hamiltonian structures and lax equations generated by matrix differential operators with polynomial dependence on a parameter we investigate a general set of equations which can be studied by the inverse scattering method

graph based change point detection we consider the testing and estimation of change points locations where the distribution abruptly changes in a data sequence a new approach based on scan statistics utilizing graphs representing the similarity between observations is proposed the graph based approach is nonparametric and can be applied to any data set as long as an informative similarity measure on the sample space can be defined accurate analytic approximations to the significance of graph based scan statistics for both the single change point and the changed interval alternatives are provided simulations reveal that the new approach has better power than existing approaches when the dimension of the data is moderate to high the new approach is illustrated on two applications the determination of authorship of a classic novel and the detection of change in a network over time

asymptotics for euclidean minimal spanning trees on random points asymptotic results for the euclidean minimal spanning tree onn random vertices inrd can be obtained from consideration of a limiting infinite forest whose vertices form a poisson process in allrd in particular we prove a conjecture of robert bland the sum of thed th powers of the edge lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofrd tends to a constant asn

a multivariate two sample test based on the number of nearest neighbor type coincidences for independent d variate random samples x 1 cdots x n 1 i i d f x y 1 cdots y n 2 i i d g x where the densities f and g are assumed to be continuous a e consider the number t of all k nearest neighbor comparisons in which observations and their neighbors belong to the same sample we show that if f g a e the limiting normal distribution of t as min n 1 n 2 rightarrow infty n 1 n 1 n 2 rightarrow tau 0 tau 1 does not depend on f an omnibus procedure for testing the hypothesis h 0 f g a e is obtained by rejecting h 0 for large values of t the result applies to a general distance generated by a norm on mathbb r d for determining nearest neighbors and it generalizes to the multisample situation

the continuous heisenberg chain and constrained harmonic motion it is shown that the equation describing the evolution of the classical continuous heisenberg ferromagnet can be regarded as one aspect of quadratically constrained harmonic motion as it is also the case in a number of other integrable systems two ingredients are used a method to solve the inverse scattering problem for second order operators using ordinary differential rather than integral equations and the equivalence of the heisenberg chain with the nonlinear schrodinger equation

backlund transformations for nonlinear sigma models with values in riemannian symmetric spaces this work deals with backlund transformations for the principal sl n sigma model together with all reduced models with values in riemannian symmetric spaces first the dressing method of zakharov mikhailov and shabat is shown for the case of a meromorphic dressing matrix to be equivalent to a backlund transformation for an associated linearly extended system comparison of this multi backlund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem a new method of solution for such multi backlund transformations mbt is developed by the introduction of a soliton correlation matrix which satisfies a riccati system equivalent to the mbt using the geometric structure of this system a linearization is achieved leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single backlund transformation through purely linear algebraic relations a systematic study of all reductions of the system by involutive automorphisms is made thereby defining the multi backlund transformations and their solution for all riemannian symmetric spaces

matrix formulation of hamiltonian structures of constrained kp hierarchy we give a matrix formulation of the hamiltonian structures of constrained kp hierarchy first we derive from the matrix formulation the hamiltonian structure of the one constraint kp hierarchy which was originally obtained by oevel and strampp we then generalize the derivation to the multiconstraint case and show that the resulting bracket is actually the second gelfand dickey bracket associated with the corresponding lax operator the matrix formulation of the hamiltonian structure of the one constraint kp hierarchy in the form introduced in the study of matrix model is also discussed

an exact distribution free test comparing two multivariate distributions based on adjacency summary a new test is proposed comparing two multivariate distributions by using distances between observations unlike earlier tests using interpoint distances the new test statistic has a known exact distribution and is exactly distribution free the interpoint distances are used to construct an optimal non bipartite matching i e a matching of the observations into disjoint pairs to minimize the total distance within pairs the cross match statistic is the number of pairs containing one observation from the first distribution and one from the second distributions that are very different will exhibit few cross matches when comparing two discrete distributions with finite support the test is consistent against all alternatives the test is applied to a study of brain activation measured by functional magnetic resonance imaging during two linguistic tasks comparing brains that are impaired by arteriovenous abnormalities with normal controls a second exact distribution free test is also discussed it ranks the pairs and sums the ranks of the cross matched pairs

asymptotic distribution free change point detection for multivariate and non euclidean data we consider the testing and estimation of change points locations where the distribution abruptly changes in a sequence of multivariate or non euclidean observations we study a nonparametric framework that utilizes similarity information among observations which can be applied to various data types as long as an informative similarity measure on the sample space can be defined the existing approach along this line has low power and or biased estimates for change points under some common scenarios we address these problems by considering new tests based on similarity information simulation studies show that the new approaches exhibit substantial improvements in detecting and estimating change points in addition under some mild conditions the new test statistics are asymptotically distribution free under the null hypothesis of no change analytic p value approximations to the significance of the new test statistics for the single change point alternative and changed interval alternative are derived making the new approaches easy off the shelf tools for large datasets the new approaches are illustrated in an analysis of new york taxi data

the random minimal spanning tree in high dimensions for the minimal spanning tree on n independent uniform points in the d dimensional unit cube the proportionate number of points of degree k is known to converge to a limit alpha k d as n to infty we show that alpha k d converges to a limit alpha k as d to infty for each k the limit alpha k arose in earlier work by aldous as the asymptotic proportionate number of vertices of degree k in the minimum weight spanning tree on k vertices when the edge weights are taken to be independent identically distributed random variables we give a graphical alternative to aldous s characterization of the alpha k

integrable nonlinear equations and liouville s theorem i a symplectic structure for stationary lax equations of the type l p 0 is constructed wherel is a matrix differential operator of the first order it is shown that the equation has a sufficient for the complete integrability amount of first integrals in involution the well known linearization of the equation by the abelian mapping is obtained in a natural manner in consequent exercising of liouville s procedure

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

optimal estimation of wasserstein distance on a tree with an application to microbiome studies the weighted unifrac distance a plug in estimator of the wasserstein distance of read counts on a tree has been widely used to measure the microbial community difference in microbiome studies ou

a new graph based two sample test for multivariate and object data abstracttwo sample tests for multivariate data and especially for non euclidean data are not well explored this article presents a novel test statistic based on a similarity graph constructed on the pooled observations from the two samples it can be applied to multivariate data and non euclidean data as long as a dissimilarity measure on the sample space can be defined which can usually be provided by domain experts existing tests based on a similarity graph lack power either for location or for scale alternatives the new test uses a common pattern that was overlooked previously and works for both types of alternatives the test exhibits substantial power gains in simulation studies its asymptotic permutation null distribution is derived and shown to work well under finite samples facilitating its application to large datasets the new test is illustrated on two applications the assessment of covariate balance in a matched observational study and the comparison of network data under different co

kac moody lie algebras and soliton equations iii stationary equations associated with a1 1 abstract the hamiltonian structure of stationary soliton equations associated with the akns eigenvalue problem is derived in two ways first it is shown to arise from the kostant kirillov symplectic structure on a coadjoint orbit in an infinite dimensional lie algebra second it is obtained as the restriction to a finite dimensional manifold of the infinite dimensional hamiltonian structure associated with a certain eigenvalue problem polynomial in the eigenvalue parameter

higher local conservation laws for nonlinear sigma models on symmetric spaces we show that for the classical two dimensional nonlinear model on a riemannian symmetric space of dimensionm and rankp there existp independent series of higher local conservation laws and we reduce the field equations of the model to a system of nonlinear partial differential equations possessing an associated lax pair and involvingm p independent variables

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

consistent and powerful graph based change point test for high dimensional data a change point detection is proposed by using a bayesian type statistic based on the shortest hamiltonian path and the change point is estimated by using ratio cut a permutation procedure is applied to approximate the significance of bayesian type statistics the change point test is proven to be consistent and an error probability in change point estimation is provided the test is very powerful against alternatives with a shift in variance and is accurate in change point estimation as shown in simulation studies its applicability in tracking cell division is illustrated

a weighted edge count two sample test for multivariate and object data abstracttwo sample tests for multivariate data and non euclidean data are widely used in many fields parametric tests are mostly restrained to certain types of data that meets the assumptions of the parametric models in this article we study a nonparametric testing procedure that uses graphs representing the similarity among observations it can be applied to any data types as long as an informative similarity measure on the sample space can be defined the classic test based on a similarity graph has a problem when the two sample sizes are different we solve the problem by applying appropriate weights to different components of the classic test statistic the new test exhibits substantial power gains in simulation studies its asymptotic permutation null distribution is derived and shown to work well under finite samples facilitating its application to large datasets the new test is illustrated through an analysis on a real dataset of network data

integrable nonlinear equations and liouville s theorem ii a symplectic structure is constructed and the liouville integration carried out for a stationary lax equation l p 0 wherel is a scalar differential operator of an arbitrary order nth order operators are included into the variety of first order matrix operators and properties of this inclusion are studied

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

the squared eigenstates of the sine gordon eigenvalue problem an analysis of the squared sine gordon eigenvalue problem in laboratory coordinates is presented it is shown that unlike the unsquared laboratory coordinate eigenvalue problem the squared laboratory coordinate eigenvalue problem may be cast into the form of a standard eigenvalue problem wherein an eigenvalue independent operator operating on an eigenfunction generates the eigenvalue with this form it becomes rather elementary to obtain the squared eigenfunction expansion of the sine gordon potentials as well as to demonstrate closure

gauge covariant theory of the generating operator i a gauge covariant formulation of the generating operator operator theory for the zakharov shabat system is proposed the operator tilde lambda corresponding to the gauge equivalent system in the pole gauge is explicitly calculated thus the unified approach to the nonlinear schrodinger type equations based on is automatically reformulated with the help of tilde lambda for the heisenberg ferromagnet type equations consequently it is established that the conserved densities for the heisenberg ferromagnet type equations are polynomial ins x and itsx derivatives special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge equivalent equations it is shown that the geometrical properties of the conjugated operator are gaugeindependent

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

hamiltonian structure of the general integrable equations under reductions the general form of the nonlinear evolution equations integrable by the linear matrix spectral problem and their hamiltonian structures are presented for potentials with nonzero constant asymptotic values the reduction problem is discussed within the framework of the akns method a method is proposed for the calculation of the hamiltonian structures for the integrable equations under reductions matrix zn and n reductions and some other reductions are considered as examples general forms of the corresponding reduced equations and their hamiltonian structures are found

on some linear integral equations generating solutions of nonlinear partial differential equations abstract two types of linear inhomogeneous integral equations which yield solutions of a broad class of nonlinear evolution equations are investigated one type is characterized by a two fold integration with an arbitrary measure and contour over a complex variable k and thier complex conjugates whereas the other one has a two fold integration over one and the same contour the inhomogeneous term which may contain an arbitrary function of k makes it possible to define a matrix structure on the solutions of the integral equations the elements of these matrices are shown to obey a system of partial differential equations the special form of which depends on the choice of the dispersion relation occurring in the integral equations for special elements of the matrices closed partial differential equations can be derived such as e g the nonlinear schrodinger equation and the real and complex modified korteweg de vries and sine gordon equations the relations between the matrix elements are shown to lead to miura transformations between the various partial differential equations

nonparametric maximum likelihood approach to multiple change point problems in multiple change point problems different data segments often follow different distributions for which the changes may occur in the mean scale or the entire distribution from one segment to another without the need to know the number of change points in advance we propose a nonparametric maximum likelihood approach to detecting multiple change points our method does not impose any parametric assumption on the underlying distributions of the data sequence which is thus suitable for detection of any changes in the distributions the number of change points is determined by the bayesian information criterion and the locations of the change points can be estimated via the dynamic programming algorithm and the use of the intrinsic order structure of the likelihood function under some mild conditions we show that the new method provides consistent estimation with an optimal rate we also suggest a prescreening procedure to exclude most of the irrelevant points prior to the implementation of the nonparametric likelihood method simulation studies show that the proposed method has satisfactory performance of identifying multiple change points in terms of estimation accuracy and computation time

integrable g strands on semisimple lie groups the present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple lie algebra it also determines the general form of hamilton s principles and hamiltonians for these systems and analyzes the linear stability of their equilibrium solutions in the examples of so 3 and sl 2 r

differential difference akns equations and homogeneous heisenberg algebras the well known evolution equations associated to the homogeneous heisenberg algebras of kac moody algebras g 1 akns systems are extended by differential difference equations that can be written in zero curvature form

on the multivariate runs test for independent d variate random variables x 1 x m with common density f and y 1 y n with common density g let r m n be the number of edges in the minimal spanning tree with vertices x 1 x m y 1 y n that connect points from different samples friedman and rafsky conjectured that a test of h 0 f g that rejects h 0 for small values of r m n should have power against general alternatives we prove that r m n is asymptotically distribution free under h 0 and that the multivariate two sample test based on r m n is universally consistent

reducing multidimensional two sample data to one dimensional interpoint comparisons the most popular technique for reducing the dimensionality in comparing two multidimensional samples of x f and y g is to analyze distributions of interpoint comparisons based on a univariate function h e g the interpoint distances we provide a theoretical foundation for this technique by showing that having both i the equality of the distributions of within sample comparisons h x 1 x 2 l h y 1 y 2 and ii the equality of these with the distribution of between sample comparisons h x 1 x 2 l h x 3 y 3 is equivalent to the equality of the multivariate distributions f g

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

gauge equivalence of exactly integrable field theoretic models exactly integrable field theoretic models are constructed which are gauge equivalent to the n component or m n component nonlinear schrodinger equations and to the o n nonlinear model we obtain the cpn heisenberg model or the grassmann heisenberg model and the generalized sine gordon model respectively consequences for the conserved quantities are discussed

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

a new approach to the self dual yang mills equations inspired by sato s new theory for soliton equations we find a new approach to the self dual yang mills equations we first establish a correspondence of solutions between the self dual yang mills equations and a new system of equations with infinitely many unknown functions it then turns out that the latter equations can be easily solved by a simple explicit procedure this leads to an explicit description of a very broad class of solutions to the self dual yang mills equations and also to a construction of transformations acting on these solutions

symplectic structure lagrangian and involutiveness of first integrals of the principal chiral field equation we deal with a form of the chiral equation for which first integrals can be written explicitly for these equations we find a symplectic structure the lagrangian and first integrals in involution

instanton like solutions in chiral models abstract general two dimensional euclidean chiral models of field theory are considered in detail it is shown that in the case when the field takes its values in an arbitrary kahler manifold the duality equations reduce to the cauchy riemann equations on this manifold for homogeneous manifolds the solutions of these equations do exist and are given by rational functions

generalized hyperbolic ernst equations for an einstein maxwell weyl field a new family of nonlinear partial differential equations is presented they represent a generalization of the hyperbolic ernst equations for an einstein mawxell weyl field in general relativity a backlund transformation for the system of equations under consideration is given and their direct relation to the complete boussinesq hierarchy of soliton equations is illustrated

general zakharov shabat equations multi time hamiltonian formalism and constants of motion we construct a hamiltonian formalism for general zakharov shabat equations zero curvature equations with rational dependence on a parameter as well as their constants of motion and prove that the latter are in involution the field theoretical multi time hamiltonian formalism is used

the direct linearization of a class of nonlinear evolution equations the paper deals with the direct linearization an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scattering transform linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1 1 and 2 1 dimensions

new multivariate tests for assessing covariate balance in matched observational studies we propose new tests for assessing whether covariates in a treatment group and matched control group are balanced in observational studies the tests exhibit high power under a wide range of multivariate alternatives some of which existing tests have little power for the asymptotic permutation null distributions of the proposed tests are studied and the p values calculated through the asymptotic results work well in simulation studies facilitating the application of the test to large data sets the tests are illustrated in a study of the effect of smoking on blood lead levels the proposed tests are implemented in an r package balancecheck

fredholm determinants and the mkdv sinh gordon hierarchies for a particular class of integral operatorsk we show that the quantity phi log det i k log det i k satisfies both the integrated mkdv hierarchy and the sinh gordon hierarchy this proves a conjecture of zamolodchikov

darboux coordinates and liouville arnold integration in loop algebras darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part tilde mathfrak g of loop algebras these are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves defined within a given matrix representation a liouville generating function is obtained in completely separated form and shown through the liouvile arnold integration method to lead to the abel map linearization of all hamiltonian flows induced by the spectral invariants as illustrative examples the caseg sl 2 together with its real forms is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics with the liouville generating function expressed in hyperellipsoidal coordinates forg sl 3 the method is applied to the computation of quasi periodic solutions of the two component coupled nonlinear schrodinger equation a case which requires further symplectic constraints in order to deal with singularities in the spectral data at

hamiltonian structure for singular isomonodromy deformation equations a hamiltonian structure is constructed explicitly for a class of isomonodromy deformation equations for ordinary differential equations with one irregular singularity the hamiltonian turns out to be closely related to a differential form which is known to be naturally associated to isomonodromy deformation equations the monodromy data are constants of motion the functional independence of the formal monodromy exponent and stokes matrices is demonstrated by an example

extensions to metric based model selection metric based methods have recently been introduced for model selection and regularization often yielding very significant improvements over the alternatives tried including cross validation all these methods require unlabeled data over which to compare functions and detect gross differences in behavior away from the training points we introduce three new extensions of the metric model selection methods and apply them to feature selection the first extension takes advantage of the particular case of time series data in which the task involves prediction with a horizon h the idea is to use at t the h unlabeled examples that precede t for model selection the second extension takes advantage of the different error distributions of cross validation and the metric methods cross validation tends to have a larger variance and is unbiased a hybrid combining the two model selection methods is rarely beaten by any of the two methods the third extension deals with the case when unlabeled data is not available at all using an estimated input density experiments are described to study these extensions in the context of capacity control and feature subset selection

the soliton correlation matrix and the reduction problem for integrable systems integrable 1 1 dimensional systems associated to linear first order matrix equations meromorphic in a complex parameter as formulated by zakharov mikhailov and shabat 1 3 zms are analyzed by a new method based upon the soliton correlation matrix m matrix the multi backlund transformation which is equivalent to the introduction of an arbitrary number of poles in the zms dressing matrix is expressed by a pair of matrix riccati equations for them matrix through a geometrical interpretation based upon group actions on grassman manifolds the solution of this system is explicitly determined in terms of the solutions to the zms linear system reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing them matrix suitably so as to preserve the class of invariant solutions

τ functions and zero curvature equations of toda akns type the connection between functions and zero curvature equations for the homogeneous construction of the basic module l 0 over the simplest affine kac moody algebra a 1 1 is studied

multivariate two sample tests based on nearest neighbors abstract a new class of simple tests is proposed for the general multivariate two sample problem based on the possibly weighted proportion of all k nearest neighbor comparisons in which observations and their neighbors belong to the same sample large values of the test statistics give evidence against the hypothesis h of equality of the two underlying distributions asymptotic null distributions are explicitly determined and shown to involve certain nearest neighbor interaction probabilities simple infinite dimensional approximations are supplied the unweighted version yields a distribution free test that is consistent against all alternatives optimally weighted statistics are also obtained and asymptotic efficiencies are calculated each of the tests considered is easily adapted to a permutation procedure that conditions on the pooled sample power performance for finite sample sizes is assessed in simulations

g strands on symmetric spaces we study the g strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces these equations are simple field theory models whose configuration space is a lie group or in this case a symmetric space in this class of systems we derive several models that are completely integrable on finite dimensional lie group g and we treat in more detail examples with symmetric space su 2 s1 and so 4 so 3 the latter model simplifies to an apparently new integrable nine dimensional system we also study the g strands on the infinite dimensional group of diffeomorphisms which gives together with the sobolev norm systems of 1 2 camassa holm equations the solutions of these equations on the complementary space related to the witt algebra decomposition are the odd function solutions

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

two dimensional exactly and completely integrable dynamical systems monopoles instantons dual models relativistic strings lund regge model generalized toda lattice etc an investigation of two dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary lie algebra mathfrak g whose grading is consistent with an arbitrary integral embedding of 3d subalgebra in mathfrak g has been carried out we have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a goursat problem a method for the construction of a wide class of infinite dimensional lie algebras of finite growth has been proposed

a nonparametric approach to high dimensional k sample comparison problems summaryhigh dimensional k sample comparison is a common task in applications we construct a class of easy to implement distribution free tests based on new nonparametric tools and unexplored connections with spectral graph theory the test is shown to have various desirable properties and a characteristic exploratory flavour that has practical consequences for statistical modelling numerical examples show that the proposed method works surprisingly well across a broad range of realistic situations

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

some classes of solutions to the toda lattice hierarchy we apply an analogue of the zakharov shabat dressing method to obtain infinite matrix solutions to the toda lattice hierarchy using an operator transformation we convert some of these into solutions in terms of integral operators and fredholm determinants others are converted into a class of operator solutions to the l periodic toda hierarchy