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

dimension reduction for nonelliptically distributed predictors sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor or when such conditions are not satisfied rely on marginal transformation or reweighting to fulfill them approximately for example a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distribution in this paper we reformulate the commonly used dimension reduction methods via the notion of central solution space so as to circumvent the requirements of such strong assumptions while at the same time preserve the desirable properties of the classical methods such as n consistency and asymptotic normality imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the curse of dimensionality but the development of this paper shows that this need not be the case the new methods will be compared with existing methods by simulation and applied to a data set

a theory of exact and approximate configurational invariants abstract a theory of integrable hamiltonian systems in two dimensions is formulated and applied the four dimensional phase space problem that is the vanishing of the poisson bracket between another invariant and the hamiltonian is here reduced to the solution of a series of two dimensional configuration space equations the theory is also applicable to more than two dimensions the constraints are found which admit to integrability of the orbits for magnetic or coriolis forces as well as for forces derivable from a potential when a system admits a given invariant the invariant is found for example by quadrature a number of examples including known and apparently previously unknown invariants are given the theory of exact integrals of the motion also can be extended to the derivation of approximate invariants the orbital structure of integrable or approximately integrable systems correlates with the degree maximum power of the velocity of a standardized invariant the theory admits a variational principle among other approximation techniques for the computation of a best approximate invariant the problem of the general cubic potential with one symmetric coordinate v 1 2 ax 2 1 2 by 2 cx 2 y 1 3 dy 3 of which the well studied henon heiles potential is the special case for a b and c d is examined in detail

a validated information criterion to determine the structural dimension in dimension reduction models summary a crucial component of performing sufficient dimension reduction is to determine the structural dimension of the reduction model we propose a novel information criterion based method for this purpose a special feature of which is that when examining the goodness of fit of the current model one needs to perform model evaluation by using an enlarged candidate model althoughtheproceduredoesnotrequireestimationundertheenlargedmodelofdimensionk 1 the decision as to how well the current model of dimensionk fits relies on the validation provided by the enlarged model thus we call this procedure the validated information criterion vic k our method is different from existing information criterion based model selection methods it breaks free from dependence on the connection between dimension reduction models and their corresponding matrix eigenstructures which relies heavily on a linearity condition that we no longer assume we prove consistency of the proposed method and its finite sample performance is demonstrated numerically

atlas of two dimensional irreversible conservative lagrangian mechanical systems with a second quadratic integral this paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant quadratic in velocities other than the hamiltonian the configuration space is in general a two dimensional riemannian or pseudo riemannian manifold and the determination of its geometry is a part of the process of solution forces acting on the system include a part derived from a scalar potential and a part derived from a vector potential associated with terms linear in velocities in the lagrangian function of the system the last caused time irreversibility of the system we constructed 41 multiparameter integrable systems of the type described in the title mostly on riemannian manifolds they are mostly new and cover all previously known systems as special cases corresponding to special values of the parameters those include all known cases of motion of a particle in the plane and all known cases in the dynamics of rigid body in the last field we introduce a n

dimension reduction and semiparametric estimation of survival models in this paper we propose a new dimension reduction method by introducing a nominal regression model with the hazard function as the conditional mean which naturally retrieves information from complete data and censored data as well moreover without requiring the linearity condition the new method can estimate the entire central subspace consistently and exhaustively the method also provides an alternative approach for the analysis of censored data assuming neither the link function nor the distribution hence it exhibits superior robustness properties numerical studies show that the method can indeed be readily used to efficiently estimate survival models explore the data structures and identify important variables

cylindrical korteweg de vries equation and painleve property the authors demonstrate that the lax representation for the cylindrical korteweg de vries equation can be obtained with the help of the painleve property of this equation

on a kdv equation with higher order nonlinearity traveling wave solutions abstract in this work the improved tanh coth method is used to obtain wave solutions to a korteweg de vries kdv equation with higher order nonlinearity from which the standard kdv and the modified korteweg de vries mkdv equations with variable coefficients can be derived as particular cases however the model studied here include other important equations with applications in several fields of physical and nonlinear sciences periodic and soliton solutions are formally derived

a constructive approach to the estimation of dimension reduction directions in this paper we propose two new methods to estimate the dimension reduction directions of the central subspace cs by constructing a regression model such that the directions are all captured in the regression mean compared with the inverse regression estimation methods e g j amer statist assoc 86 1991 328 332 j amer statist assoc 86 1991 316 342 j amer statist assoc 87 1992 1025 1039 the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable and have better performance than the inverse regression estimation methods for finite samples compared with the direct regression estimation methods e g j amer statist assoc 84 1989 986 995 ann statist 29 2001 1537 1566 j r stat soc ser b stat methodol 64 2002 363 410 which can only estimate the directions of cs in the regression mean the new methods can detect the directions of cs exhaustively consistency of the estimators and the convergence of corresponding algorithms are proved

fourier methods for estimating the central subspace and the central mean subspace in regression in regression with a high dimensional predictor vector it is important to estimate the central and central mean subspaces that preserve sufficient information about the response and the mean response using the fourier transform we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively under the normality assumption of the predictors explicit estimates of the central and central mean subspaces are derived bootstrap procedures are used for determining dimensionality and choosing tuning parameters simulation results and an application to a real data are reported our methods demonstrate competitive performance compared with sir save and other existing methods the approach proposed in the article provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future

hamiltonians with high order integrals and the weak painleve concept we examine the singularity structure of the equations of motion associated to integrable two dimensional hamiltonians with second integrals of order higher than 2 we show in these specific examples that the integrability is associated to a singularity expansion of the weak painleve type new cases of integrability are discovered with still higher order integrals which are explicitly computed

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

painleve analysis yoshida s theorems and direct methods in the search for integrable hamiltonians treating as an example two dimensional similarity invariant hamiltonians with polynomial velocity dependent potentials the authors show how painleve analysis yoshida s theorems and direct methods can be combined in the search for integrable hamiltonians new integrable systems are found one of which has a polynomial second invariant of fifth order in the momenta most of the necessary calculations have been performed by applying various reduce packages developed for that purpose

non integrability of homogeneous two dimensional hamiltonians with velocity dependent potential the non integrability of classes of homogeneous two dimensional hamiltonian systems with a polynomial velocity dependent potential is shown on the basis of ziglin s theorem an analytic expression for the trace of the relevant monodromy matrices is presented which fits the numerical data perfectly an application is made to fokker planck hamiltonians with quadratic and with cubic drift terms

comment on integrable potentials with logarithmic integrals of motion the authors analyse and complete the results of an article by ichtiaroglou and voyatzis see ibid vol 21 p 3537 1988 concerning the existence of integrals of motion logarithmic in the momenta for two dimensional velocity dependent hamiltonians they find a new integrable case and establish the generality of some results found by ichtiaroglou and voyatzis which were presented as particular cases

exact solutions to kdv equations with variable coefficients and or nonuniformities abstract a class of kdv equations with variable coefficients and or nonuniformities is discussed we derive easily the constraint conditions which must apply to this class in order to obtain exact solutions by a direct method we obtain solutions in a number of generic as well as specific cases

on sliced inverse regression with high dimensional covariates sliced inverse regression is a promising method for the estimation of the central dimension reduction subspace cdr space in semiparametric regression models it is particularly useful in tackling cases with high dimensional covariates in this article we study the asymptotic behavior of the estimate of the cdr space with high dimensional covariates that is when the dimension of the covariates goes to infinity as the sample size goes to infinity strong and weak convergence are obtained we also suggest an estimation procedure of the bayes information criterion type to ascertain the dimension of the cdr space and derive the consistency a simulation study is conducted

sufficient dimension reduction for censored regressions methodology of sufficient dimension reduction sdr has offered an effective means to facilitate regression analysis of high dimensional data when the response is censored however most existing sdr estimators cannot be applied or require some restrictive conditions in this article we propose a new class of inverse censoring probability weighted sdr estimators for censored regressions moreover regularization is introduced to achieve simultaneous variable selection and dimension reduction asymptotic properties and empirical performance of the proposed methods are examined

a variable coefficient korteweg de vries equation similarity analysis and exact solution ii a korteweg de vries kdv equation with time dependent coefficients is studied in this paper the similarity transformation for this system is investigated and an exact solution in a particular case is obtained the ablowitz ramani segur ars conjecture is used to identify the integrability of the system it is found that in some special cases the system may be integrable

new exact explicit solutions of the generalized kdv equations abstract as kdv equations are characterized by the painleve equations we show here that generalized kdv equations are characterized by euler painleve equations a plethora of exact explicit similarity solutions are found which include inverse of hypergeometric function

n dimensional integrability from two photon coalgebra symmetry a wide class of hamiltonian systems with n degrees of freedom and endowed with at least n 2 functionally independent integrals of motion in involution is constructed by making use of the two photon lie poisson coalgebra h6 the set of n 2 constants of the motion is shown to be a universal one for all these hamiltonians irrespective of the dependence of the latter on several arbitrary functions and n free parameters within this large class of quasi integrable n dimensional hamiltonians new families of completely integrable systems are identified by finding explicitly a new independent integral through the analysis of the sub coalgebra structure of h6 in particular new completely integrable n dimensional hamiltonians describing natural systems geodesic flows and static electromagnetic hamiltonians are presented

new exact solutions for a generalized variable coefficient kdv equation new exact solutions for a generalized variable coefficient kdv equation were obtained using the generalized expansion method r sabry m a zahran e g fan phys lett a 326 2004 93 the obtained solutions include solitary wave solutions besides jacobi and weierstrass doubly periodic wave solutions

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

auto backlund transformation lax pairs and painleve property of a variable coefficient korteweg de vries equation i using the painleve property of partial differential equations the auto backlund transformation and lax pairs for a korteweg de vries kdv equation with time dependent coefficients are obtained the lax pair criterion also makes it possible for some new models of the variable coefficient kdv equation to be found that can represent nonsoliton dynamical systems this can explain the wave breaking phenomenon in variable depth shallow water

new integrable hamiltonians with transcendental invariants on presente 3 hamiltoniens integrables a 2 dimensions dont le second invariant i 2 est rationnel ou transcendant en impulsion on trouve un troisieme invariant i 3

integrable nonlinear evolution equations with time dependent coefficients a simple and straightforward method for generating completely integrable nonlinear evolution equations with time dependent coefficients is presented herein for the equations under consideration the solutions to those given by vector fields which are independent of time are given thus explicit links between equations are obtained as an application of the proposed method it is shown that the linear superposition with arbitrary time dependent coefficients of different members of an integrable hierarchy is again integrable furthermore it turns out that for some integrable equations like the korteweg de vries kdv the benjamin ono bo or the kadomtsev petviashvili kp the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow those flows demonstrated for the kdv which can be generated by lie homomorphisms coming from first order problems are completely classified many well known equations which can be found in the literature are of that type as an application of such a first order link a direct link is given from kdv to the cylindrical kdv and from there to the kp with nontrivial dependence on the second spatial variable

integrals of motion for toda systems with unequal masses we present new integrals of motion for the toda lattice chain of particles in one dimension with exponential interaction for two special cases of boundary conditions the free end lattice with three non equal mass particles and the fixed end lattice for two particles in both cases we use two distinct approaches in order to identify the integrable cases direct search of the integral of motion and group theoretical methods our results are in agreement with the predictions of painleve analysis

a feasible method for optimization with orthogonality constraints minimization with orthogonality constraints e g x top x i and or spherical constraints e g vert x vert 2 1 has wide applications in polynomial optimization combinatorial optimization eigenvalue problems sparse pca p harmonic flows 1 bit compressive sensing matrix rank minimization etc these problems are difficult because the constraints are not only non convex but numerically expensive to preserve during iterations to deal with these difficulties we apply the cayley transform a crank nicolson like update scheme to preserve the constraints and based on it develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics the efficiency of the proposed algorithms is demonstrated on a variety of test problems in particular for the maxcut problem it exactly solves a decomposition formulation for the sdp relaxation for polynomial optimization nearest correlation matrix estimation and extreme eigenvalue problems the proposed algorithms run very fast and return solutions no worse than those from their state of the art algorithms for the quadratic assignment problem a gap 0 842 to the best known solution on the largest problem tai256c in qaplib can be reached in 5 min on a typical laptop

painleve property auto backlund transformation lax pairs and reduction to the standard form for the korteweg de vries equation with nonuniformities it is demonstrated that the kdv equation with nonuniformities ut a t u b x t u x c t uux d t uxxx e x t 0 has the painleve property if the compatibility condition among the coefficients of it holds bt a lc b bbx dbxxx 2ah hl d c2 dh dt ce x 2a2 al d3 c4 da dt l d c l d c2 d dt l d c where l d dt lg and h t is an arbitrary function of t the auto backlund transformation and lax pairs for this equation are obtained by truncating the laurent expansion furthermore assuming the compatibility condition then the kdv equation with nonuniformities is transformable via suitable variable transformations to the standard kdv

counting process based dimension reduction methods for censored outcomes we propose counting process based dimension reduction methods for right censored survival data semiparametric estimating equations are constructed to estimate the dimension reduction subspace for the failure time model our methods address two limitations of existing approaches first using the counting process formulation they do not require estimation of the censoring distribution to compensate for the bias in estimating the dimension reduction subspace second the nonparametric estimation involved adapts to the structural dimension so our methods circumvent the curse of dimensionality asymptotic normality is established for the estimators we propose a computationally efficient approach that requires only a singular value decomposition to estimate the dimension reduction subspace numerical studies suggest that our new approaches exhibit significantly improved performance the methods are implemented in the formula see text package formula see text

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

dimension reduction for non elliptically distributed predictors second order methods summary many classical dimension reduction methods especially those based on inverse conditional moments require the predictors to have elliptical distributions or at least to satisfy a linearity condition such conditions however are too strong for some applications li and dong 2009 introduced the notion of the central solution space and used it to modify first order methods such as sliced inverse regression so that they no longer rely on these conditions in this paper we generalize this idea to second order methods such as sliced average variance estimation and directional regression in doing so we demonstrate that the central solution space is a versatile framework we can use it to modify essentially all inverse conditional moment based methods to relax the distributional assumption on the predictors simulation studies and an application show a substantial improvement of the modified methods over their classical counterparts

complete integrability and analytic solutions of a kdv type equation the complete integrability of the variable coefficient version of a kdv equation via the painleve approach is analyzed through the painleve backlund equations its auto backlund transformation lax pairs symmetry strong symmetry bilinear form and analytic solutions are obtained

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research

how to construct integrable fokker planck and electromagnetic hamiltonians from ordinary integrable hamiltonians integrable hamiltonians with velocity dependent potentials including those of fokker planck type h 1 2 p2x p2y kxpx kypy are constructed from integrable hamiltonians of type h 1 2 p2x p2y v x y using certain canonical and noncanonical transformations some of the hamiltonians obtained this way are integrable only for zero energy candidates for the potential which is of interest for fokker planck models are constructed in several cases

on the kdv type equation with variable coefficients in this paper the kdv type equations with variable coefficients appearing a lot in the literature are discussed generally the explicit transformations which transform various kdv mkdv and kp type nonlinear evolution equations into their canonical forms with constant coefficients are given the results presented here tie up many of the investigations scattered in the literature

universal integrals for superintegrable systems on n dimensional spaces of constant curvature an infinite family of classical superintegrable hamiltonians defined on the n dimensional spherical euclidean and hyperbolic spaces are shown to have a common set of 2n 3 functionally independent constants of the motion among them two different subsets of n integrals in involution including the hamiltonian can always be explicitly identified as particular cases we recover in a straightforward way most of the superintegrability properties of the smorodinsky winternitz and generalized kepler coulomb systems on spaces of constant curvature and we introduce as well new classes of quasi maximally superintegrable potentials on these spaces results presented here are a consequence of the poisson coalgebra symmetry of all the hamiltonians together with an appropriate use of the phase spaces associated with poincare and beltrami coordinates

dimension reduction for censored regression data without parametric assumptions high dimensional regression analysis is already complex this is made even harder when data are subject to censoring in this article we seek ways of reducing the dimensionality of the regressor before applying nonparametric smoothing techniques if the censoring time is independent of the lifetime then the method of sliced inverse regression can be applied directly otherwise modification is needed to adjust for the censoring bias a key identity leading to the bias correction is derived and the root n consistency of the modified estimate is established patterns of censoring can also be studied under a similar dimension reduction framework some simulation results and an application to a real data set are reported 1 introduction survival data are often subject to censoring when this occurs the incompleteness of the observed data may induce a substantial bias in the sample several approaches have been suggested to overcome the associated difficulties in regression including the accelerated failure time model censored linear regression the cox proportional hazard model and many others survival analysis becomes even more intricate when the dimension of the regressor increases to apply any of the aforementioned methods users are required to specify a functional form which relates the outcome variables to the input ones however in reality knowledge needed for an appropriate model specification is often inadequate as a matter of fact the acquisition of such information may well turn out to be one of the primary goals of the study itself under such circumstances it seems preferable to have exploratory tools that rely less on such model specification this is the issue to be addressed in this article the dimension reduction approach of li 1991 will be extended to settings which allow for censoring in the data we shall offer methods of finding low dimensional projections of the data for visually examining the censoring pattern we shall show how censored regression data can still be analyzed without assuming the functional form a priori

accelerated failure time models for counting processes summary we present a natural extension of the conventional accelerated failure time model for survival data to formulate the effects of covariates on the mean function of the counting process for recurrent events a class of consistent and asymptotically normal rank estimators is developed for estimating the regression parameters of the proposed model in addition a nelson aalen type estimator for the mean function of the counting process is constructed which is consistent and properly normalised converges weakly to a zeromean gaussian process we assess the finite sample properties of the proposed estimators and the associated inference procedures through monte carlo simulation and provide an application to a well known bladder cancer study

a semiparametric approach to dimension reduction we provide a novel and completely different approach to dimension reduction problems from the existing literature we cast the dimension reduction problem in a semiparametric estimation framework and derive estimating equations viewing this problem from the new angle allows us to derive a rich class of estimators and obtain the classical dimension reduction techniques as special cases in this class the semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact the common assumption of linearity and or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression the semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example this article has online supplementary material