three component nonlinear dynamical system generated by the new third order discrete spectral problem we propose a nonlinear model on a regular infinite one dimensional lattice it describes the three component dynamical system with modulated on site masses and is shown to admit a zero curvature representation the associated auxiliary spectral problem is basically of third order and gives rise to fairly complicated subdivision into domains of regularity of jost functions in the plane of complex spectral parameter as a result both the direct and the inverse scattering problems turn out to be substantially nontrivial the caudrey version of the direct and inverse scattering technique for the needs of model integration is adapted the simplest soliton solution is found

nonparametric quantile estimations for dynamic smooth coefficient models in this article quantile regression methods are suggested for a class of smooth coefficient time series models we use both local polynomial and local constant fitting schemes to estimate the smooth coefficients in a quantile framework we establish the asymptotic properties of both the local polynomial and local constant estimators for mixing time series also a bandwidth selector based on the nonparametric version of the akaike information criterion is suggested together with a consistent estimate of the asymptotic covariance matrix furthermore the asymptotic behaviors of the estimators at boundaries are examined a comparison of the local polynomial quantile estimator with the local constant estimator is presented a simulation study is carried out to illustrate the performance of estimates an empirical application of the model to real data further demonstrates the potential of the proposed modeling procedures

on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

discrete integrable systems and deformations of associative algebras interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed theory of deformations for associative algebras is presented closed left ideal generated by the elements representing the multiplication table plays a central role in this theory deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations it is demonstrated that many discrete equations such as discrete boussinesq equation discrete wdvv equation discrete schwarzian kp and bkp equations discrete hirota miwa equations for kp and bkp hierarchies are particular realizations of the central system an interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful an interpretation of the menelaus relation discrete schwarzian kp equation discrete hirota miwa equation for kp hierarchy consistency around the cube as the associativity conditions and the concept of gauge equivalence for instance between the menelaus and kp configurations are particular examples

equivalence of various pseudopotential approaches for einstein maxwell fields in the literature various systems of liner eigenvalue equations from which the einstein maxwell equations for stationary axisymmetric exterior fields follow as the integrability conditions were derived in the present paper these linear systems are shown to be equivalent the explicit transformations mapping one form to another are given

a variational formula for the free energy of an interacting many particle system we consider n bosons in a box in r d with volume n rho under the influence of a mutually repellent pair potential the particle density p is an element of 0 infinity is kept fixed our main result is the identification of the limiting free energy f beta p at positive temperature 1 beta in terms of an explicit variational formula for any fixed rho if beta is sufficiently small and for any fixed beta if rho is sufficiently small the thermodynamic equilibrium is described by the symmetrized trace of e beta h n where h n denotes the corresponding hamilton operator the well known feynman kac formula reformulates this trace in terms of n interacting brownian bridges due to the symmetrization the bridges are organized in an ensemble of cycles of various lengths the novelty of our approach is a description in terms of a marked poisson point process whose marks are the cycles this allows for an asymptotic analysis of the system via a large deviations analysis of the stationary empirical field the resulting variational formula ranges over random shift invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process in our proof of the lower bound for the free energy we drop all interaction involving infinitely long cycles and their possible presence is signalled by a loss of mass of the finitely long cycles in the variational formula in the proof of the upper bound we only keep the mass on the finitely long cycles we expect that the precise relationship between these two bounds lies at the heart of bose einstein condensation and intend to analyze it further in future

integrable dynamics of toda type on square and triangular lattices in a recent paper we constructed an integrable generalization of the toda law on the square lattice we construct other examples of integrable dynamics of toda type on the square lattice as well as on the triangular lattice as nonlinear symmetries of the discrete laplace equations on square and triangular lattices we also construct the ensuremath tau function formulations and darboux b acklund transformations of these dynamics

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

integrable three dimensional lattices following the work of zakharov and shabat 1974 the authors derive nonlinear equations which are integrable through a discrete gel fand levitan integral equation in two resp one resp zero continuous and one resp two resp three discrete variables

hexagonal circle patterns and integrable systems patterns with constant angles hexagonal circle patterns with constant intersection angles are introduced and studied it is shown that they are described by discrete integrable systems of toda type conformally symmetric patterns are classified circle pattern analogs of holomorphic mappings z sp c and log z are constructed as special isomonodromic solutions circle patterns studied in the paper include schramm s circle patterns with the combinatorics of the square grid as a special case

the equivalence of ensembles for classical systems of particles for systems of particles in classical phase space with standard hamiltonian we consider spatially averaged microcanonical gibbs distributions in finite boxes we show that infinite volume limits along suitable subsequences exist and are grand canonical gibbs measures on the way we establish a variational formula for the thermodynamic entropy density as well as a variational characterization of grand canonical gibbs measures

quenched asymptotics of the ground state energy of random schrodinger operators with scaled gibbsian potentials this article describes the almost sure infinite volume asymptotics of the ground state energy of random schrodinger operators with scaled gibbsian potentials the random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered gibbs measure with a superstable pair interaction there is no restriction on the strength of the pair interaction it may be taken e g at a critical point the potential is scaled with the box size in a critical way i e the scale is determined by the typical size of large deviations in the gibbsian cloud the almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities the qualitative behaviour of the ground state energy asymptotics is analysed depending on the dimension and on the holder exponents of the free energy density it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy

existence of gibbsian point processes with geometry dependent interactions we establish the existence of stationary gibbsian point processes for interactions that act on hyperedges between the points for example such interactions can depend on delaunay edges or triangles cliques of voronoi cells or clusters of k nearest neighbors the classical case of pair interactions is also included the basic tools are an entropy bound and stationarity

direct linearizing transform for three dimensional discrete integrable systems the lattice akp bkp and ckp equations a unified framework is presented for the solution structure of three dimensional discrete integrable systems including the lattice akp bkp and ckp equations this is done through the so called direct linearizing transform which establishes a general class of integral transforms between solutions as a particular application novel soliton type solutions for the lattice ckp equation are obtained

hydrodynamical limit for a hamiltonian system with weak noise starting from a general hamiltonian system with superstable pairwise potential we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles we prolve that in the scaling limit the time conserved quantities energy momenta and density satisfy the euler equation of conservation laws up to a fixed timet provided that the euler equation has a smooth solution with a given initial data up to timet the strength of the noise term is chosen to be very small but nonvanishing so that it disappears in the scaling limit

r local delaunay inhibition model unlike in the classical framework of gibbs point processes usually acting on the complete graph in the context of nearest neighbour gibbs point processes the nonnegativeness of the interaction functions do not ensure the local stability property this paper introduces domain wise but not pointwise inhibition stationary gibbs models based on some tailor made delaunay subgraphs all of them are subgraphs of the r local delaunay graph defined as the delaunay subgraph specifically not containing the edges of delaunay triangles with circumscribed circles of radii greater than some large positive real value r the usual relative compactness criterion for point processes needed for the existence result is directly derived from the ruelle bound of the correlation functions furthermore assuming only the nonnegativeness of the energy function we have managed to prove the existence of the existence of r local delaunay stationary gibbs states based on nonnegative interaction functions thanks to the use of the compactness of sublevel sets of the relative entropy

lattice equations hierarchies and hamiltonian structures iii the 2d toda and kp hierarchies abstract we continue the investigation of the continuum limit properties of discrete versions of the kadomtsev petviashvili kp equation using a multiple time scale expansion in which one of the discrete variables is replaced by an infinite number of continous variables we can derive a hierarchy of integrable equations in this paper hierarchies associated with the 2d toda equation and the kp equation are treated the direct linearization of the hierarchy is obtained applying the expansion mentioned above to the free wave function in the integral equation the hamiltonian structure of the hierarchy is established via a relation between the generator of the hierarchy and an appropriate monodromy factor of the scattering problem

burmann expansion and test for additivity summary we propose a lagrange multiplier test for additivity based on the btirmann expansion of a conditional mean function the asymptotic null distribution of the test is shown to be x2 under some regularity conditions in contrast the lagrange multiplier test proposed by chen et al 1995 is based on the volterra expansion of the conditional mean function we discuss some desirable advantages of the bt rmann expansion over the volterra expansion for nonlinear time series modelling we also reported an empirical study which shows that in terms of empirical power the lagrange multiplier test motivated by the biirmann expansion outperforms the test of chen et al 1995 for the cases for which the lagrange multiplier test is designed for other cases for which none of the tests is specifically designed the empirical powers of the two tests are comparable finally we illustrated the use of the lagrange multiplier test with a blowfly experimental system

toda chain and nonlinear schro dinger equation nonlocalities abstract the problem of non localities in the theory of 2d toda chain is discussed our approach is based on direct generalization of the notion of first integrals to the 2d case we formulate also special geometrical properties of the coordinates system formed by non local dynamical variables applications to 2 1 integrable pde s are considered

recursion operator and bi hamiltonian structure for integrable multidimensional lattices the recursion operator and the bi hamiltonian structure for an integrable two dimensional version of the toda chain are algorithmically derived from the corresponding linear problem the intimate relation between two dimensional theories and non abelian one dimensional theories is emphasized the analogies and differences between discrete and continuous integrable two dimensional systems are pointed out and discussed

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

an example of problem arising in a finite difference context direct and inverse problem for the discrete analog of the equation ψxx uψ σψy the direct and inverse spectral problem for the discrete analog of the equation xx u y is solved in the framework of theory the time evolution of the spectral data for the simplest nonlinear differential difference equations associated to this linear problem is derived

complete equivalence of the gibbs ensembles for one dimensional markov systems we show the equivalence of the gibbs ensembles at the level of measures for one dimensional markov systems with arbitrary boundary conditions that is the limit of the microcanonical gibbs ensemble is a gibbs measure with an interaction depending on the microcanonical constraint in fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins including the boundary spins are fixed when conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble we show that the limit is a gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures for this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions we establish analogous results for finite range potentials

nonparametric tests of linearity for time series summary we introduce tests of linearity for time series based on nonparametric estimates of the conditional mean and the conditional variance the tests are compared to a number of parametric tests and to nonparametric tests based on the bispectrum asymptotic expressions give bad approximations and the null distribution under linearity is constructed using resampling of the best linear approximation the new tests perform well on the examples tested

on generalized loewner systems novel integrable equations in 2 1 dimensions a reinterpretation and generalization of a class of infinitesimal backlund transformations originally introduced in a gas dynamics context by loewner in 1952 leads to a linear representation for a novel class of 2 1 dimensional nonlinear equations the latter may be parametrized in terms of a triad of eigenfunctions moreover like the well known integrable davey stewartson and nizhnik novikov veselov equations the nonlinear systems typically contain the two spatial variables on an equal footing the dressing method is outlined for these generalized loewner systems it is noted that basic reductions lead to 2 1 dimensional versions of the principal chiral fields model toda lattice system and notably the classical sine gordon equation loewner systems on grassmannian the projective cpn and rpn manifolds are considered in particular the 2 1 dimensional integrable sine gordon system in which x and y occur in a symmetric manner arises naturally in this context a gauge equivalent system likewise eme

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

on the lattice geometry and birational group of the six point multi ratio equation the inherent self consistency properties of the six point multi ratio equation allow it to be considered on a domain associated with a t shaped coxeter dynkin diagram this extends the kadomtsev pe

nonlinear additive models for environmental time series with applications to ground level ozone data analysis abstract environmental time series usually vary systematically in response to meteorological conditions and thus often are not stationary in this article a class of additive models are introduced for environmental time series in which both mean levels and variances of the series are nonlinear functions of relevant meteorological variables backfitting algorithms in nonlinear regression are adopted to estimate the unknown functions in the model and the maximum likelihood method is used to estimate the parameters in the noise component asymptotic properties of the parameter estimates including consistency and limiting distribution are derived under mild conditions the model is applied to daily maxima of ground level ozone concentrations in the chicago area for possible long term trend assessment compared to alternative models the proposed models gave more accurate estimations for the 95th and 99th percentiles of the ozone distribution

analytic bilinear approach to integrable hierarchies i generalized kp hierarchy an analytic bilinear approach for construction and study of integrable hierarchies in particular the kp hierarchy is proposed it starts with the generalized hirota identity for the cauchy baker akhiezer cba function and leads to a generalized kp hierarchy in the form of compact functional equations containing a special shift operator a generalized kp hierarchy incorporates the basic kp hierarchy modified kp hierarchy kp singularity manifold equation hierarchy and corresponding hierarchies of linear problems different vertical levels of the generalized kp hierarchy are connected via invariants of the combescure symmetry group the resolution of functional equations also gives rise to the function and the addition formulas for it

integrable hierarchies of nonlinear difference difference equations and symmetries we construct the hierarchy of nonlinear difference difference equations associated with the discrete schrodinger spectral problem as examples of equations contained in this hierarchy we obtain the discrete time toda and volterra lattice equations in the case of the time discrete toda lattice we construct its lie point and generalized symmetries finally we present its backlund transformations and relate it to the already constructed symmetries

nonparametric identification of nonlinear time series selecting significant lags abstract in this article we suggest a nonparametric procedure for selecting significant lags in the model description of a general nonlinear stationary time series the procedure can be applied to both the conditional mean and the conditional variance and is valid for heteroscedastic series the procedure is illustrated by simulations and sunspot data lynx data and blowfly data are analyzed it is indicated that projectors can be used in conjunction with the procedure for selecting significant lags to check the adequacy of an additive time series model

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

nonparametric transfer function models in this paper a class of nonparametric transfer function models is proposed to model nonlinear relationships between input and output time series the transfer function is smooth with unknown functional forms and the noise is assumed to be a stationary autoregressive moving average arma process the nonparametric transfer function is estimated jointly with the arma parameters by modelling the correlation in the noise the transfer function can be estimated more efficiently the parsimonious arma structure improves the estimation efficiency in finite samples the asymptotic properties of the estimators are investigated the finite sample properties are illustrated through simulations and one empirical example

a consistent characteristic function based test for conditional independence abstract y is conditionally independent of z given x if pr f y x z f y x 1 for all y on its support where f denotes the conditional density of y given x z or x this paper proposes a nonparametric test of conditional independence based on the notion that two conditional distributions are equal if and only if the corresponding conditional characteristic functions are equal we extend the test of su and white 2005 a hellinger metric nonparametric test for conditional independence discussion paper department of economics ucsd in two directions 1 our test is less sensitive to the choice of bandwidth sequences 2 our test has power against deviations on the full support of the density of x y z we establish asymptotic normality for our test statistic under weak data dependence conditions simulation results suggest that the test is well behaved in finite samples applications to stock market data indicate that our test can reveal some interesting nonlinear dependence that a traditional linear granger causality test fails to detect

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

application of a least absolute shrinkage and selection operator to aeroelastic flight test data identification of non linear systems involves estimating unknown parameters and model regressor selection selection of a subset of candidate terms that best describes the observed output model selection is an important step in black box modelling of any observed process this procedure is concerned with selecting a subset of parameters to give a parsimonious description of the system which may afford greater insight into the functionality of the system or a simpler controller design in this study a least absolute shrinkage and selection operator lasso technique is investigated for computing efficient model descriptions of non linear aeroelastic systems the lasso minimises the residual sum of squares by the addition of an l1 penalty term on the parameter vector of the traditional l2 minimisation problem its use for model selection is a natural extension of this constrained minimisation approach to pseudolinear regression problems which produces some model parameters that are exactly zero and the

large deviations and the maximum entropy principle for marked point random fields we establish large deviation principles for the stationary and the individual empirical fields of poisson and certain interacting random fields of marked point particles in d the underlying topologies are induced by a class of not necessarily bounded local functions and thus finer than the usual weak topologies our methods yield further that the limiting behaviour of conditional poisson distributions as well as certain distributions of gibbsian type is governed by the maximum entropy principle we also discuss various applications and examples

analytic bilinear approach to integrable hierarchies ii multicomponent kp and 2d toda lattice hierarchies an analytic bilinear approach for the construction and study of integrable hierarchies is discussed generalized multicomponent kp and 2d toda lattice hierarchies are considered this approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations a generalized hierarchy incorporates basic hierarchy modified hierarchy singularity manifold equation hierarchy and corresponding linear problems different levels of generalized hierarchy are connected via invariants of combescure symmetry transformation the resolution of functional equations also leads to the function and addition formulas to it

large deviations and the equivalence of ensembles for gibbsian particle systems with superstable interaction for gibbsian systems of particles inr d we investigate large deviations of the translation invariant empirical fields in increasing boxes the particle interaction is given by a superstable regular pair potential the large deviation principle is established for systems with free or periodic boundary conditions and under a stronger stability hypothesis on the potential for systems with tempered boundary conditions and for tempered infinite volume gibbs measures as a by product we obtain the gibbs variational formula for the pressure we also prove the asymptotic equivalence of microcanonical and grand canonical gibbs distributions and establish a variational expression for the thermodynamic entropy density

positivity of the self diffusion matrix of interacting brownian particles with hard core we prove the positivity of the self diffusion matrix of interacting brownian particles with hard core when the dimension of the space is greater than or equal to 2 here the self diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle we will do this for all activities z 0 of gibbs measures in particular for large z the case of high density particles a typical example of such a particle system is an infinite amount of hard core brownian balls

lattice equations hierarchies and hamiltonian structures ii kp type of hierarchies on 2d lattices abstract we investigate the continuum limit properties of an integrable lattice version of the kadomtsev petviashvili kp equation by applying continuum limits with vertex operators involving an infinite number of continous variables to the lattice kp we obtain hierarchies of integrable equations for fields depending on one continous variable at the sites of a two dimensional lattice the direct linearization of the hierarchies is obtained applying the same limit to the free wave function in the integral equation for lattice kp on the basis of the relation between the generator of the hierarchy and the monodromy factor for the scattering problem on a line which is not involved in the continuum limit we establish the hamiltonian structure of the equations of the hierarchy

phase transitions for dilute particle systems with lennard jones potential we consider a classical dilute particle system in a large box with pair interaction given by a lennard jones type potential the inverse temperature is picked proportionally to the logarithm of the particle density we identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity loosely speaking the particle system separates into spatially distant components in such a way that within each phase all components are of the same size which is the larger the lower the temperature the main tool in our proof is a new large deviation principle for sparse point configurations

additivity tests for nonlinear autoregression summary additivity is commonly used in the statistical literature to simplify data analysis especially in analysis of variance and in multivariate smoothing in this paper we propose three procedures for testing additivity in nonlinear time series analysis the first procedure combines some smoothing techniques with analysis of variance the second is a lagrange multiplier test using nonparametric estimation and the third is a permutation test which uses smoothing techniques to obtain the test statistic and its reference distribution we investigate properties of the proposed tests and use simulation to check their performance in finite samples applications of the tests to nonlinear time series analysis are discussed and illustrated by real examples

memory universal prediction of stationary random processes we consider the problem of one step ahead prediction of a real valued stationary strongly mixing random process xi sub i spl infin sup spl infin the best mean square predictor of x sub 0 is its conditional mean given the entire infinite past x sub i sub i spl infin sup 1 given a sequence of observations x sub 1 x sub 2 x sub n we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension for example neural networks and legendre polynomials the proposed estimators select both the model memory and the model dimension in a data driven fashion by minimizing certain complexity regularized least squares criteria when the underlying predictor function has a finite memory we establish that the proposed estimators are memory universal the proposed estimators which do not know the true memory deliver the same statistical performance rates of integrated mean squared error as that delivered by estimators that know the true memory furthermore when the underlying predictor function does not have a finite memory we establish that the estimator based on legendre polynomials is consistent

existence of delaunay pairwise gibbs point process with superstable component the present stuffy deals with the existence of delaunay pairwise gibbs point process with superstable component by using the well known preston theorem in particular we prove the stability the lower regularity and the quasilocality properties of the delaunay model

identification of nonlinear time series first order characterization and order determination summary we study the possibility of identifying nonlinear time series using nonparametric estimates of the conditional mean and conditional variance it is shown that most nonlinear models satisfy the assumptions needed to apply nonparametric asymptotic theory sampling variations of the conditional quantities are studied by simulation and explained by asymptotic arguments for a number of first order nonlinear autoregressive processes the conditional mean and variance can be used for identification purposes but one must be aware of bias and misspecification effects we also propose a criterion for determining the order of a general nonlinear model the criterion is justified in parts by heuristics but encouraging results are obtained from a limited set of simulation experiments several

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

large deviations and maximum entropy principle for interacting random fields on mathbb z d we present a new approach to the principle of large deviations for the empirical field of a gibbsian random field on the integer lattice zd this approach has two main features first we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities and thus obtain estimates which are finer and more widely applicable second we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field the limits being those predicted by the maximum entropy principle this result implies a general version of the equivalence of gibbs ensembles stating that every microcanonical limiting state is a grand canonical equilibrium state we also prove a converse to the last statement and discuss some applications 0 introduction as is well known the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles the principle of large deviations and the maximum entropy principle the former provides the exact rate of exponential decay of the fluctuation probabilities whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large it is obvious that these principles are intimately related in this paper we investigate these principles in the case of interacting random fields on the integer lattice 7zd the setup is the following first we let e 4 be any measurable space we shall assume throughout that e a is standard borel but we shall avoid making any explicit topological assumptions on e so we do not assume that e is polish next we let s zd be the d dimensional integer lattice and q f e gas the associated product space in lattice models of statistical mechanics e is called the state space or single spin space and q f the configuration space we let j s f denote the set of all probability

phase transition in the nearest neighbor continuum potts model in the present study we establish a phase transition in the nearest neighbor continuum potts model the repulsion between particles of different type acts only on a nearest neighbor graph more precisely a subgraph of the delaunay graph this work is an adaptation of the lebowitz and lieb soft core continuum potts model

a flexible coefficient smooth transition time series model we consider a flexible smooth transition autoregressive star model with multiple regimes and multiple transition variables this formulation can be interpreted as a time varying linear model where the coefficients are the outputs of a single hidden layer feedforward neural network this proposal has the major advantage of nesting several nonlinear models such as the self exciting threshold autoregressive setar the autoregressive neural network ar nn and the logistic star models furthermore if the neural network is interpreted as a nonparametric universal approximation to any borel measurable function our formulation is directly comparable to the functional coefficient autoregressive far and the single index coefficient regression models a model building procedure is developed based on statistical inference arguments a monte carlo experiment showed that the procedure works in small samples and its performance improves as it should in medium size samples several real examples are also addressed

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

the tangential map and associated integrable equations the tangential map is a map on the set of smooth planar curves it satisfies the 3d consistency property and is closely related to some well known integrable equations

consistency of likelihood estimation for gibbs point processes strong consistency of the maximum likelihood estimator mle for parametric gibbs point process models is established the setting is very general it includes pairwise pair potentials finite and infinite multibody interactions and geometrical interactions where the range can be finite or infinite the gibbs interaction may depend linearly or non linearly on the parameters a particular case being hardcore parameters and interaction range parameters as important examples we deduce the consistency of the mle for all parameters of the strauss model the hardcore strauss model the lennard jones model and the area interaction model

application of an incidence theorem for conics cauchy problem and integrability of the dckp equation we demonstrate that pascal s classical theorem for hexagons inscribed in conics allows one to define in a compact manner maps which are governed algebraically by the integrable discrete ckp equation a theorem for conics on oriented triangulated surfaces is used to construct a well posed cauchy problem for these dckp maps moreover the same theorem is exploited to construct in a purely geometric manner a backlund transformation for dckp maps thus the integrability of dckp maps and their underlying nonlinear soliton equation is shown to be encoded in an incidence theorem of projective geometry

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

discrete holomorphic geometry i darboux transformations and spectral curves finding appropriate notions of discrete holomorphic maps and more generally conformal immersions of discrete riemann surfaces into 3 space is an important problem of discrete differential geometry and computer visualization we propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over discrete surfaces by which we mean the vertex sets of triangulated surfaces with bi colored set of faces the resulting theory of discrete conformality is simultaneously mobius invariant and based on linear equations in the special case of maps into the 2 sphere we obtain a reinterpretation of the theory of complex holomorphic functions on discrete surfaces introduced by dynnikov and novikov as an application of our theory we introduce a darboux transformation for discrete surfaces in the conformal 4 sphere this darboux transformation can be interpreted as the space and time discrete davey stewartson flow of konopelchenko and schief for a generic map of a discrete torus with regular combinatorics the space of all darboux transforms has the structure of a compact riemann surface the spectral curve this makes contact to the theory of algebraically completely integrable systems and is the starting point of a soliton theory for triangulated tori in 3 and 4 space devoid of special assumptions on the geometry of the surface

on laplace darboux type sequences of generalized weingarten surfaces two novel classes of integrable surfaces generalized weingarten surfaces of class 1 and 2 which may be defined in a coordinate independent manner or equivalently via an extension of the classical lelieuvre formulas are introduced the classes 1 and 2 generalize bianchi surfaces and surfaces of harmonic inverse mean curvature respectively and turn out to be governed by analogs of the ernst weyl equation descriptive of gravitational and neutrino fields in general relativity the latter connection is exploited to derive laplace darboux type transformations for class 2 via the associated loewner konopelchenko rogers representation the laplace darboux type transformations are shown to have remarkable geometric properties in terms of sphere congruences connections with the pohlmeyer lund regge vortex model and an inhomogeneous heisenberg spin equation are recorded a backlund transformation for a particular painleve iii equation is derived in a purely geometric manner and a geometric interpretation of a

nonlinear additive arx models abstract we consider in this article a class of nonlinear additive autoregressive models with exogenous variables for nonlinear time series analysis and propose two modeling procedures for building such models the procedures proposed use two backfitting techniques the ace and bruto algorithms to identify the nonlinear functions involved and use the methods of best subset regression and variable selection in regression analysis to determine the final model simulated and real examples are used to illustrate the analysis

on a 2 1 dimensional integrable ernst type equation a strong 2 1 dimensional integrable extension of ernst s equation of general relativity is proposed its richness is demonstrated by means of various canonical dimensional reductions and specializations which lead to formal analogues of well known 1 1 and 2 1 dimensional integrable systems such as the self induced transparency equations the konopelchenko rogers equations a 2 1 dimensional darboux system descriptive of conjugate coordinate systems a single 2 1 dimensional sine gordon equation and the equations representing its backlund transformation a darboux levi type transformation is given and its compatibility with the above mentioned reductions is shown

phase transition in continuum potts models we establish phase transitions for a class of continuum multi type particle systems with finite range repulsive pair interaction between particles of different type this proves an old conjecture of lebowitz and lieb a phase transition still occurs when we allow a background pair interaction between all particles which is superstable and has sufficiently short range of repulsion our approach involves a random cluster representation analogous to the fortuin kasteleyn representation of the potts model in the course of our argument we establish the existence of a percolation transition for gibbsian particle systems with random edges between the particles and also give an alternative proof for the existence of gibbs measures with supperstable interaction

a novel generalization of clifford s classical point circle configuration geometric interpretation of the quaternionic discrete schwarzian kadomtsev petviashvili equation the algebraic and geometric properties of a novel generalization of clifford s classical 4 pointcircle configuration are analysed a connection with the integrable quaternionic discrete schwarzian

lattice and q difference darboux zakharov manakov systems via delta dressing method a general scheme is proposed for the introduction of lattice and q difference variables to integrable hierarchies in the frame of the delta dressing method using this scheme lattice and q difference darboux zakharov manakov systems of equations are derived darboux backlund and combescure transformations and exact solutions for these systems are studied

lattice equations hierarchies and hamiltonian structures by now many integrable systems with soliton solutions are known in two dimensions for fields u n1 n2 defined at the sites nln2 of a two dimensional 2d lattice for time dependent fields u n t defined at the sites n of a one dimensional chain and for fields u x t depending on two continuous variables x and t the relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method 1 2 which is based on a linear integral equation with arbitrary measure and contour 3 4 in the treatment use is made of backlund transformations bt s which are generated by scalar multiplications of the free wave function and or measure in the integral equation 5 the integrable lattice equations are obtained in the form of bianchi identities expressing the commutativity of bt s the integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time

cellular automata in 1 1 2 1 and 3 1 dimensions constants of motion and coherent structures abstract extending a procedure introduced in a previous paper we develop a general method to construct nonlinear cellular automata in arbitrary space dimensions from the compatibility of pairs of linear operators giving examples in 1 1 2 1 and 3 1 dimensions these cellular automata possess a lot of constants of motion and exhibit a vast array of coherent structures both particle like i e localized in space and wave like not localized

a note on the uniqueness result for the inverse henderson problem the inverse henderson problem of statistical mechanics is the theoretical foundation for many bottom up coarse graining techniques for the numerical simulation of complex soft matter physics this inverse problem concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles roughly stated it asks for the interaction potential given the equilibrium pair correlation function of the system in 1974 henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system here we provide a rigorous proof of a slightly more general version of the latter statement using georgii s variant of the gibbs variational principle the inverse henderson problem of statistical mechanics is the theoretical foundation for many bottom up coarse graining techniques for the numerical simulation of complex soft matter physics this inverse problem concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles roughly stated it asks for the interaction potential given the equilibrium pair correlation function of the system in 1974 henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system here we provide a rigorous proof of a slightly more general version of the latter statement using georgii s variant of the gibbs variational principle

asymptotic properties of multivariate nonstationary processes with applications to autoregressions asymptotic properties of multivariate time series with characteristic roots on the unit circle are considered for a vector autoregressive moving average process we derive the limiting distributions of certain statistics which are useful in understanding nonstationary processes these distributions are derived in a unified manner for all types of characteristic roots and are expressed in terms of stochastic integrals of brownian motions we use the limiting distributions to establish the consistency properties of the ordinary least squares estimates of various autoregressions of a vector process

menelaus theorem clifford configurations and inversive geometry of the schwarzian kp hierarchy it is shown that the integrable discrete schwarzian kp dskp equation which constitutes an algebraic superposition formula associated with for instance the schwarzian kp hierarchy the classical darboux transformation and quasi conformal mappings encapsulates nothing but a fundamental theorem of ancient greek geometry thus it is demonstrated that the connection with menelaus theorem and more generally clifford configurations renders the dskp equation a natural object of inversive geometry on the plane the geometric and algebraic integrability of dskp lattices and their reductions to lattices of menelaus darboux schwarzian kdv schwarzian boussinesq and schramm type are discussed the dskp and discrete schwarzian boussinesq equations are shown to represent discretizations of families of quasi conformal mappings

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given