the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

absorbing boundary in one dimensional anomalous transport in this paper we study the space time probability distribution q x t of a random walk subject to an absorbing boundary at the origin x 0 for motion controlled by l evy flights and l evy walks characterized by the exponent ensuremath gamma we find that the method of images usually applicable to brownian motion may break down for l evy processes we calculate the distribution q x t to be at xg0 after time tg0 having started at the origin assuming that the boundary is effective at time tg0 we show that q x t depends on the details of the underlying process q x t ensuremath sim mathit x ensuremath gamma 2 mathit t 1 1 ensuremath gamma 1 ensuremath le ensuremath gamma ensuremath le 2 for small x while total survival is independent of the spatial realization of motion and displays a universal behavior we also discuss the related smoluchowski boundary condition problem

modeling record breaking stock prices abstract we study the statistics of record breaking events in daily stock prices of 366 stocks from the standard and poor s 500 stock index both the record events in the daily stock prices themselves and the records in the daily returns are discussed in both cases we try to describe the record statistics of the stock data with simple theoretical models the daily returns are compared to i i d rvs and the stock prices are modeled using a biased random walk for which the record statistics are known these models agree partly with the behavior of the stock data but we also identify several interesting deviations most importantly the number of records in the stocks appears to be systematically decreased in comparison with the random walk model considering the autoregressive ar 1 process we can predict the record statistics of the daily stock prices more accurately we also compare the stock data with simulations of the record statistics of the more complicated garch 1 1 model which in combination with the ar 1 model gives the best agreement with the observational data to better understand our findings we discuss the survival and first passage times of stock prices on certain intervals and analyze the correlations between the individual record events after recapitulating some recent results for the record statistics of ensembles of n stocks we also present some new observations for the weekly distributions of record events

differential algebraic systems as differential equations on manifolds based on the theory of differential equations on manifolds existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications both the autonomous and nonautonomous case are considered moreover a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower dimensional manifolds this class includes systems for which the application of ode solvers is known to lead to difficulties finally some solution approach based on continuation techniques is outlined

record statistics for biased random walks with an application to financial data we consider the occurrence of record breaking events in random walks with asymmetric jump distributions the statistics of records in symmetric random walks was previously analyzed by majumdar and ziff phys rev lett 101 050601 2008 and is well understood unlike the case of symmetric jump distributions in the asymmetric case the statistics of records depends on the choice of the jump distribution we compute the record rate p n c defined as the probability for the nth value to be larger than all previous values for a gaussian jump distribution with standard deviation that is shifted by a constant drift c for small drift in the sense of c n 1 2 the correction to p n c grows proportional to arctan n and saturates at the value c 2 for large n the record rate approaches a constant which is approximately given by 1 2 c exp c 2 2 2 for c 1 these asymptotic results carry over to other continuous jump distributions with finite variance as an application we compare our analytical results to the record statistics of 366 daily stock prices from the standard poor s 500 index the biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices and after detrending the number of upper records is in good agreement with the symmetric random walk however the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

record statistics for multiple random walks we study the statistics of the number of records r n n for n identical and independent symmetric discrete time random walks of n steps in one dimension all starting at the origin at step 0 at each time step each walker jumps by a random length drawn independently from a symmetric and continuous distribution we consider two cases i when the variance ensuremath sigma 2 of the jump distribution is finite and ii when ensuremath sigma 2 is divergent as in the case of l evy flights with index 0l ensuremath mu l2 in both cases we find that the mean record number ensuremath langle r n n ensuremath rangle grows universally as ensuremath sim ensuremath alpha n sqrt n for large n but with a very different behavior of the amplitude ensuremath alpha n for ng1 in the two cases we find that for large n ensuremath alpha n ensuremath approx 2 sqrt mathrm ln n independently of ensuremath sigma 2 in case i in contrast in case ii the amplitude approaches to an n independent constant for large n ensuremath alpha n ensuremath approx 4 sqrt ensuremath pi independently of 0l ensuremath mu l2 for finite ensuremath sigma 2 we argue and this is confirmed by our numerical simulations that the full distribution of r n n sqrt n ensuremath 2 sqrt mathrm ln n sqrt mathrm ln n converges to a gumbel law as n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty in case ii our numerical simulations indicate that the distribution of r n n sqrt n converges for n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty to a universal nontrivial distribution independently of ensuremath mu we discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the standard poor s 500 index

breaking records in the evolutionary race abstract we explore some aspects of the relationship between biological evolution processes and the mathematical theory of records for eigen s quasispecies model with an uncorrelated fitness landscape we show that the evolutionary trajectories traced out by a population initially localized at a randomly chosen point in sequence space can be described in close analogy to record dynamics with two complications first the increasing number of genotypes that become available with increasing distance from the starting point implies that fitness records are more frequent than for the standard case of independent identically distributed random variables second fitness records can be bypassed which strongly reduces the number of genotypes that take part in an evolutionary trajectory for exponential and gaussian fitness distributions this number scales with sequence length n as n and it is of order unity for distributions with a power law tail this is in strong contrast to the number of records which is of order n for any fitness distribution

levy flights on the half line we study the probability distribution function pdf of the position of a levy flight of index 0 2 in the presence of an absorbing wall at the origin the solution of the associated fractional fokker planck equation can be constructed using a perturbation scheme around the brownian solution corresponding to 2 as an expansion in e 2 we obtain an explicit analytical solution exact at the first order in e which allows us to conjecture the precise asymptotic behavior of this pdf including the first subleading corrections for any careful numerical simulations as well as an exact computation for 1 confirm our conjecture

painleve test and exact similarity solutions of a class of nonlinear diffusion equations for 1 1 dimensional diffusion equations where the diffusion coefficient is an arbitrary power of the dependent variable a painleve test is performed on both the partial differential equation and the similarity reductions the invariance of the painleve property under a large class of similarity reductions is shown for the reductions with the painleve property the explicit similarity solutions are given it is demonstrated how results of a painleve analysis can be used to obtain further exact similarity solutions even in cases without the painleve property

correlations of record events as a test for heavy tailed distributions a record is an entry in a time series that is larger or smaller than all previous entries if the time series consists of independent identically distributed random variables with a superimposed linear trend record events are positively negatively correlated when the tail of the distribution is heavier lighter than exponential here we use these correlations to detect heavy tailed behavior in small sets of independent random variables the method consists of converting random subsets of the data into time series with a tunable linear drift and computing the resulting record correlations

record statistics and persistence for a random walk with a drift we study the statistics of records of a one dimensional random walk of n steps starting from the origin and in presence of a constant bias c at each time step the walker makes a random jump of length eta drawn from a continuous distribution f eta which is symmetric around a constant drift c we focus in particular on the case were f eta is a symmetric stable law with a levy index 0 mu leq 2 the record statistics depends crucially on the persistence probability which as we show here exhibits different behaviors depending on the sign of c and the value of the parameter mu hence in the limit of a large number of steps n the record statistics is sensitive to these parameters c and mu of the jump distribution we compute the asymptotic mean record number after n steps as well as its full distribution p r n we also compute the statistics of the ages of the longest and the shortest lasting record our exact computations show the existence of five distinct regions in the c 0 mu leq 2 strip where these quantities display qualitatively different behaviors we also present numerical simulation results that verify our analytical predictions

correlations between record events in sequences of random variables with a linear trend the statistics of records in sequences of independent identically distributed random variables is a classic subject of study one of the earliest results concerns the stochastic independence of record events recently records statistics beyond the case of i i d random variables have received much attention but the question of independence of record events has not been addressed systematically in this paper we study this question in detail for the case of independent non identically distributed random variables specifically for random variables with a linearly moving mean we find a rich pattern of positive and negative correlations and show how their asymptotics is determined by the universality classes of extreme value statistics

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

persistence in nonequilibrium surface growth persistence probabilities of the interface height in 1 1 and 2 1 dimensional atomistic solid on solid stochastic models of surface growth are studied using kinetic monte carlo simulations with emphasis on models that belong to the molecular beam epitaxy mbe universality class both the initial transient and the long time steady state regimes are investigated we show that for growth models in the mbe universality class the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady state regimes for the mbe universality class the positive and negative persistence exponents in the steady state are found to be 0 66 0 02 and 0 78 0 02 respectively in 1 1 dimensions and 0 76 0 02 and 0 85 0 02 respectively in 2 1 dimensions the noise reduction technique is applied on some of the 1 1 dimensional models in order to obtain accurate values of the persistence exponents we show analytically that a relation between the steady state persistence exponent and the dynamic growth exponent found earlier to be valid for linear models should be satisfied by the smaller of the two steady state persistence exponents in the nonlinear models our numerical results for the persistence exponents are consistent with this prediction we also find that the steady state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state the dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

universal first passage properties of discrete time random walks and levy flights on a line statistics of the global maximum and records abstract in these lecture notes i will discuss the universal first passage properties of a simple correlated discrete time sequence x 0 0 x 1 x 2 x n up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently at each time step a random jump length from an arbitrary symmetric and continuous distribution it includes e g the levy flights i will focus on the statistics of two extreme observables associated with the sequence i its global maximum and the time step at which the maximum occurs and ii the number of records in the sequence and their ages i will demonstrate how the universal statistics of these observables emerge as a consequence of pollaczek spitzer formula and the associated sparre andersen theorem

leapover lengths and first passage time statistics for levy flights exact results for the first passage time and leapover statistics of symmetric and one sided levy flights lfs are derived lfs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one sided lfs and surprisingly with an index alpha 2 for symmetric lfs the first passage time distribution scales like a power law with an index 1 2 as required by the sparre andersen theorem for symmetric lfs whereas one sided lfs have a narrow distribution of first passage times the exact analytic results are confirmed by extensive simulations

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

statistics of superior records we study statistics of records in a sequence of random variables these identical and independently distributed variables are drawn from the parent distribution the running record equals the maximum of all elements in the sequence up to a given point we define a superior sequence as one where all running records are above the average record expected for the parent distribution we find that the fraction of superior sequences s n decays algebraically with sequence length n s n n in the limit n interestingly the decay exponent is nontrivial being the root of an integral equation for example when is a uniform distribution with compact support we find 0 450265 in general the tail of the parent distribution governs the exponent we also consider the dual problem of inferior sequences where all records are below average and find that the fraction of inferior sequences i n decays algebraically albeit with a different decay exponent i n n we use the above statistical measures to analyze earthquake data language en

lie transformations similarity reduction and solutions for the nonlinear madelung fluid equations with external potential the application of lie group methods to a system of coupled nonlinear partial differential equations representing what is usually called a madelung fluid is shown the generating operators of the transformation group that depends on five arbitrary group constants will be constructed and all subclasses of systems of ordinary differential equations derived by similarity reduction will be presented in tabular form two subclasses of physical interest are investigated in detail and the similarity solutions are compared with solutions found earlier by the application of inverse scattering transform techniques to the cubic nonlinear schrodinger equation similarity solutions for the madelung equations with linear external potential x f0x are presented

symmetries and integrability of generalized diffusion reaction equations using lie group methods and the painleve test the authors analyse nonlinear diffusion reaction equations ut d u u f u with power law diffusion coefficients d approximately uv and arbitrary nonlinear reaction terms f u which have a wide spectrum of applications in many areas of science the lie group based similarity method leads to a classification of the reaction terms according to its symmetry properties with the help of the adjoint representation the optimal system of similarity reductions is calculated to check the integrability of the partial differential equation the existence of generalized lie backlund symmetries is investigated apart from three known cases no further cases with third order symmetries exist examining the integrability of the second order ordinary differential equations resulting from the reductions only a few parameter combinations can be found for which the painleve property is given however they are able to construct unknown integrals of motion for a much larger range of parameter values from the integrals exact similarity solutions may be derived this is demonstrated by examples corresponding to the important moving wave reduction

when nonlinear differential equations are equivalent to linear differential equations a necessary and sufficient condition is established for the existence of a 1 1 transformation of a system of nonlinear differential equations to a system of linear equations the obtained theorems enable one to construct such transformations from the invariance groups of differential equations the hodograph transformation the legendre transformation and lie s transformation of the monge ampsre equation are shown to be special cases noninvertible transformations are also considered examples include burgers equation a nonlinear diffusion equation and the liouville equation introduction in this work we study transformations mapping nonlinear differen tial equations to linear differential equations in a 1 1 manner based upon the group analysis of differential equations we obtain necessary and sufficient conditions for the existence of such transformations the established theorems not only allow us to determine the existence of the transformations but also enable us to actually construct these transformations from invariance groups of the nonlinear equations in the following analysis two types of transformations are considered 1 the invariance groups of differential equations and 2 the mappings which transform nonlinear differential equations to linear differential equations theorems will be proved based upon the following observations clearly if there exists a 1 1 mapping between any two differential equations it must inject properties of one equation into the other including their invariance properties for this reason the often ignored fact that any linear differential equation admits an invariance group related to the superpo sition principle becomes very significant the generator of this particular group depends upon an arbitrary solution of the linear equation it follows then that any nonlinear equation transformable to a linear equation by a 1 1 mappingmust admit an invariance group whose generator depends upon an arbitrary solution of some linear differential equation the idea of comparing invariance groups of differential equations in the search of mappings connecting the equations was first used by bluman in his study of burgers equation 1 and it was applied to the study of the mappings of one dimensional linear parabolic equations to the heat equation 2 in the first section of this paper some basic properties of various transformations are summarized the theorems are presented in the second section the examples in the third section include the hodograph transformation the legendre transformation lie s transformation of themonge ampre equation and the equation ux uxx uyy 0 in the last section we examine properties of noninvertible mappings of burgers equation the diffusion equation u 2ux uy 0 and the liouville equation u y e u respectively into linear equations

punctuated evolution for the quasispecies model abstract biological evolution in a sequence space with random fitnesses is studied within eigen s quasispecies model a strong selection limit is employed in which the population resides at a single sequence at all times evolutionary trajectories start at a randomly chosen sequence and proceed to the global fitness maximum through a small number of intermittent jumps the distribution of the total evolution time displays a universal power law tail with exponent 2 simulations show that the evolutionary dynamics is very well represented by a simplified shell model in which the sub populations at local fitness maxima grow independently the shell model allows for highly efficient simulations and provides a simple geometric picture of the evolutionary trajectories

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

on the discretization error of parametrized nonlinear equations many applications lead to nonlinear parameter dependent equations h y t y 0 where h y times t to y y 0 in operatorname rge h and the state space y is infinite dimensional while the parameter space t has finite dimension the case dim t 1 is of special interest in connection with continuation methods for this case a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite dimensional approximations and which allows for an assessment of the error between these paths a principal tool in this analysis is the theory of nonlinear fredholm operators the results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted

feedback stabilization of control systems described by a class of nonlinear differential algebraic equations a theoretical framework for the stabilization of con trol systems defined by a class of nonlinear differential alge braic equations is established assumptions are stated which guarantee that the nonlinear differential algebraic equations can be described by a nonlinear control system defined on a smooth manifold a procedure for obtaining a local state realization is developed conditions for local stabilization of a single equilibrium solution including one set of conditions which can easily be checked using standard computations are indicated

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 observing nonlinear descriptor systems we study the problem of observing the state of continuous nonlinear descriptor systems in quasilinear form and present a method to construct a state observer our approach is based on rewriting the descriptor system as an equivalent system of explicit differential equations on a restricted manifold finally the restrictions are replaced by measurement equations thus an observer for the descriptor system can be constructed by common state space techniques for explicit systems

universal statistics of longest lasting records of random walks and levy flights we study the record statistics of random walks after n steps x0 x1 xn with arbitrary symmetric and continuous distribution p of the jumps i xi xi 1 we consider the age of the records i e the time up to which a record survives depending on how the age of the current last record is defined we propose three distinct sequences of ages indexed by i ii iii associated to a given sequence of records we then focus on the longest lasting record which is the longest element among this sequence of ages to characterize the statistics of these longest lasting records we compute i the probability that the record of the longest age is broken at step n denoted by q n which we call the probability of record breaking and ii the duration of the longest lasting record we show that both q n and the full statistics of are universal i e independent of the jump distribution p we compute exactly the large n asymptotic behaviors of q n as well as when it exists and show that each case gives rise to a different universal constant associated to random walks including levy flights while two of them appeared before in the excursion theory of brownian motion for which we provide here a simpler derivation the third case gives rise to a non trivial new constant ciii 0 241 749 associated to the records of random walks other observables characterizing the ages of the records exhibiting an interesting universal behavior are also discussed

observer design for unknown input nonlinear descriptor systems via convex optimization this paper treats the design problem of full order observers for nonlinear descriptor systems with unknown input ui depending on the available knowledge on the ui dynamics two cases are considered first a ui proportional observer uipo is proposed when the spectral domain of the ui is unknown second a pio is proposed when the spectral domain of the ui is in the low frequency range sufficient conditions for the existence and stability of such observers are given and proved based on the linear matrix inequality lmi approach an algorithm is presented to compute the observer gain matrix that achieves the asymptotic stability objective an example is included to illustrate the method

state observation by on line minimization a new approach is presented to observing the state of a nonlinear control system the class considered consists of nonlinear observable systems described by an ordinary differential equation in combination with a measurement equation the system functions are required to be twice differentiable based on newton s method an iterative process of state approximation is derived the algorithm computes a piecewise continuous function which is an approximation of the true state the convergence of the proposed algorithm is proved finally several examples show the performance characteristics of the method

singular point analysis and integrals of motion for coupled nonlinear schrodinger equations for a system of two coupled nonlinear schrodinger equations and the corresponding madelung fluid equations we calculate the similarity transformation and reductions to ordinary differential equations the symmetries can be used to classify several types of solutions besides a system of coupled painleve ii equations we obtain a two dimensional quartic hamiltonian system using noether s theorem and performing a painleve test we identify those parameters which lead to regular motion on tori using the results of the p test we obtain heretofore undiscovered integrals of motion for the madelung fluid by direct calculations investigating the dynamics of the system for parameters that are different from those obtained by the painleve test we calculate numerically the surface of section kam tori and study the transition into chaotic behaviour

isogroup and general similarity solution of a nonlinear diffusion equation for a class of one dimensional nonlinear diffusion equations where the diffusion coefficient varies as some power of the dependent variable the invariance group and its lie algebra are given the isovector fields which generate the isogroup are then used to derive a general similarity solution for a particular case the solution can be reduced to a one parameter group solution which is in full agreement with previously published results

persistence of random walk records we study records generated by brownian particles in one dimension specifically we investigate an ordinary random walk and define the record as the maximal position of the walk we compare the record of an individual random walk with the mean record obtained as an average over infinitely many realizations we term the walk superior if the record is always above average and conversely the walk is said to be inferior if the record is always below average we find that the fraction of superior walks s decays algebraically with time s t in the limit t and that the persistence exponent is nontrivial 0 382 258 the fraction of inferior walks i also decays as a power law i t but the persistence exponent is smaller 0 241 608 both exponents are roots of transcendental equations involving the parabolic cylinder function to obtain these theoretical results we analyze the joint density of superior walks with a given record and position while for inferior walks it suffices to study the density as a function of position

an observer design for a class of implicit systems this paper deals with the problem of observer design for a class of implicit nonlinear systems of index one to solve this problem an implicit observer which is given by an implicit system is usually used the implementation of this kind of observer requires an optimization algorithm to solve at each time the algebraic constraints alternatively an explicit observer which is described by an ordinary differential equation can be used the advantage of this approach is that no optimization algorithm is required moreover the observer initialization may be done outside the constraints in this paper it is shown that under some conditions the existence of an implicit observer implies the existence of an explicit one then an explicit observer is given for implicit state affine systems whose dynamic involves an additional nonlinear term depending on the input the output and the implicit variable

order statistics of 1 fα signals order statistics of periodic gaussian noise with 1 f power spectrum is investigated using simulations and phenomenological arguments we find three scaling regimes for the average gap d k x k x k 1 between the kth and k 1 st largest values of the signal the result d k k 1 known for independent identically distributed variables remains valid for 0 5 the spectra of average ordered values e k x 1 x k k is also examined the exponent is derived from the gap scaling as well as by relating e k to the density of near extreme states known results for the density of near extreme states combined with scaling suggest that 2 1 2 4 3 2 and 2 are exact values we also show that parallels can be drawn between e k and the quantum mechanical spectra of a particle in power law potentials

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

universal survival probability for a correlated random walk and applications to records we consider a model of space continuous one dimensional random walk with simple correlation between the steps the probability that two consecutive steps have same sign is q with 0 leq q leq 1 the parameter q allows thus to control the persistence of the random walk we compute analytically the survival probability of a walk of n steps showing that it is independent of the jump distribution for any finite n this universality is a consequence of the sparre andersen theorem for random walks with uncorrelated and symmetric steps we then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk which show the same universality in particular we show that the distribution of the number of records for a walk of n gg 1 steps is the same as for a random walk with n rm eff q n 2 1 q uncorrelated and symmetrically distributed steps we also show that in the regime where n to infty and q to 1 with y n 1 q this model converges to the run and tumble particle a persistent random walk often used to model the motion of bacteria our theoretical results are confirmed by numerical simulations

universal survival probability for a d dimensional run and tumble particle we consider an active run and tumble particle rtp in d dimensions and compute exactly the probability s t that the x component of the position of the rtp does not change sign up to time t when the tumblings occur at a constant rate we show that s t is independent of d for any finite time t and not just for large t as a consequence of the celebrated sparre andersen theorem for discrete time random walks in one dimension moreover we show that this universal result holds for a much wider class of rtp models in which the speed v of the particle after each tumbling is random drawn from an arbitrary probability distribution we further demonstrate as a consequence the universality of the record statistics in the rtp problem

on a random area variable arising in discrete time queues and compact directed percolation a well known discrete time single server queueing system with mean arrival rate and mean departure rate is considered from the perspective of the area a swept out by the queue occupation process during a busy period we determine the exact form of the tail of the distribution pr a x in particular we show that pr a x cx 1 4 exp dx1 2 for all 1 where and expressions for c and d are given for the critical case 1 we show that pr a x c x 1 3 with c also given a simple mapping used in the derivation establishes a connection with compact directed percolation on a square lattice as a corollary therefore we are also able to specify the large area asymptotic behaviour of this model at all points in the phase diagram this extends previous scaling results which are only valid close to the percolation threshold

record statistics of a strongly correlated time series random walks and levy flights we review recent advances on the record statistics of strongly correlated time series whose entries denote the positions of a random walk or a l evy flight on a line after a brief survey of the theory of records for independent and identically distributed random variables we focus on random walks during the last few years it was indeed realized that random walks are a very useful laboratory to test the effects of correlations on the record statistics we start with the simple one dimensional random walk with symmetric jumps both continuous and discrete and discuss in detail the statistics of the number of records as well as of the ages of the records i e the lapses of time between two successive record breaking events then we review the results that were obtained for a wide variety of random walk models including random walks with a linear drift continuous time random walks constrained random walks like the random walk bridge and the case of multiple independent random walkers finally we discuss further observables related to records like the record increments as well as some questions raised by physical applications of record statistics like the effects of measurement error and noise

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

tracking in nonlinear differential algebraic control systems with applications to constrained robot systems we consider the design of a feedback control law for control systems described by a class of nonlinear differential algebraic equations so that certain desired outputs track given reference inputs the nonlinear differential algebraic control system being considered is not in state variable form assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential algebraic equations is expressed in a familiar normal form a nonlinear feedback control law is then proposed which ensures under appropriate assumptions that the tracking error in the closed loop differential algebraic system approaches zero exponentially applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints constrained robot systems with joint flexibility and constrained robot systems with significant actuator dynamics are discussed

brief paper an observer for non linear differential algebraic systems in this paper we consider design of observers for non linear models containing both dynamic and algebraic equations so called differential algebraic equations dae of index 1 the observer is formulated as a dae that by construction has index 1 the main results of the paper include conditions that ensure local stability of the observer error dynamics design methodology is presented and illustrated using a small simulation study

stabilization of non linear differential algebraic equation systems in this paper the feedback stabilization problem is investigated for non linear differential algebraic equation systems the paper is composed of two algorithms namely a regularization algorithm and a stabilization algorithm it is shown that the feasibility of the regularization algorithm guarantees that there exists a feedback so that the closed loop system admits a unique impulse free solution the application of the stabilization algorithm produces a set of new coordinates in which the original system can be expressed as a standard form based on the standard form a stabilizing feedback is constructed which guarantees that the closed loop system admits a unique impulse free solution and is locally asymptotically stable at the origin

brief paper a generalized framework for robust nonlinear h filtering of lipschitz descriptor systems with parametric and nonlinear uncertainties in this paper a generalized robust nonlinear h filtering method is proposed for a class of lipschitz descriptor systems in which the nonlinearities appear both in the state and measured output equations the system is assumed to have norm bounded uncertainties in the realization matrices as well as nonlinear uncertainties we synthesize the h filter through semidefinite programming and strict lmis the admissible lipschitz constants of the nonlinear functions are maximized through lmi optimization the resulting h filter guarantees asymptotic stability of the estimation error dynamics with prespecified disturbance attenuation level and is robust against time varying parametric uncertainties as well as lipschitz nonlinear additive uncertainty explicit bound on the tolerable nonlinear uncertainty is derived based on a norm wise robustness analysis