nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

precise large deviations for dependent regularly varying sequences we study a precise large deviation principle for a stationary regularly varying sequence of random variables this principle extends the classical results of nagaev theory probab appl 14 51 64 193 208 1969 and nagaev ann probab 7 745 789 1979 for iid regularly varying sequences the proof uses an idea of jakubowski stoch proc appl 44 291 327 1993 68 1 20 1997 in the context of central limit theorems with infinite variance stable limits we illustrate the principle for stochastic volatility models real valued functions of a markov chain satisfying a polynomial drift condition and solutions of linear and non linear stochastic recurrence equations

proliferation scheme for kaup boussinesq system abstract the proliferation scheme for kaup boussinesq system dispersive water wave system is discussed the recurrent procedure gives the series of integrable nonlinear equations coupled by miura maps and its lax pairs new integrable nonlinear systems are found in this series

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

monte carlo renormalisation group approach to percolation on a continuum test of universality it is shown that a monte carlo renormalisation group technique can be employed for direct renormalisation of a system on a continuum i e without restriction to a periodic lattice for the problem of overlapping discs the authors find the critical area fraction sc 0 688 or 0 005 and the correlation length critical exponent nu 1 33 or 0 07 the latter result indicates the regular irregular lattice universality

integrable equations in 2 1 dimensions associated with symmetric and homogeneous spaces generalizations of the n wave davey stewartson and kadomtsev petviashvili equations associated with homogeneous and symmetric spaces are presented these equations are 2 1 dimensional generalizations of those presented by fordy and kulish commun math phys 89 427 1983 and athorne and fordy j phys a 20 1377 1987 examples are explicitly presented that are associated with the simplest spaces in particular a new single component 2 1 dimensional generalization of the kdv equation is presented

large deviations for solutions to stochastic recurrence equations under kesten s condition in this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation we assume kesten s condition acta math 131 1973 207 248 under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails while the noise sequence of the equations can have light tails the results of the paper are analogs to those obtained by a v nagaev theory probab appl 14 1969 51 64 193 208 and s v nagaev ann probab 7 1979 745 789 in the case of partial sums of i i d random variables in the latter case the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum for the solution to a stochastic recurrence equation the magnitude of the large deviation probabilities is again given by the tail of the maximum summand but the exact asymptotic tail behavior is also influenced by clusters of extreme values due to dependencies in the sequence we apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model

elementary backlund transformations nonlinear superposition formulae and algebraic construction of solutions for the nonlinear evolution equations solvable by the zakharov shabat spectral transform abstract novel nonlinear superposition formulae are given for solutions of the class of nonlinear evolution equations solvable via the spectral transform associated with the zakharov shabat spectral problem and an algebraic procedure is provided to construct solutions of these equations

limit theory for the sample covariance and correlation functions of moving averages abstract document describes a moving average process which have regularly varying tail probabilities with index alpha 0 the limit distribution of the sample covariance function is derived in the case that the process has a finite variance but an infinite variance but an infinite fourth moment furthermore in the infinite variance case 0 alpha 2 the sample correlation function is shown to converge in distribution to the ratio of two independent stable random variables with indices alpha and alpha 2 respectively this result immediately gives the limit distribution for the least squares estimates of the parameters in an autoregressive process author

explicit auto transformations of integrable chains a construction scheme for explicit auto transformations of integrable discrete differential equations chains is presented these transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems on the other hand they exemplify new integrable discrete mappings their group properties are also of great interest the scheme is illustrated by several examples of integrable systems which contains the nonlinear schrodinger system and the landau lifshits model

systematic method of generating new integrable systems via inverse miura maps we provide a new natural interpretation of the lax representation for an integrable system that is the spectral problem is the linearized form of a miura transformation between the original system and a modified version of it on the basis of this interpretation we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by miura transformations thus this method can be used to generate new integrable systems from known systems through inverse miura maps it can be applied to both continuous and discrete systems in 1 1 dimensions as well as in 2 1 dimensions the effectiveness of the method is illustrated using examples such as the nonlinear schrodinger nls system the zakharov ito system two component kdv the three wave interaction system the yajima oikawa system the ablowitz ladik lattice integrable space discrete nls and two 2 1 dimensional nls systems

linear integral equations and multicomponent nonlinear integrable systems i abstract a systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations pde s is developed the method starts from a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k the spectral parameter special k dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term in this way the direct linearization is obtained of multicomponent generalizations of various nonlinear pde s such as the nonlinear schrodinger equation the derivative nonlinear schrodinger equations the isotropic heisenberg spin chain equation the complex sine gordon equation and the massive thirring model equations in the present paper i we present the general framework to derive finite matrix pde s and we also discuss backlund transformations and multicomponent lattice versions in a subsequent paper ii we treat a variety of examples of multicomponent pde s and discuss miura transformations and gauge equivalences

limit theory for moving averages of random variables with regularly varying tail probabilities let z k infty k infty be iid where the z k s have regularly varying tail probabilities under mild conditions on a real sequence c j j geq 0 the stationary process x n sum infty j 0 c jz n j n geq 1 exists a point process based on x n converges weakly and from this a host of weak limit results for functionals of x n ensue we study sums extremes excedences and first passages as well as behavior of sample covariance functions

large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy tailed innovations in this paper we consider the stochastic recurrence equation y t a t y t 1 b t for an i i d sequence of pairs a t b t of nonnegative random variables where we assume that b t is regularly varying with index k 0 and ea k t x c 1 np y 1 x as x holds for some constant c 1 0 for k 1 we also study the large deviation probabilities p s n es n x x x n for some sequence x n whose growth depends on the heaviness of the tail of the distribution of y 1 we show that the relation p s n es n x c 2 np y 1 x holds uniformly for x x n and some constant c 2 0 then we apply the large deviation results to derive bounds for the ruin probability u p sup n 1 s n es n n u for any 0 we show that u c 3 up y 1 u 1 1 1 for some constant c 3 0 in contrast to the case of i i d regularly varying y t s when the above results hold with c 1 c 2 c 3 1 the constants c 1 c 2 and c 3 are different from 1

extreme values for stationary and markov sequences soit x n n 1 2 une suite strictement stationnaire de variables aleatoires a valeurs reelles soit m i j max x i 1 x j et m n m 0 n soit c n une suite de nombres reels on montre sous des circonstances generales que p m n c n p x 1 c n np mu 1 u u p u n cu n xu 1 cu n o pour toute suite pn satisfaisant certaines conditions de taux de croissance

on some estimates based on sample behavior near high level excursions let j be a stationary sequence of weakly dependent random variables and letm n k be thek th largest value of j 1 j n the estimation of the parameters of the asymptotic distribution ofm n k is considered using a procedure motivated by a limit theorem pertaining to the point process sum nolimits j delta j n n bar f xi j a number of statistical issues concerning the procedure including how to select the tuning parameters are addressed the second problem that we consider is the estimation of the filter of a moving average process with heavy tails in particular the investigation covers the moving average stable process motivated by ideas in rootzen 1978 our estimator uses information contained in the sample behavior of the process near the largest excursion

classification of polynomial integrable systems of mixed scalar and vector evolution equations i we perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries we consider polynomial systems that are homogeneous under a suitable weighting of variables this paper deals with the kdv weighting the burgers or potential kdv or modified kdv weighting the ibragimov shabat weighting and two unfamiliar weightings the case of other weightings will be studied in a subsequent paper making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems we give the complete lists of second order systems with a third order or a fourth order symmetry and third order systems with a fifth order symmetry for all but a few systems in the lists we show that the system or at least a subsystem of it admits either a lax representation or a linearizing transformation a thorough comparison with recent work of foursov and olver is made

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

modified hamiltonian systems and canonical transformations arising from the relationship between generalized zakharov shabat and energy dependent schrodinger operators a new family of completely integrable infinite dimensional hamiltonian systems is found two canonical maps are deduced which transform this family into the families of hamiltonian systems associated with the generalized zakharov shabat and energy dependent schrodinger operators through these maps several relevant hamiltonian structures are connected in particular a simple explanation of the second hamiltonian structure for akns equations is given

a new family of integrable models in 2 1 dimensions associated with hermitian symmetric spaces in a series of papers fordy and his collaborators have studied families of integrable models in 1 1 dimensions associated with hermitian symmetric spaces these models were also generalized to 2 1 dimensions however the generalization used is not unique and in this paper a different generalization is considered resulting in new families of integrable models in 2 1 dimensions

constructing hierarchy of sinh gordon type equations from soliton hierarchy with self consistent source abstract a method for constructing hierarchy of sinh gordon shg type equations from soliton hierarchy with self consistent source is proposed the lax representation for the shg type equation is naturally deduced from that for soliton equation with self consistent sources several new hierarchies of shg type equations are presented

busemann functions and the speed of a second class particle in the rarefaction fan in this paper we will show how the results found in probab theory related fields 154 2012 89 125 about the busemann functions in last passage percolation can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan in either the totally asymetric exclusion process or the hammersley interacting particle process the method will be to use the well known last passage percolation description of the exclusion process and of the hammersley process and then the well known connection between second class particles and competition interfaces

zero curvature representation bi hamiltonian structure and an integrable hierarchy for the zakharov ito system in the present paper we present an integrable hierarchy for the zakharov ito system we first construct the lenard recursion sequence and zero curvature representation for the zakharov ito system following cao s method as significantly generalized by other authors we then construct the bi hamiltonian structures employing variational trace identities but woven together with the lenard recursion sequences from this we are in a position to construct an integrable hierarchy of equations from the zakharov ito system and we obtain the recursion operator and poisson brackets for constructing this hierarchy finally we demonstrate that the obtained hierarchy is indeed liouville integrable

classical lattice w algebras and integrable systems several aspects of the lattice algebra are studied motivated by the fact that the lotka volterra model can be written in terms of a current of the lattice virasoro algebra the faddeev takhtajan volkov algebra integrable dynamical models on the lattice have been formulated as a model associated with the lattice algebra

modified equations rational solutions and the painleve property for the kadomtsev petviashvili and hirota satsuma equations we propose a method for finding the lax pairs and rational solutions of integrable partial differential equations that is when an equation possesses the painleve property a backlund transformation is defined in terms of an expansion about the singular manifold this backlund transformation obtains 1 a type of modified equation that is formulated in terms of schwarzian derivatives and 2 a miura transformation from the modified to the original equation by linearizing the ricati type miura transformation the lax pair is found on the other hand consideration of the distinct backlund transformations of the modified equations provides a method for the iterative construction of rational solutions this also obtains the lax pairs for the modified equations in this paper we apply this method to the kadomtsev petviashvili equation and the hirota satsuma equations

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

a large deviations approach to limit theory for heavy tailed time series in this paper we propagate a large deviations approach for proving limit theory for generally multivariate time series with heavy tails we make this notion precise by introducing regularly varying time series we provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin we study a variety of such functionals including large deviations of random walks their suprema the ruin functional and further derive weak limit theory for maxima point processes cluster functionals and the tail empirical process one of the main results of this paper concerns bounds for the ruin probability in various heavy tailed models including garch stochastic volatility models and solutions to stochastic recurrence equations

from akns to derivative nls hierarchies via deformations of associative products using deformations of associative products derivative nonlinear schrodinger dnls hierarchies are recovered as akns type hierarchies since the latter can also be formulated as gelfand dickey type lax hierarchies a recently developed method to obtain functional representations can be applied we actually consider hierarchies with dependent variables in any possibly noncommutative associative algebra e g an algebra of matrices of functions this also covers the case of hierarchies of coupled derivative nls equations

extremes and local dependence in stationary sequences extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction a a weak mixing condition restricting long range dependence b a local condition restricting the clustering of high level exceedances

on a new hierarchy of hamiltonian soliton equations a method is suggested for studying the hamiltonian structure of the nonlinear partial differential equations that can be solved by the use of the spectral transform soliton equations the method is applied to a new hierarchy of n 1 coupled partial differential equations related to a schrodinger like spectral problem it is shown that these soliton equations are integrable hamiltonian equations with commuting flows for n 1 and n 2 a miura like transformation is computed and the corresponding modified equations are explicitly given

the sample autocorrelations of heavy tailed processes with applications to arch we study the sample acvf and acf of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying this includes linear and bilinear processes with regularly varying noise and arch processes their squares and absolute values we show that the distributional limits of the sample acf can be random provided that the variance of the marginal distribution is infinite and the process is nonlinear this is in contrast to infinite variance linear processes if the process has a finite second but infinite fourth moment then the sample acp is consistent with scaling rates that grow at a slower rate than the standard root n consequently asymptotic confidence bands are wider than those constructed in the classical theory we demonstrate the theory in full detail far an arch 1 process

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation