators is developed and thus a limit asl tends to infinity is possible a correspondence between graded manifolds and supermanifolds is constructed extending results of 5 and 8