rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

distributional limit theorems in infinite ergodic theory we present a unified approach to the darling kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator we illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps we also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

integrable counterparts of the d kaup newell soliton hierarchy two integrable counterparts of the d kaup newell soliton hierarchy are constructed from a matrix spectral problem associated with the three dimensional special orthogonal lie algebra so 3 r an application of the trace identity presents hamiltonian or quasi hamiltonian structures of the resulting counterpart soliton hierarchies thereby showing their liouville integrability i e the existence of infinitely many commuting symmetries and conserved densities the involved hamiltonian and quasi hamiltonian properties are shown by computer algebra systems

computational experience with confidence regions and confidence intervals for nonlinear least squares we present the results of a monte carlo study of the leading methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares we examine three variants of the linearization method the likelihood method and the lack of fit method the linearization method is computationally inexpensive produces easily understandable results and is widely used in practice the likelihood and lack of fit methods are much more expensive and more difficult to report in our tests both the likelihood and lack of fit methods perform very reliably all three variants of the linearization method however often grossly underestimate confidence regions and sometimes significantly underestimate confidence intervals the linearization method variant based solely on the jacobian matrix appears preferable to the two variants that use the full hessian matrix because it is less expensive more numerically stable and at least as accurate the bates and watts curvat

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

a spectral problem based on so 3 r and its associated commuting soliton equations we present a spectral problem based on the three dimensional real special orthogonal lie algebra so 3 r and construct a hierarchy of commuting bi hamiltonian soliton equations by zero curvature equations associated with the spectral problem an illustrative example of soliton equations is computed together with its associated bi hamiltonian structure

the algebraic structures of isospectral lax operators and applications to integrable equations for a general spectral operator the author establishes types of algebraic structures of the spaces of the corresponding isospectral lax operators which essentially form the theoretical basis of the lax operator method furthermore the author introduces the concepts of tau algebras and master algebras to describe time polynomial dependent symmetries of nonlinear integrable equations finally the author applies the theory of lax operators to the kp hierarchy of integrable equations as an illustrative example and thus obtain the master symmetry algebra of the kp hierarchy

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

a soliton hierarchy associated with a new spectral problem and its hamiltonian structure a hierarchy of soliton equations together with its hamiltonian structure is constructed from a new spectral problem associated with the three dimensional special orthogonal real lie algebra so 3 the liouville integrability of the presented soliton hierarchy is proved based on the hamiltonian structure

analysis of regression confidence intervals and bayesian credible intervals for uncertainty quantification 1 confidence intervals based on classical regression theories augmented to include prior information and credible intervals based on bayesian theories are conceptually different ways to quantify parametric and predictive uncertainties because both confidence and credible intervals are used in environmental modeling we seek to understand their differences and similarities this is of interest in part because calculating confidence intervals typically requires tens to thousands of model runs while bayesian credible intervals typically require tens of thousands to millions of model runs given multi gaussian distributed observation errors our theoretical analysis shows that for linear or linearized nonlinear models confidence and credible intervals are always numerically identical when consistent prior information is used for nonlinear models nonlinear confidence and credible intervals can be numerically identical if parameter confidence regions defined using the approximate likelihood method and parameter credible regions estimated using markov chain monte carlo realizations are numerically identical and predictions are a smooth monotonic function of the parameters both occur if intrinsic model nonlinearity is small while the conditions of gaussian errors and small intrinsic model nonlinearity are violated by many environmental models heuristic tests using analytical and numerical models suggest that linear and nonlinear confidence intervals can be useful approximations of uncertainty even under significantly nonideal conditions in the context of epistemic model error for a complex synthetic nonlinear groundwater problem the linear and nonlinear confidence and credible intervals for individual models performed similarly enough to indicate that the computationally frugal confidence intervals can be useful in many circumstances experiences with these groundwater models are expected to be broadly applicable to many environmental models we suggest that for environmental problems with lengthy execution times that make credible intervals inconvenient or prohibitive confidence intervals can provide important insight during model development when frequent calculation of uncertainty intervals is important to understanding the consequences of various model construction alternatives and data collection strategies strategic use of both confidence and credible intervals can be critical

exact scheffe type confidence intervals for output from groundwater flow models 2 combined use of hydrogeologic information and calibration data calibration data observed values corresponding to model computed values of dependent variables are incorporated into a general method of computing exact scheffe type confidence intervals analogous to the confidence intervals developed in part 1 cooley this issue for a function of parameters derived from a groundwater flow model parameter uncertainty is specified by a distribution of parameters conditioned on the calibration data this distribution was obtained as a posterior distribution by applying bayes theorem to the hydrogeologically derived prior distribution of parameters from part 1 and a distribution of differences between the calibration data and corresponding model computed dependent variables tests show that the new confidence intervals can be much smaller than the intervals of part 1 because the prior parameter variance covariance structure is altered so that combinations of parameters that give poor model fit to the data are unlikely the confidence intervals of part 1 and the new confidence intervals can be effectively employed in a sequential method of model construction whereby new information is used to reduce confidence interval widths at each stage

combining independent studies in a calibration problem abstract the problem of calibration in which the response variable is measured by k different methods or using different instruments is considered it is well known that the usual classical estimator for the unknown explanatory variable has infinite mean and mean squared error when k 1 in this article a linear combination of the classical estimators is proposed it is shown that the combined estimator has finite mean provided that k 2 and finite mean squared error provided that k 3 expressions for asymptotic bias and mean squared error are given also two confidence sets for the unknown exploratory variable are developed sufficient conditions under which they will be finite intervals are given the results are illustrated by a practical example

practical scheffe type credibility intervals for variables of a groundwater model simultaneous scheffe type credibility intervals the bayesian version of confidence intervals for variables of a groundwater flow model calibrated using a bayesian maximum a posteriori procedure were derived by cooley 1993b it was assumed that variances reflecting the expected differences between observed and model computed quantities used to calibrate the model are known whereas they would often be unknown for an actual model in this study the variances are regarded as unknown and variance variability from observation to observation is approximated by grouping the data so that each group is characterized by a uniform variance the credibility intervals are calculated from the posterior distribution which was developed by considering each group variance to be a random variable about which nothing is known a priori then eliminating it by integration numerical experiments using two test problems illustrate some characteristics of the credibility intervals nonlinearity of the statistical model greatly affected some of the credibility intervals indicating that credibility intervals computed using the standard linear model approximation may often be inadequate to characterize uncertainty for actual field problems the parameter characterizing the probability level for the credibility intervals was however accurately computed using a linear model approximation as compared with values calculated using second order and fully nonlinear formulations this allows the credibility intervals to be computed very efficiently

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

role of infinite invariant measure in deterministic subdiffusion statistical properties of the transport coefficient for deterministic subdiffusion are investigated from the viewpoint of infinite ergodic theory we find that the averaged diffusion coefficient is characterized by the infinite invariant measure of the reduced map we also show that when the time difference is much smaller than the total observation time the time averaged mean square displacement depends linearly on the time difference furthermore the diffusion coefficient becomes a random variable and its limit distribution is characterized by the universal law called the mittag leffler distribution

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

approximate predictive likelihood summary a predictive likelihood is given which approximates both bayes and maximum likeli hood predictive inference by expansion of a posterior likelihood this synthesizes and extends previous results and is widely applicable the approximation usually differs from exact bayes posterior predictive density by op n 2 and from exact predictive likelihood by op n 1 but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found the results are applied to the prediction various treatments of the likelihood function form the basis of parametric inference but until recently few non bayesian attempts have been made to define versions of it suitable for prediction however lauritzen 1974 hinkley 1979 and more recently butler 1986 have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic unfortunately they apply only when sufficiency provides a genuine reduction of the data moreover even if for a given problem an exact predictive likelihood exists the calculations needed to derive it can be onerous this paper provides approximations to hinkley lauritzen predictive likelihood accur ate to op n 1 in many cases which are applicable in situations where maximum likelihood estimation is regular they also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data the approximations are derived from a posterior predictive density and may if prior informa tion is available be regarded as bayesian procedures their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum and is fast and accurate using usual numerical maximization methods there is a close connexion with the approximations to posterior moments and marginal densities given by tierney kadane 1986 who suggest the same approxima tion as used here but give no detailed results for prediction section 2 of this paper shows how laplace s method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood section 3 discusses a connection with conditional inference essentially the same approximation

two integrable couplings of a generalized d kaup newell hierarchy and their hamiltonian and bi hamiltonian structures abstract we enlarge the spectral problem of a generalized d kaup newell d kn spectral problem solving the enlarged zero curvature equations we produce integrable couplings a reduction of the spectral matrix leads to a second integrable coupling system next bilinear forms that are symmetric ad invariant and non degenerate on the given non semisimple matrix lie algebra are computed to employ the variational identity the variational identity is then applied to the original enlarged spectral problem of a generalized d kn hierarchy and the reduced problem hamiltonian structures are presented as well as a bi hamiltonian formulation of the reduced problem both hierarchies have infinitely many commuting high order symmetries and conserved densities i e are liouville integrable

generalized lyapunov exponent as a unified characterization of dynamical instabilities the lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits a positive lyapunov exponent exponential dynamical instability is a manifestation of chaos here we propose the lyapunov pair which is based on the generalized lyapunov exponent as a unified characterization of nonexponential and exponential dynamical instabilities in one dimensional maps chaos is classified into three different types i e superexponential exponential and subexponential chaos using one dimensional maps we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the lyapunov pair in subexponential chaos we show superweak chaos which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth the scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process which is related to infinite ergodic theory

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

a soliton hierarchy associated with so 3 r we generate a hierarchy of soliton equations from zero curvature equations associated with the real lie algebra so 3 r and show that each equation in the resulting hierarchy has a bi hamiltonian structure and thus integrable in the liouville sense

renewal sequences and intermittency in this paper we examine the generating function z of a renewal sequence arising from the distribution of return times in the turbulent region for a class of piecewise affine interval maps introduced by gaspard and wang and studied by several authors we prove that it admits a meromorphic continuation to the entire complex z plane with a branch cut along the ray 1 moreover we compute the asymptotic behavior of the coefficients of its taylor expansion at z 0 from this we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite

improved parameter inference in catchment models 2 combining different kinds of hydrologic data and testing their compatibility often some of the parameters of catchment models fitted to runoff data are poorly determined thereby making the task of developing useful regionalization relationships more difficult the bayesian methodology developed in part 1 is extended to utilize several kinds of hydrologic data in parameter inference the goal being to improve the precision of poorly determined parameters the concept of compatibility is developed using statistical hypothesis tests different kinds of data are said to be compatible if differences between their fitted parameters are not statistically significant the pooling of incompatible data may undermine the model s ability to predict runoff and also induce bias in the parameters a hierarchy of three levels of information is introduced to enable systematic checking for compatibility finally a case study is presented using data on runoff soil moisture and interception it is shown that substantial reductions in parameter uncertainty can be realized also the importance of compatibility testing is demonstrated

distributional behavior of time averages of non l 1 observables in one dimensional intermittent maps with infinite invariant measures in infinite ergodic theory two distributional limit theorems are well known one is characterized by the mittag leffler distribution for time averages of l 1 m functions i e integrable functions with respect to an infinite invariant measure the other is characterized by the generalized arc sine distribution for time averages of non l 1 m functions here we provide another distributional behavior of time averages of non l 1 m functions in one dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure observation functions considered here are non l 1 m functions which vanish at the indifferent fixed point we call this class of observation functions weak non l 1 m function our main result represents a first step toward a third distributional limit theorem i e a distributional limit theorem for this class of observables in infinite ergodic theory to prove our proposition we propose a stochastic process induced by a renewal process to mimic a birkoff sum of a weak non l 1 m function in the one dimensional intermittent maps

on inverse recursion operator and tri hamiltonian formulation for a kaup newell system of dnls equations an inverse of the recursion operator is computed and a clear presentation of a tri hamiltonian formulation is provided for a kaup newell system of derivative nonlinear schrodinger dnls equations therefore all kaup newell systems in the whole hierarchy of dnls equations are tri hamiltonian and have an inverse hierarchy of common commuting symmetries

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

evaluation of prediction intervals for expressing uncertainties in groundwater flow model predictions we tested the accuracy of 95 individual prediction intervals for hydraulic heads streamflow gains and effective transmissivities computed by groundwater models of two danish aquifers to compute the intervals we assumed that each predicted value can be written as the sum of a computed dependent variable and a random error testing was accomplished by using a cross validation method and by using new field measurements of hydraulic heads and transmissivities that were not used to develop or calibrate the models the tested null hypotheses are that the coverage probability of the prediction intervals is not significantly smaller than the assumed probability 95 and that each tail probability is not significantly different from the assumed probability 2 5 in all cases tested these hypotheses were accepted at the 5 level of significance we therefore conclude that for the groundwater models of two real aquifers the individual prediction intervals appear to be accurate

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

bihamiltonian geometry darboux coverings and linearization of the kp hierarchy we use ideas of the geometry of bihamiltonian manifolds developed by gel fand and zakharevich to study the kp equations in this approach they have the form of local conservation laws and can be traded for a system of ordinary differential equations of riccati type which we call the central system we show that the latter can be linearized by means of a darboux covering and we use this procedure as an alternative technique to construct rational solutions of the kp equations

generalized arcsine law and stable law in an infinite measure dynamical system limit theorems for the time average of some observation functions in an infinite measure dynamical system are studied it is known that intermittent phenomena such as the rayleigh benard convection and belousov zhabotinsky reaction are described by infinite measure dynamical systems we show that the time average of the observation function which is not the l 1 m function whose average with respect to the invariant measure m is finite converges to the generalized arcsine distribution this result leads to the novel view that the correlation function is intrinsically random and does not decay moreover it is also numerically shown that the time average of the observation function converges to the stable distribution when the observation function has the infinite mean

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

an integrable generalization of the d kaup newell soliton hierarchy and its bi hamiltonian reduced hierarchy abstract we present a new spectral problem a generalization of the d kaup newell spectral problem associated with the lie algebra sl 2 r zero curvature equations furnish the soliton hierarchy the trace identity produces the hamiltonian structure for the hierarchy and shows its liouville integrability lastly a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi hamiltonian structure the major motivation of this paper is to present spectral problems that generate two soliton hierarchies with infinitely many conservation laws and high order symmetries

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

simultaneous confidence and prediction intervals for nonlinear regression models with application to a groundwater flow model methods are presented for computing three types of simultaneous confidence and prediction intervals exact likelihood ratio and linearized on output from nonlinear regression models with normally distributed residuals the confidence intervals can be placed on individual regression parameters or on the true regression function at any number of points in the domain of the independent variables and the prediction intervals can be placed on any number of future observations the confidence intervals are analogous to simultaneous scheffe intervals for linear models and the prediction intervals are analogous to the prediction intervals of hahn 1972 all three types of intervals can be computed efficiently by using the same straightforward lagrangian optimization scheme the prediction intervals can be treated in the same computational framework as the confidence intervals by including the random errors as pseudoparameters in the lagrangian scheme the methods are applied to a hypothetical groundwater model for flow to a well penetrating a leaky aquifer three different data sets are used to demonstrate the effect of sampling strategies on the intervals for all three data sets the linearized confidence intervals are inferior to the exact and likelihood ratio intervals with the latter two being very similar however all three types of prediction intervals yielded similar results the third data set time drawdown data at only a single observation well points out many of the problems that can arise from extreme nonlinear behavior of the regression model

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

the weibull log weibull distribution for interoccurrence times of earthquakes abstract by analyzing the japan meteorological agency jma seismic catalog for different tectonic settings we have found that the probability distributions of time intervals between successive earthquakes interoccurrence times can be described by the superposition of the weibull distribution and the log weibull distribution in particular the distribution of large earthquakes obeys the weibull distribution with the exponent 1 1 indicating the fact that the sequence of large earthquakes is not a poisson process it is found that the ratio of the weibull distribution to the probability distribution of the interoccurrence time gradually increases with increase in the threshold of magnitude our results infer that weibull statistics and log weibull statistics coexist in the interoccurrence time statistics and that the change of the distribution is considered as the change of the dominant distribution in this case the dominant distribution changes from the log weibull distribution to the weibull distribution allowing us to reinforce the view that the interoccurrence time exhibits the transition from the weibull regime to the log weibull regime

aging generates regular motions in weakly chaotic systems using intermittent maps with infinite invariant measures we investigate the universality of time averaged observables under aging conditions according to aaronson s darling kac theorem in non aged dynamical systems with infinite invariant measures the distribution of the normalized time averages of integrable functions converges to the mittag leffler distribution this well known theorem holds when the start of observations coincides with the start of the dynamical processes introducing a concept of the aging limit where the aging time t a and the total measurement time t goes to infinity while the aging ratio t a t is kept constant we obtain a novel distributional limit theorem of time averaged observables integrable with respect to the infinite invariant density applying the theorem to the lyapunov exponent in intermittent maps we find that regular motions and a weakly chaotic behavior coexist in the aging limit this mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in hamiltonian systems the probability of finding regular motions in non aged processes is zero while in the aging regime it is finite and it increases when system ages

exact scheffe type confidence intervals for output from groundwater flow models 1 use of hydrogeologic information a new method is developed to efficiently compute exact scheffe type confidence intervals for output or other function of parameters g beta derived from a groundwater flow model the method is general in that parameter uncertainty can be specified by any statistical distribution having a log probability density function log pdf that can be expanded in a taylor series however for this study parameter uncertainty is specified by a statistical multivariate beta distribution that incorporates hydrogeologic information in the form of the investigator s best estimates of parameters and a grouping of random variables representing possible parameter values so that each group is defined by maximum and minimum bounds and an ordering according to increasing value the new method forms the confidence intervals from maximum and minimum limits of g beta on a contour of a linear combination of 1 the quadratic form for the parameters used by cooley and vecchia 1987 and 2 the log pdf for the multivariate beta distribution three example problems are used to compare characteristics of the confidence intervals for hydraulic head obtained using different weights for the linear combination different weights generally produced similar confidence intervals whereas the method of cooley and vecchia 1987 often produced much largermore confidence intervals less

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

on liouville integrability of zero curvature equations and the yang hierarchy sufficient conditions for a zero curvature equation ut vx u v 0 being liouville integrable are investigated in the case that the equation is integrable an explicit formula of the poisson bracket h lambda h mu for hamiltonians h is proposed the yang hierarchy is derived and shown to be liouville integrable

distributional response to biases in deterministic superdiffusion we report on a novel response to biases in deterministic superdiffusion for its reduced map we show using infinite ergodic theory that the time averaged velocity tav is intrinsically random and its distribution obeys the generalized arcsine distribution a distributional limit theorem indicates that the tav response to a bias appears in the distribution which is an example of what we term a distributional response induced by a bias although this response in single trajectories is intrinsically random the ensemble averaged tav response is linear

a coupled akns kaup newell soliton hierarchy a coupled akns kaup newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from hamiltonian pairs zero curvature representations and tri hamiltonian structures are established for all coupled akns kaup newell systems in the hierarchy therefore all systems have infinitely many commuting symmetries and conservation laws two reductions of the systems lead to the akns hierarchy and the kaup newell hierarchy and thus those two soliton hierarchies also possess tri hamiltonian structures