new reductions of the kadomtsev petviashvili and two dimensional toda lattice hierarchies via symmetry constraints new types of reductions of the kadomtsev petviashvili kp hierarchy and the two dimensional toda lattice 2dtl hierarchy are considered on the basis of sato s approach within this approach these hierarchies are represented by infinite sets of equations for potentials u1 u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp and the 2dtl hierarchies are studied under constraints of the following type n 1n nsn u1 u2 u3 x where sn are symmetries for these hierarchies n are arbitrary constants and is an arbitrary linear functional of the quantity it is shown that for the kp hierarchy these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 u3 un many known systems and several new ones are among them symmetry reductions for the 2dtl hierarchy give rise both to finite dimensional dynamical systems and 1 1 dimensional discrete systems some few results for the modified kp hierarc

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

the cauchy problem associated to the benjamin equation in weighted sobolev spaces abstract we study the initial value problem associated to the benjamin equation our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted sobolev spaces

conserved quantities and symmetries of kp hierarchy conserved quantities and symmetries of the kp equation from the point of view of the sato theory that provides a unifying approach to soliton equations is studied conserved quantities are derived from the generalized lax equations some reductions of the kp hierarchy such as kdv boussinesq a coupled kdv and sawada kotera equation are also considered by expansion of the squared eigenfunctions of the lax equations in terms of the function symmetries of the kp equations are obtained the relationship of this procedure to the two dimensional recursion operator newly found by fokas and santini is discussed

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

linear interpolation with a nonparametric accelerated failure time model abstract assuming a nonparametric accelerated failure time model a method is proposed for extrapolating low stress response probabilities on negative sloping line segments in the stress failure time plane the method analogous to linear interpolation in dose response studies results in simultaneous extrapolation ahead in time and down in stress in the two sample two stress level setting a goodness of fit test of the nonparametric model is considered and two hypothesis tests are proposed each pertaining to a condition that in addition to the assumed model must be met for the method to be valid the method is illustrated with censored strand life data and some simulated examples are used to consider power and robustness the failure time model is a scale family with curvature constraints the failure probability is assumed to be a sigmoid function of the product of time and a sigmoid scaling function of stress time percentiles as functions of stress are assumed to be convex the goodness of fit

the well posedness of the korteweg de vries benjamin ono equation abstract the well posedness of the cauchy problem to the generalized korteweg de vries benjamin ono equation is considered local results for data in h s r s 1 8 and the global well posedness for data in l 2 r are obtained if l 2 moreover for l 3 the problem is locally well posed for data in hs s 1 4 the main idea is to use the fourier restriction norm method

global well posedness for the benjamin equation in low regularity abstract in this paper we consider the initial value problem of the benjamin equation t u h x 2 u x 3 u x u 2 0 where u r 0 t r and the constants r 0 we use the i method to show that it is globally well posed in sobolev spaces h s r for s 3 4 moreover we use some argument to obtain a good estimative for the lifetime of the local solution and employ some multiplier decomposition argument to construct the almost conserved quantities

the constraint on potential and decomposition for 2 1 dimensional integrable systems besides a sufficient condition a necessary condition is given to determine some kind of consistent constraint on the potential of a 2 1 dimensional integrable system which cannot be obtained from the sufficient condition under this kind of constraint on the potential a 2 1 dimensional integrable system can be decomposed into two commuting 1 1 dimensional integrable systems and can be further decomposed into three commuting finite dimensional integrable hamiltonian systems the bkp hierarchy is taken as an illustrative example

projection pursuit density estimation abstract the projection pursuit methodology is applied to the multivariate density estimation problem the resulting nonparametric procedure is often less biased than the kernel and near neighbor methods in addition graphical information is produced that can be used to help gain geometric insight into the multivariate data distribution

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

bump hunting with non gaussian kernels it is well known that the number of modes of a kernel density estimator is monotone nonincreasing in the bandwidth if the kernel is a gaussian density there is numerical evidence of nonmonotonicity in the case of some non gaussian kernels but little additional information is available the present paper provides theoretical and numerical descriptions of the extent to which the number of modes is a nonmonotone function of bandwidth in the case of general compactly supported densities our results address popular kernels used in practice for example the epanechnikov biweight and triweight kernels and show that in such cases nonmonotonicity is present with strictly positive probability for all sample sizes n 3 in the epanechnikov and biweight cases the probability of nonmonotonicity equals 1 for all n 2 nevertheless in spite of the prevalence of lack of monotonicity revealed by these results it is shown that the notion of a critical bandwidth the smallest bandwidth above which the number of modes is guaranteed to be monotone is still well defined moreover just as in the gaussian case the critical bandwidth is of the same size as the bandwidth that minimises mean squared error of the density estimator these theoretical results and new numerical evidence show that the main effects of nonmonotonicity occur for relatively small bandwidths and have negligible impact on many aspects of bump hunting jj

l2global well posedness of the initial value problem associated to the benjamin equation abstract we study well posedness for the initial value problem associated to the benjamin equation and the periodic benjamin equation global results are established for data inl2 r andl2 t respectively we apply the recent theory developed by kenig ponce and vega and bourgain to deal with low regularity data for the initial value problem associated to the korteweg de vries equation

integrable hamiltonian systems related to the polynomial eigenvalue problem the independent integrals of motion in involution for the hamiltonian system related to the second order polynomial eigenvalue problem are constructed by using relevant recursion formula the hierarchy of hamiltonian systems obtained from the above problem and the time part of the lax pair are shown to be completely integrable and they are shown to commute with each other furthermore their solution solves the evolution equation associated with the lax pair

mode testing in difficult cases usually when testing the null hypothesis that a distribution has one mode against the alternative that it has two the null hypothesis is interpreted as entailing that the density of the sampling distribution has a unique point of zero slope which is a local maximum in this paper we argue that a more appropriate null hypothesis is that the density has two points of zero slope of which one is a local maximum and the other is a shoulder we show that when a test for a mode with shoulder is properly calibrated so that it has asymptotically correct level it is generally conservative when applied to the case of a mode without a shoulder we suggest methods for calibrating both the bandwidth and dip excess mass tests in the setting of a mode with a shoulder we also provide evidence in support of the converse a test calibrated for a single mode without a shoulder tends to be anticonservative when applied to a mode with a shoulder the calibration method involves resampling from a template density with exactly one mode and one shoulder it exploits the following asymptotic factorization property for both the sample and resample forms of the test statistic all dependence of these quantities on the sampling distribution cancels asymptotically from their ratio in contrast to other approaches the method has very good adaptivity properties

sharp well posedness for the benjamin equation abstract having the ill posedness in the range s 3 4 of the cauchy problem for the benjamin equation with an initial h s r data we prove that the already established local well posedness in the range s 3 4 of this initial value problem is extendable to s 3 4 and also that such a well posed property is globally valid for s 3 4

constraints of the 2 1 dimensional integrable soliton systems the authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2 1 dimensions can be constrained to integrable hierarchies in 1 1 dimensions such that submanifolds solutions of the given systems in 2 1 can be obtained by solving the resulting integrable systems in 1 1 dimensions the constraints of the kp hierarchy to the akns and burgers hierarchies respectively are shown in detail and the results of these for the modified kp and 2 1 dimensional analogue of the caudrey dodd gibbon kotera sawata equations to several integrable systems in 1 1 are given

an approach to the integrability of hamiltonian systems obtained by reduction a hierarchy of finite dimensional integrable hamiltonian systems can be obtained in a straightforward way by restricting a hierarchy of integrable evolution equations to the invariant subspace of their recursion operator the independent integrals of motion and hamiltonian functions for these hamiltonian systems can be constructed by using the recursion formula and can be shown to be in involution so these hamiltonian systems are completely integrable in the sense of liouville and commute with each other

exploratory projection pursuit abstract a new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods a number of practical issues concerning its application are addressed a connection to multivariate density estimation is established and its properties are investigated through simulation studies and application to real data the goal of exploratory projection pursuit is to use the data to find low one two or three dimensional projections that provide the most revealing views of the full dimensional data with these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance since linear effects are directly captured by the covariance structure of the variable pairs which are straightforward to estimate the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables although arbitrary

darboux theorems and wronskian formulas for integrable systems i constrained kp flows abstract generalizations of the classical darboux theorem are established for pseudo differential scattering operators of the form l i 0 n u i i i 1 m i 1 i i iteration of the darboux transformations leads to a gauge transformed operator with coefficients given by wronskian formulas involving a set of eigenfunctions of l nonlinear integrable partial differential equations are associated with the scattering operator l which arise as a symmetry reduction of the multicomponent kp hierarchy with a suitable linear time evolution for the eigenfunctions the darboux transformation is used to obtain solutions of the integrable equations in terms of wronskian determinants

multimodality exploration by an unsupervised projection pursuit neural network graphical inspection of multimodality is demonstrated using unsupervised lateral inhibition neural networks three projection pursuit indexes are compared on low dimensional simulated and real world data principal components legendre polynomial and projection pursuit network

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

on polynomial based projection indices for exploratory projection pursuit we develop asymptotic theory for two polynomial based methods of estimating orientation in projection pursuit density approximation one of the techniques uses legendre polynomials and has been proposed and implemented by friedman 1 the other employs hermite functions issues of smoothing parameter choice and robustness are addressed it is shown that each method can be used to construct sqrt n consistent estimates of the projection which maximizes distance from normality although the former can only be employed in that manner when the underlying distribution has extremely light tails the former can be used very generally to measure low frequency departure from normality

the constraints of potentials and the finite dimensional integrable systems restricting potential to the space spanned by the eigenvectors of the recursion operator leads to a natural constraint of potential and a finite dimensional integrable hamiltonian system the general method for proving the consistency of the two systems stemming from the lax pair and obtaining the constants of the motion for the hamiltonian system is illustrated by the classical boussinesq and akns hierarchies by using gauge transformation similar results for the jaulent miodek and kaup newell hierarchies are presented

c neumann and bargmann systems associated with the coupled kdv soliton hierarchy under two different constraints between the potentials and the eigenfunctions the eigenvalue problem associated with the coupled kdv hierarchy is nonlinearised to be a completely integrable c neumann system on the tangent bundle of sphere tsn 1 with the hamiltonian h 1 2 lambda q p 1 2 q q p p 1 2 q p 2 1 2 q p lambda p p and a completely integrable bargmann system r2n dpdq h with the hamiltonian h 1 2 lambda q p 1 2 p p q p 1 2 q q respectively the involutive solutions of the coupled kdv equation associated with the two systems are given

excess mass estimates and tests for multimodality abstract we propose and study a method for analyzing the modality of a distribution the method is based on the excess mass functional this functional measures excessive empirical mass in comparison with multiples of uniform distribution by the excess mass approach we separate the investigation about the number of modes from questions concerning their location for distributions on the line the excess mass functional can be estimated at a square root rate the asymptotic behavior of estimators is studied and tests for multimodality based on the excess mass are derived

almost global existence for hamiltonian semilinear klein gordon equations with small cauchy data on zoll manifolds this paper is devoted to the proof of almost global existence results for klein gordon equations on zoll manifolds e g spheres of arbitrary dimension with hamiltonian nonlinearities when the cauchy data are smooth and small the proof relies on birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on zoll manifolds 2007 wiley periodicals inc

dimensional reductions of bkp and ckp hierarchies a discussion of dimensional reductions which are not classical symmetry reductions is made for the bkp and ckp hierarchies of integrable evolution equations a novel direct method for testing pfaffian solutions to bilinear identities is presented and applied to these reductions

on the propagation of regularity and decay of solutions to the benjamin equation in this paper we investigate some special regularities and decay properties of solutions to the initial value problem of the benjamin equation the main result shows that for initial datum u0 hs r with s 3 4 if the restriction of u0 belongs to hl x0 for some l z and x0 r then the restriction of the corresponding solution u t belongs to hl for any r and any t 0 t consequently this type of regularity travels with infinite speed to its left as time evolves

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

reductions of 2 1 dimensional integrable systems via mixed potential eigenfunction constraints new types of reductions of 2 1 dimensional integrable systems which are associated with mixed potential eigenfunction constraints are discussed necessary and sufficient conditions for the admissibility of such constraints are presented classification results for integrable equations will be used for applying these conditions several typical examples are considered including the kp mkp 2 1 dimensional savada kotera kaup kupershmidt harry dym and nizhnik veselov novikov equations