on a class of three dimensional integrable lagrangians we characterize non degenerate lagrangians of the form such that the corresponding euler lagrange equations are integrable by the method of hydrodynamic reductions the integrability conditions constitute an over determined system of fourth order pdes for the lagrangian density f which is in involution and possesses interesting differential geometric properties the moduli space of integrable lagrangians factorized by the action of a natural equivalence group is three dimensional familiar examples include the dispersionless kadomtsev petviashvili dkp and the boyer finley lagrangians f u x 3 3 u y 2 u x u t and respectively a complete description of integrable cubic and quartic lagrangians is obtained up to the equivalence transformations the list of integrable cubic lagrangians reduces to three examples there exists a unique integrable quartic lagrangian we conjecture that these examples exhaust the list of integrable polynomial lagrangians which are essentially three dimensional it was verified that there exist no polynomial integrable lagrangians of degree five we prove that the euler lagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless lax pair

classification of integrable hydrodynamic chains and generating functions of conservation laws a new approach to classification of integrable hydrodynamic chains is established generating functions of conservation laws are classified by the method of hydrodynamic reductions the n parametric family of explicit hydrodynamic reductions allows us to reconstruct corresponding hydrodynamic chains many new hydrodynamic chains are found

lower dimensional reductions of gl m c self dual yang mills equation solutions with break of wave profiles any matrix self dual type s integrable partial differential equation pde possesses a family of lower dimensional reductions represented by the matrix quasilinear first order pdes solvable by the method of characteristics these pdes admit two types of particular solutions a explicit solutions and b implicit solutions represented by a system of nondifferential equations the latter solutions exhibit the break of wave profile 2 1 and 3 1 dimensional gl m c self dual yang mills equations have been considered as examples

linearly degenerate hierarchies of quasiclassical sdym type we demonstrate that sdym self dual yang mills equations for the lie algebra of one dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy we define the reduction in terms of wave functions and introduce a generating relation lax sato equations and the dressing scheme for the reduced hierarchy a multidimensional case is also discussed

solvable vector nonlinear riemann problems exact implicit solutions of dispersionless pdes and wave breaking we have recently solved the inverse spectral problem for integrable partial differential equations pdes in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter the associated inverse problem in particular can be formulated as a nonlinear riemann hilbert nrh problem on a given contour of the complex plane the most distinguished examples of integrable pdes of this type like the dispersionless kadomtsev petviashivili dkp the heavenly and the two dimensional dispersionless toda equations are real pdes associated with hamiltonian vector fields the corresponding nrh data satisfy suitable reality and symplectic constraints in this paper generalizing the examples of solvable nrh problems illustrated in manakov and santini 2009 j phys a math theor 42 095203 2008 j phys a math theor 41 055204 2009 j phys a math theor 42 404013 we present a general procedure to construct solvable nrh problems for integrable real pdes associated with hamiltonian vector fields allowing one to construct exact implicit solutions of such pdes parametrized by an arbitrary number of real functions of a single variable then we illustrate this theory on few distinguished examples for the dkp and heavenly equations for the dkp case we characterize a class of similarity solutions of solutions constant on their parabolic wave front and breaking simultaneously on it of localized solutions whose breaking point travels with constant speed along the wave front and of localized solutions breaking in a point of the x y plane for the heavenly equation we characterize two classes of symmetry reductions

hydrodynamic chains and the classification of their poisson brackets infinite component poisson brackets of the dubrovin novikov type sov math dokl 27 665 669 1983 are considered the corresponding jacobi identity is significantly simplified in the liouville coordinates since the skew symmetry condition is automatically satisfied the concept of m poisson bracket connected with hydrodynamic chains is introduced then the jacobi identity is a nonlinear system of equations in partial derivatives which can be completely integrated in such a case a classification of infinite component poisson brackets of the dubrovin novikov type can be obtained two simplest examples m 0 and m 1 are considered also infinite component poisson brackets of the ferapontov am math soc transl 170 38 58 1995 type can be simplified in the liouville coordinates the jacobi identity for infinite component poisson brackets of the ferapontov mokhov type russ math surveys 45 218 219 1990 is presented in the liouville coordinates

classical r matrix theory of dispersionless systems ii 2 1 dimension theory a systematic way of constructing 2 1 dimensional dispersionless integrable hamiltonian systems is presented the method is based on the so called central extension procedure and classical r matrix applied to the poisson algebras of formal laurent series results are illustrated with the known and new 2 1 dimensional dispersionless systems

a search for bilinear equations passing hirota s three soliton condition ii mkdv type bilinear equations in this paper the results of a search for complex bilinear equations with two soliton solutions are presented the following basic types are discussed a the nonlinear schrodinger equation b dx g f 0 a dx dt f f gg and b the benjamin ono equation p dx f f 0 it is found that the existence of two soliton solutions is not automatic but introduces conditions that are like the usual three and four soliton conditions the search was limited by the degree of a 2 and by degree of p 4 the main results are the following 1 iad3x dxdt idy b g f 0 d2xf f gg 2 d2x ad2y idt b g f 0 dxdy f f gg 3 iad3x d2x idt f f 0 and 4 dxdt i adx bdt f f 0

coisotropic deformations of associative algebras and dispersionless integrable hierarchies the paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies like the dispersionless kp and modified kp hierarchies and the universal whitham hierarchy of genus zero it stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras it discusses the link between the structure constants and hirota s tau function and shows that the dispersionless hirota bilinear equations are within this approach a way of writing the associativity conditions for the structure constants in terms of the tau function it also suggests a simple interpretation of the algebro geometric construction of the universal whitham equations of genus zero due to krichever

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

on the integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

prediction via orthogonalized model mixing abstract we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors we focus on the choice of predictors in linear models and mix over possible subsets of candidate predictors our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix advantages are both statistical and computational statistically orthogonalization often leads to a reduction in the number of competing models by eliminating correlations computationally large model spaces cannot be enumerated recent approaches are based on sampling models with high posterior probability via markov chains based on orthogonalization of the space of candidate predictors we can approximate the posterior probabilities of models by products of predictor specific terms this leads to an importance sampling function for sampling directly from the joint distribution over the model space without resorting to markov chains compared to the latter ortho

towards classification of dimensional integrable equations integrability conditions i in this paper we attempt to extend the symmetry approach well developed in the case of 1 1 dimensional equations to the 2 1 dimensional case presence of nonlocal terms in symmetries and conservation laws is the main feature of integrable 2 1 dimensional equations we have introduced a concept of quasi local functions to characterize nonlocalities we have found a few first integrability conditions for a class of scalar equations in terms of quasi local functions and have demonstrated that they are suitable for testing integrability

the inverse spectral transform for the dunajski hierarchy and some of its reductions i cauchy problem and longtime behavior of solutions in this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations pdes arising from the commutation condition of pairs of one parameter families of vector fields recently developed by s v manakov and one of the authors to one distinguished class of equations the so called dunajski hierarchy we concentrate for concreteness i on the system of pdes characterizing a general anti self dual conformal structure in neutral signature ii on its first commuting flow and iii on some of their basic and novel reductions we formally solve their cauchy problem and we use it to construct the longtime behavior of solutions showing in particular that unlike the case of soliton pdes different dispersionless pdes belonging to the same hierarchy of commuting flows evolve in time in very different ways exhibiting either a smooth dynamics or a gradient catastrophe at finite time

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

differential geometric approach to the integrability of hydrodynamic chains the haantjes tensor the integrability of an m component system of hydrodynamic type u t v u u x by the generalized hodograph method requires the diagonalizability of the m m matrix v u this condition is known to be equivalent to the vanishing of the corresponding haantjes tensor we generalize this approach to hydrodynamic chains infinite component systems of hydrodynamic type for which the matrix v u is sufficiently sparse for such systems the haantjes tensor is well defined and the calculation of its components involves finite summations only we illustrate our approach by classifying broad classes of conservative and hamiltonian hydrodynamic chains with the zero haantjes tensor we prove that the vanishing of the haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi hamiltonian hydrodynamic reductions thus providing an easy to verify necessary condition for the integrability

three dimensional reductions of four dimensional quasilinear systems in this paper we show that four dimensional quasilinear systems of first order integrable by the method of two dimensional hydrodynamic reductions possess infinitely many three dimensional hydrodynamic reductions which are also integrable systems these three dimensional multi component integrable systems are irreducible to two dimensional hydrodynamic reductions in a generic case

conditional symmetries and riemann invariants for inhomogeneous hydrodynamic type systems a new approach to the solution of quasilinear nonelliptic first order systems of inhomogeneous partial differential equations in many dimensions is presented it is based on a version of the conditional symmetry and riemann invariant methods we discuss in detail the necessary and sufficient conditions for the existence of rank 2 and rank 3 solutions expressible in terms of riemann invariants we perform the analysis using the cayley hamilton theorem for a certain algebraic system associated with the initial system the problem of finding such solutions has been reduced to expanding a set of trace conditions on wave vectors and their profiles which are expressible in terms of riemann invariants a couple of theorems useful for the construction of such solutions are given these theoretical considerations are illustrated by the example of inhomogeneous equations of fluid dynamics which describe motion of an ideal fluid subjected to gravitational and coriolis forces several new rank 2 solutions are obtained some physical interpretation of these results is given

nonparametric regression using bayesian variable selection abstract this paper estimates an additive model semiparametrically while automatically selecting the significant independent variables and the appropriate power transformation of the dependent variable the nonlinear variables are modeled as regression splines with significant knots selected from a large number of candidate knots the estimation is made robust by modeling the errors as a mixture of normals a bayesian approach is used to select the significant knots the power transformation and to identify outliers using the gibbs sampler to carry out the computation empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares favorably with a kernel weighted local linear smoother the variable selection algorithm in the paper is substantially faster than previous bayesian variable selection algorithms

on integrability in grassmann geometries integrable systems associated with fourfolds in gr 3 5 let gr d n be the grassmannian of d dimensional linear subspaces of an n dimensional vector space v n a submanifold x gr d n gives rise to a differential system x that governs d dimensional submanifolds of v n whose gaussian image is contained in x systems of the form x appear in numerous applications in continuum mechanics theory of integrable systems general relativity and differential geometry they include such wellknown examples as the dispersionless kadomtsev petviashvili equation the boyer finley equation plebansky s heavenly equations and so on in this paper we concentrate on the particularly interesting case of this construction where x is a fourfold in gr 3 5 our main goal is to investigate differential geometric and integrability aspects of the corresponding systems x we demonstrate the equivalence of several approaches to dispersionless integrability such as the method of hydrodynamic reductions the method of dispersionless lax pairs integrability on solutions based on the requirement that the characteristic variety of system x defines an einstein weyl geometry on every solution integrability on equation meaning integrability in twistor theoretic sense of the canonical gl 2 r structure induced on a fourfold x gr 3 5 all these seemingly different approaches lead to one and the same class of integrable systems x we prove that the moduli space of such systems is 6 dimensional we give a complete description of linearisable systems the corresponding fourfold x is a linear section of gr 3 5 and linearly degenerate systems the corresponding fourfold x is the image of a quadratic map p4 99k gr 3 5 the fourfolds corresponding to generic integrable systems are not algebraic and can be parametrised by generalised hypergeometric functions

the initial value problem for the dkp hierarchy the solution corresponding to each initial condition for the dispersionless kp hierarchy can be found from the integration of a hamilton jacobi equation by means of a transformation of coordinates the solution is explicitly determined in a parametric form as well as a power series in a deformation parameter in addition a twistor formulation of the solution is given

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

on the central quadric ansatz integrable models and painleve reductions it was observed by tod 1995 class quantum grav 12 1535 47 and later by dunajski and tod 2002 phys lett a 303 253 64 that the boyer finley bf and the dispersionless kadomtsev petviashvili dkp equations possess solutions whose level surfaces are central quadrics in the space of independent variables the so called central quadric ansatz it was demonstrated that generic solutions of this type are described by painlev equations piii and pii respectively the aim of our paper is threefold 1 based on the method of hydrodynamic reductions we classify integrable models possessing the central quadric ansatz this leads to the five canonical forms including bf and dkp 2 applying the central quadric ansatz to each of the five canonical forms we obtain all painlev equations pi pvi with pvi corresponding to the generic case of our classification 3 we argue that solutions coming from the central quadric ansatz constitute a subclass of two phase solutions provided by the method of hydrodynamic reductions

bayesian regression modeling with interactions and smooth effects abstract there have been many recent suggestions as to how to build and estimate flexible bayesian regression models using constructs such as trees neural networks and gaussian processes although there is much to commend these methods their implementation and interpretation can be daunting for practitioners this article presents a spline based methodology for flexible bayesian regression that is quite simple in terms of computation and interpretation smooth bivariate interactions are modeled in an economical and apparently novel way and prior distributions that penalize complexity are used predictions can be based on either model selection or model averaging taking computation interpretation and predictive performance into account the method is seen to perform well when applied to simulated and real data

dispersionless 3 1 dimensional integrable hierarchies in this paper we introduce a multi dimensional version of the r matrix approach to the construction of integrable hierarchies applying this method to the case of the lie algebra of functions with respect to the contact bracket we construct integrable hierarchies of 3 1 dimensional dispersionless systems of the type recently introduced in sergyeyev 2014 http arxiv org abs 1401 2122

whitham modulation theory for the kadomtsev petviashvili equation the genus 1 kadomtsev petviashvili kp whitham system is derived for both variants of the kp equation namely the kpi and kpii equations the basic properties of the kp whitham system including symmetries exact reductions and its possible complete integrability together with the appropriate generalization of the one dimensional riemann problem for the korteweg de vries equation are discussed finally the kp whitham system is used to study the linear stability properties of the genus 1 solutions of the kpi and kpii equations it is shown that all genus 1 solutions of kpi are linearly unstable while all genus 1 solutions of kpii are linearly stable within the context of whitham theory

a reference bayesian test for nested hypotheses and its relationship to the schwarz criterion abstract to compute a bayes factor for testing h 0 0 in the presence of a nuisance parameter priors under the null and alternative hypotheses must be chosen as in bayesian estimation an important problem has been to define automatic or reference methods for determining priors based only on the structure of the model in this article we apply the heuristic device of taking the amount of information in the prior on equal to the amount of information in a single observation then after transforming to be null orthogonal to we take the marginal priors on to be equal under the null and alternative hypotheses doing so and taking the prior on to be normal we find that the log of the bayes factor may be approximated by the schwarz criterion with an error of order o p n rather than the usual error of order o p 1 this result suggests the schwarz criterion should provide sensible approximate solutions to bayesian testing problems at least when the hypotheses are nested when

variable selection and function estimation in additive nonparametric regression using a data based prior abstract a hierarchical bayesian approach is proposed for variable selection and function estimation in additive nonparametric gaussian regression models and additive nonparametric binary regression models the prior for each component function is an integrated wiener process resulting in a posterior mean estimate that is a cubic smoothing spline each of the explanatory variables is allowed to be in or out of the model and the regression functions are estimated by model averaging to allow variable selection and model averaging data based priors are used for the smoothing parameter and the slope at 0 of each component function a two step markov chain monte carlo method is used to efficiently obtain the data based prior and to carry out variable selection and function estimation it is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously the methodology is illustrated for a binary regressio

conditional symmetries and riemann invariants for hyperbolic systems of pdes this paper contains an analysis of rank k solutions in terms of riemann invariants obtained from interrelations between two concepts that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of pdes in many dimensions a variant of the conditional symmetry method for obtaining this type of solutions is proposed a lie module of vector fields which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints is constructed it is shown that this overdetermined system admits rank k solutions expressible in terms of riemann invariants finally examples of applications of the proposed approach to the fluid dynamics equations in k 1 dimensions are discussed in detail several new soliton like solutions among them kinks bumps and multiple wave solutions have been obtained these solutions remain bounded even when the riemann invariants admit a gradient catastrophe some physical interpretation of these results is discussed

dispersive deformations of hamiltonian systems of hydrodynamic type in 2 1 dimensions abstract we develop a theory of integrable dispersive deformations of 2 1 dimensional hamiltonian systems of hydrodynamic type following the scheme proposed by dubrovin and his collaborators in 1 1 dimensions our results show that the multi dimensional situation is far more rigid and generic hamiltonians are not deformable as an illustration we discuss a particular class of two component hamiltonian systems establishing the triviality of first order deformations and classifying hamiltonians possessing nontrivial deformations of the second order

the dispersionless 2d toda equation dressing cauchy problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse spectral problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation arising as commutation of vector fields in this paper we make use of the above theory i to construct the nonlinear riemann hilbert dressing for the so called two dimensional dispersionless toda equation elucidating the spectral mechanism responsible for wave breaking ii we present the formal solution of the cauchy problem for the wave form of it exp tt xx yy iii we obtain the longtime behaviour of the solutions of such a cauchy problem showing that it is essentially described by the longtime breaking formulae of the dkp solutions confirming the expected universal character of the dkp equation as prototype model in the description of the gradient catastrophe of two dimensional waves iv we finally characterize a class of spectral data allowing one to linearize the rh problem corresponding to a class of implicit solutions of the pde

on the classification of scalar evolutionary integrable equations in 2 1 dimensions we consider evolutionary equations of the form ut f u w where w dx 1dyu is the nonlocality and the right hand side f is polynomial in the derivatives of u and w the recent paper ferapontov moro and novikov j phys a math theor 52 18 2009 provides a complete list of integrable third order equations of this kind here we extend the classification to fifth order equations besides the known examples of kadomtsev petviashvili veselov novikov and harry dym equations as well as fifth order analogs and modifications thereof our list contains a number of equations which are apparently new we conjecture that our examples exhaust the list of scalar polynomial integrable equations with the nonlocality w the classification procedure consists of two steps first we classify quasilinear systems which may potentially occur as dispersionless limits of integrable scalar evolutionary equations after that we reconstruct dispersive terms based on the requirement of the inheritance of hydrodynamic

classification of integrable two component hamiltonian systems of hydrodynamic type in 2 1 dimensions hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics the whitham averaging procedure and the theory of frobenius manifolds in 1 1 dimensions the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable hamiltonians depend on a certain number of arbitrary functions of two variables on the contrary in 2 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions which is a natural analogue of the generalised hodograph transform in higher dimensions leads to finite dimensional moduli spaces of integrable hamiltonians in this paper we classify integrable two component hamiltonian systems of hydrodynamic type for all existing classes of differential geometric poisson brackets in 2d establishing a parametrisation of integrable hamiltonians via elliptic hypergeometric functions our approach is based on the godunov type representation of hamiltonian systems and u

looped cotangent virasoro algebra and non linear integrable systems in dimension 2 1 we consider a lie algebra generalizing the virasoro algebra to the case of two space variables we study its coadjoint representation and calculate the corresponding euler equations in particular we obtain a bi hamiltonian system that leads to an integrable non linear partial differential equation this equation is an analogue of the kadomtsev petviashvili of type b equation

benchmark priors for bayesian model averaging in contrast to a posterior analysis given a particular sampling model posterior model probabilities in the context of model uncertainty are typically rather sensitive to the speciflcation of the prior in particular difiuse priors on model speciflc parameters can lead to quite unexpected consequences here we focus on the practically relevant situation where we need to entertain a large number of sampling models and we have or wish to use little or no subjective prior information we aim at providing an automatic or benchmark prior structure that can be used in such cases we focus on the normal linear regression model with uncertainty in the choice of re gressors we propose a partly noninformative prior structure related to a natural conjugate g prior speciflcation where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j the consequences of difierent choices for g0j are examined we investigate theoretical properties such as consistency of the implied bayesian procedure links with classical information criteria are provided in addition we examine the flnite sample implications of several choices of g0j in a simulation study the use of the mc 3 algorithm of madigan and york 1995 combined with e cient coding in fortran makes it feasible to conduct large simulations in addition to posterior criteria we shall also compare the predictive performance of difierent priors a classic exam ple concerning the economics of crime will also be provided and contrasted with results in the literature the main flndings of the paper will lead us to propose a benchmark prior speciflcation in a linear regression context with model uncertainty

on the solutions of the dkp equation the nonlinear riemann hilbert problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation we showed in particular that the associated inverse problems can be expressed in terms of nonlinear riemann hilbert problems on the real axis in this paper we make use of the nonlinear riemann hilbert problem of dkp i to construct the longtime behaviour of the solutions of its cauchy problem ii to characterize a class of implicit solutions iii to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dkp at finite time as well as in the longtime regime and the corresponding universal behaviours near breaking

hodograph transformation and differential constraints for wave solutions to 2 2 quasilinear hyperbolic nonhomogeneous systems the differential constraint method is used to work out a reduction approach to determine solutions in a closed form to the highly nonlinear hodograph system arising from 2 2 hyperbolic nonhomogeneous models these solutions inherit all of the features of the standard wave solutions obtainable via the classical hodograph transformation and in the meantime incorporate the dissipative effects induced on wave processes by the source like term involved in the governing equations within such a theoretical framework the problem of integrating the standard linear hodograph system associated with 2 2 homogeneous models is also revisited and a number of results obtained elsewhere of relevant interest in wave problems are recovered as a particular case along the lines of the proposed reduction approach different examples of 2 2 governing models are analysed thoroughly in order to highlight the flexibility of the provided solutions to describe hyperbolic dissipative wave processes

the cauchy problem for the pavlov equation with large data abstract we prove a local solvability of the cauchy problem for the pavlov equation with large initial data by the inverse scattering method the pavlov equation arises in studies einstein weyl geometries and dispersionless integrable models our theory yields a local solvability of cauchy problems for a quasi linear wave equation with a characteristic initial hypersurface

a bayesian approach to nonparametric bivariate regression abstract this article outlines a general bayesian approach to estimating a bivariate regression function in a nonparametric manner it models the function using a bivariate regression spline basis with many terms binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression the regression function is estimated using an estimate of its posterior mean smoothing over the distribution of these binary indicator variables to make the computations tractable all estimates are obtained using markov chain monte carlo sampling extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data driven bivariate surface estimators prominent in the literature it is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings the first

the haantjes tensor and double waves for multi dimensional systems of hydrodynamic type a necessary condition for integrability an invariant differential geometric approach to the integrability of 2 1 dimensional systems of hydrodynamic type is developed we prove that the existence of special solutions known as double waves is equivalent to the diagonalizability of an arbitrary matrix of the two parameter familysince the diagonalizability can be effectively verified by calculating the haantjes tensor this provides a simple necessary condition for integrability

bayesian cart model search abstract in this article we put forward a bayesian approach for finding classification and regression tree cart models the two basic components of this approach consist of prior specification and stochastic search the basic idea is to have the prior induce a posterior distribution that will guide the stochastic search toward more promising cart models as the search proceeds such models can then be selected with a variety of criteria such as posterior probability marginal likelihood residual sum of squares or misclassification rates examples are used to illustrate the potential superiority of this approach over alternative methods

on the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields it was shown recently that frobenius reduction of the matrix fields reveals interesting relations among the nonlinear partial differential equations pdes integrable by the inverse spectral transform method s integrable pdes linearizable by the hopf cole substitution c integrable pdes and integrable by the method of characteristics ch integrable pdes however only two classes of s integrable pdes have been involved soliton equations such as korteweg de vries nonlinear schrodinger kadomtsev petviashvili and davey stewartson equations and gl n c self dual type pdes such as yang mills equation in this paper we consider the simple five dimensional nonlinear pde from another class of s integrable pdes namely scalar nonlinear pde which is the commutativity condition of the pair of vector fields we show its origin from the 1 1 dimensional hierarchy of ch integrable pdes after certain composition of frobenius type and differential reductions imposed on the matrix fields matrix generaliza

algebro geometric approach in the theory of integrable hydrodynamic type systems the algebro geometric approach for integrability of semi hamiltonian hydrodynamic type systems is presented the class of symmetric hydrodynamic type systems is defined and the calculation of the associated riemann surfaces is greatly simplified for this class many interesting and physically motivated examples are investigated

a bayesian cart algorithm a stochastic search form of classification and regression tree cart analysis breiman et al 1984 is proposed motivated by a bayesian model an approximation to a probability distribution over the space of possible trees is explored using reversible jump markov chain monte carlo methods green 1995

killing vectors in self dual euclidean einstein spaces using the formalism of complex h spaces we show that all real euclidean self dual spaces that admit at least one killing vector may be gauged so that only two distinct types of killing vectors appear in kahler coordinates these are the generators of a translational or a rotational symmetry we give explicit forms both for the killing vectors and for the constraint on the kahler potential function which allows for such a killing vector in the translational case we show how all such spaces are determined by the general solution of the three dimensional flat laplace s equation and how these are related to the multi taub nut metrics of gibbons and hawking in the rotational case we simplify the equation determining but this is not sufficient to obtain the general solution

hydrodynamic reductions of multidimensional dispersionless pdes the test for integrability a d 1 dimensional dispersionless pde is said to be integrable if its n component hydrodynamic reductions are locally parametrized by d 1 n arbitrary functions of one variable the most important examples include the four dimensional heavenly equation descriptive of self dual ricci flat metrics and its six dimensional generalization arising in the context of sdiff 2 self dual yang mills equations given a multidimensional pde which does not pass the integrability test the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which when added to the equation make it an integrable system in fewer dimensions as an example of this phenomenon we discuss the second commuting flow of the dispersionless kp hierarchy considered separately this is a four dimensional pde which does not pass the integrability test however the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full 2 1 dimensional dis

the characterization of two component 2 1 dimensional integrable systems of hydrodynamic type we obtain the necessary and sufficient conditions for a two component 2 1 dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions these conditions are in involution implying that the systems in question are locally parametrized by 15 arbitrary constants it is proved that all such systems possess three conservation laws of hydrodynamic type and therefore are symmetrizable in godunov s sense moreover all such systems are proved to possess a scalar pseudopotential which plays the role of the dispersionless lax pair we demonstrate that the class of two component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions thus establishing the equivalence of the two possible definitions of integrability explicit linearly degenerate examples are constructed

simple waves in quasilinear hyperbolic systems ii riemann invariants for the problem of simple wave interactions in this paper a generalization of the riemann invariant method to the case of a nonhomogeneous system of equations has been formulated we have discussed in detail the necessary and sufficient conditions for the existence of riemann invariants we perform the analysis using the apparatus of differential forms and cartan theory of systems in involution the problem has been reduced to examining pfaff forms we have considered the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions depending on k arbitrary functions of one variable these solutions can be interpreted physically as the interactions between k simple waves on a simple state we have proven that in the case of interaction of many simple waves described by riemann invariants a conservation law for the type and quantity of waves holds it has been also shown that such a solution resulting from the interaction of many simple waves propagating on the simple state

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

variational inference for latent variables and uncertain inputs in gaussian processes the gaussian process latent variable model gp lvm provides a flexible approach for non linear dimensionality reduction that has been widely applied however the current approach for training gp lvms is based on maximum likelihood where the latent projection variables are maximised over rather than integrated out in this paper we present a bayesian method for training gp lvms by introducing a non standard variational inference framework that allows to approximately integrate out the latent variables and subsequently train a gp lvm by maximising an analytic lower bound on the exact marginal likelihood we apply this method for learning a gp lvm from i i d observations and for learning non linear dynamical systems where the observations are temporally correlated we show that a benefit of the variational bayesian procedure is its robustness to overfitting and its ability to automatically select the dimensionality of the non linear latent space the resulting framework is generic flexible and easy to extend for other purposes such as gaussian process regression with uncertain or partially missing inputs we demonstrate our method on synthetic data and standard machine learning benchmarks as well as challenging real world datasets including high resolution video data

on classification of integrable davey stewartson type equations this paper is devoted to the classification of integrable davey stewartson type equations a list of potentially deformable dispersionless systems is obtained through the requirement that such systems must be generated by a polynomial dispersionless lax pair a perturbative approach based on the method of hydrodynamic reductions is employed to recover the integrable systems along with their lax pairs some of the found systems seem to be new

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

numerical study of shock formation in the dispersionless kadomtsev petviashvili equation and dispersive regularizations abstract the formation of singularities in solutions to the dispersionless kadomtsev petviashvili dkp equation is studied numerically for different classes of initial data the asymptotic behavior of the fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity the approach is first tested in detail in 1 1 dimensions for the known case of the hopf equation where it is shown that the break up of the solution can be identified with prescribed accuracy for dissipative regularizations of this shock formation as the burgers equation and for dispersive regularizations as the korteweg de vries equation the fourier coefficients indicate as expected global regularity of the solutions the kadomtsev petviashvili kp equation can be seen as a dispersive regularization of the dkp equation the behavior of kp solutions for small dispersion parameter 1 near a break up of corresponding dkp solutions is studied it is found that the difference between kp and dkp solutions for the same initial data at the critical point scales roughly as 2 7 as for the korteweg de vries equation

integrable equations in 2 1 dimensions deformations of dispersionless limits we classify integrable third order equations in 2 1 dimensions which generalize the examples of kadomtsev petviashvili veselov novikov and harry dym equations our approach is based on the observation that dispersionless limits of integrable systems in 2 1 dimensions possess infinitely many multi phase solutions coming from the so called hydrodynamic reductions in this paper we adopt a novel perturbative approach to the classification problem based on the method of hydrodynamic reductions we first classify integrable quasilinear systems which may potentially occur as dispersionless limits of soliton equations in 2 1 dimensions to reconstruct dispersive deformations we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart this procedure leads to a complete list of integrable third order equations some of which are apparently new

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found