temporal process regression summary we consider regression for response and covariates which are temporal processes observed over intervals a functional generalised linear model is proposed which includes extensions of standard models in multi state survival analysis simple nonparametric esti mators of time indexed parameters are developed using working independence estimating equations and are shown to be uniformly consistent and to converge weakly to gaussian processes the procedure does not require smoothing or a markov assumption unlike approaches based on transition intensities the usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy the estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters providing novel goodness of fit tests the methodology s practical utility is illustrated in a data analysis

impulsive boundary value problems with nonlinear boundary conditions abstract in this paper the upper and lower solution method and schauder s fixed point theorem are employed in the study of boundary value problems for a class of second order impulsive ordinary differential equations with nonlinear boundary conditions we prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

solitons from dressing in an algebraic approach to the constrained kp heirachy the algebraic matrix hierarchy approach based on affine lie sl n algebras leads to a variety of 1 1 soliton equations by varying the rank of the underlying sl n algebra as well as its gradation in the affine setting one encompasses the set of the soliton equations of the constrained kp hierarchy the soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl n algebras realized in the unconventional gradations such soliton solutions exhibit non trivial dependence on the kdv odd time flows and kp odd and even time flows which distinguishes them from the conventional structure of the darboux backlund wronskian solutions of the constrained kp hierarchy

ordinary differential equations with nonlinear boundary conditions of antiperiodic type abstract the quasilinearization method is successfully applied to differential problems with nonlinear boundary conditions of antiperiodic type given are sufficient conditions under which the monotone iterations converge quadratically to a unique solution of our problem the advantage of this approach lies in the fact that the approximate solutions are obtained iteratively as solutions of the above mentioned iterations

constrained kp models as integrable matrix hierarchies we formulate the constrained kp hierarchy denoted by ckp k 1 m as an affine sl m k 1 matrix integrable hierarchy generalizing the drinfeld sokolov hierarchy using an algebraic approach including the graded structure of the generalized drinfeld sokolov hierarchy we are able to find several new universal results valid for the ckp hierarchy in particular our method yields a closed expression for the second bracket obtained through dirac reduction of any untwisted affine kac moody current algebra an explicit example is given for the case sl m k 1 for which a closed expression for the general recursion operator is also obtained we show how isospectral flows are characterized and grouped according to the semisimple non regular element e of sl m k 1 and the content of the center of the kernel of e

current algebra of classical non linear sigma models the current algebra of classical non linear sigma models on arbitrary riemannian manifolds is analyzed it is found that introducing in addition to the noether currentj associated with the global symmetry of the theory a composite scalar fieldj the algebra closes under poisson brackets

first order impulsive ordinary differential equations with anti periodic and nonlinear boundary conditions the monotone iterative technique is a powerful method that have been used to approximate the extremal solutions of several problems 1 2 on the other hand impulsive differential equations are a basic tool to study some problems of biology medicine engineering and physics 3 in this paper we deal with the following nonlinear boundary value problem for first order differential equation with impulses at fixed points

dressing method and quadratic bundles related to symmetric spaces vanishing boundary conditions we consider quadratic bundles related to hermitian symmetric spaces of the type su m n s u m u n the simplest representative of the corresponding integrable hierarchy is given by a multi component kaup newell derivative nonlinear schrodinger equation which serves as a motivational example for our general considerations we extensively discuss how one can apply zakharov shabat s dressing procedure to derive reflectionless potentials obeying zero boundary conditions those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy one can distinguish between generic soliton type solutions and rational solutions

the gauge equivalence of the nls and the schrodinger flow of maps in 2 1 dimensions the gauge equivalence of the analogues of the nonlinear schrodinger equation nls and those of the schrodinger flow of maps into h2 the minkowski hf model in 2 1 dimensions are proved respectively combining these with the already known results we obtain a complete understanding of the gauge equivalence of the analogous models of the nonlinear schrodinger equation for 1 or 1 and those of the heisenberg ferromagnet model for euclidean or minkowski in 2 1 dimensions

coupling integrable couplings and bi hamiltonian structure associated with the boiti pempinelli tu hierarchy a kind of new lie algebra is introduced whose corresponding loop algebra is followed to define from which two isospectral problems are established whose compatibility condition gives rise to a coupling integrable coupling of the boiti pempinelli tu bpt hierarchy by making use of the quadratic form identity its bi hamiltonian structure is obtained finally we reduce the coupling integrable coupling to obtain two integrable couplings of the bpt hierarchy furthermore the well known mkdv equation and a new coupled equation with two dependent variables are produced respectively

regular conjugacy classes in the weyl group and integrable hierarchies generalized kdv hierarchies associated by drinfeld sokolov reduction with grade 1 regular semisimple elements from non equivalent heisenberg subalgebras of a loop algebra g x c lambda lambda 1 are studied the graded heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the weyl group w g of the simple lie algebra g a representative w epsilon w g of a regular conjugacy class can be lifted to an inner automorphism of g given by w exp 2i pi ad i0 m where i0 is the defining vector of an sl2 subalgebra of g the grading is then defined by the operator d m i0 m lambda d d lambda ad i0 and any grade 1 regular element lambda from the heisenberg subalgebra associated with w takes the form lambda c lambda c where i0 c m 1 c and c is included in an sl2 subalgebra containing i0 the largest eigenvalue of adi0 is m 1 except for some cases in f4 e6 7 8 we explain how these lie algebraic results follow from known results and apply them to construct integrable systems if the largest ad i0 eigenvalue is m 1 then using any grade 1 regular element from the heisenberg subalgebra associated with w we can construct a kdv system possessing the standard w algebra defined by i0 as its second poisson bracket algebra for g a classical lie algebra we derive pseudo differential lax operators for those non principal kdv systems that can be obtained as discrete reductions of kdv systems related to gln non abelian toda systems are also considered

extensions of the matrix gelfand dickey hierarchy from generalized drinfeld sokolov reduction thep p matrix version of ther kdv hierarchy has been recently treated as the reduced system arising in a drinfeld sokolov type hamiltonian symmetry reduction applied to a poisson submanifold in the dual of the lie algebra widehat gl pr 1 here a series of extensions of this matrix gelfand dickey system is derived by means of a generalized drinfeld sokolov reduction defined for the lie algebra widehat gl pr s 1 using the natural embeddinggl pr glpr s fors any positive integer the hierarchies obtained admit a description in terms of ap p matrix pseudo differential operator comprising anr kdv type positive part and a non trivial negative part this system has been investigated previously in thep 1 case as a constrained kp system in this paper the previous results are considerably extended and a systematic study is presented on the basis of the drinfeld sokolov approach that has the advantage that it leads to local poisson brackets and makes clear the conformal w algebra structures related to the kdv type hierarchies

monotone method for first order functional differential equations with retardation and anticipation abstract this paper discusses first order functional differential equations with retardation and anticipation by using the method of upper and lower solutions coupled with monotone iterative technique new existence results are obtained which improve known results in the literature

varying coefficient models we explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables general algorithms are presented for estimating the models flexibly and some examples are given this class of models ties together generalized additive models and dynamic generalized linear models into one common framework when applied to the proportional hazards model for survival data this approach provides a new way of modelling departures from the proportional hazards assumption

the generalised zakharov shabat system and the gauge group action the generalized zakharov shabat systems with complex valued non regular cartan elements and the systems studied by caudrey beals and coifman cbc systems and their gauge equivalent are studied this study includes the properties of fundamental analytical solutions for the gauge equivalent to cbc systems and the minimal set of scattering data the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator related to such systems the hierarchies of hamiltonian structures the results are illustrated in the example of the multi component nonlinear schrodinger equations and the corresponding gauge equivalent multi component heisenberg ferromagnetic type models related to so 5 c algebra

integrable equations in 2 1 dimensions associated with symmetric and homogeneous spaces generalizations of the n wave davey stewartson and kadomtsev petviashvili equations associated with homogeneous and symmetric spaces are presented these equations are 2 1 dimensional generalizations of those presented by fordy and kulish commun math phys 89 427 1983 and athorne and fordy j phys a 20 1377 1987 examples are explicitly presented that are associated with the simplest spaces in particular a new single component 2 1 dimensional generalization of the kdv equation is presented

local and non local conserved quantities for generalized non linear schrodinger equations it is shown how to construct infinitely many conserved quantities for the classical non linear schrodinger equation associated with an arbitrary hermitian symmetric spaceg k these quantities are non local in general but include a series of local quantities as a special case their poisson bracket algebra is studied and is found to be a realization of the half kac moody algebrakr consisting of polynomials in positive powers of a complex parameter which have coefficients in the compact real form ofk the lie algebra ofk

vortex filament on symmetric lie algebras and generalized bi schrodinger flows in this article we display an evolving model on symmetric lie algebras from a purely geometric way by using the hamiltonian or para hamiltonian gradient flow of a fourth order functional called generalized bi schrodinger flows which corresponds to the fukumoto moffatt s model in the theory of moving curves or the vortex filament in physical words in mathbb r 3 the theory of vortex filament in mathbb r 3 or mathbb r 2 1 up to the third order approximation is shown to be generalized to symmetric lie algebras in a unified way

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

modeling partly conditional means with longitudinal data abstract we propose a general modeling approach to longitudinal data that is a hybrid of the marginal regression models of zeger and liang and of the classical transition models such as used in time series analyses rather than conditioning at time t only on covariate values as is typical with the marginal approach or on the entire history of the process up to t as is typical with the transition model approach we suggest models that condition on a subset of the process history estimation proceeds using generalized estimating equation methodology but with the restriction that the working covariance matrix is diagonal the proposed regression models share common features with cox regression models for failure time data in that they are composed of a nuisance baseline function of time and a simple parametric function of the covariates two illustrative examples are presented

quadratic pseudopotentials for gl n c principal sigma models abstract matrix pseudopotentials for the gl n c principal sigma model in two dimensional minkowski space are derived through the method of closed differential ideals the structure of the pseudopotential equations is chosen to contain only quadratic nonlinearities of the matrix riccati type a set of further constraints is deduced which leads to inner backlund transformations depending on two complex parameters through a geometrical interpretation of such matrix riccati equations and the associated constraints a linearization procedure is derived and the equations are reduced to those of zakharov mikhailov and shabat 7 10 an inductive procedure is applied to explicitly solve an iterated sequence of such backlund transformations through a purely algebraic transformation of solutions to the first step the resulting nonlinear superposition formula is used to demonstrate a permutability theorem and to analyse the appearance of singularities fr nous deduisons les pseudo potentiels matriciels pour le modele sigma principal a valeurs dans gl n c defini sur l espace de minkowski bi dimensionnel en utilisant la methode des ideaux differentiels fermes on restreint la structure des equations des pseudopotentiels de facon a ce que n apparaissent que des non linearites quadratiques du type riccati matriciel nous deduisons un ensemble de contraintes supplementaires menant a des transformations de backlund internes dependantes de deux parametres complexes gr ce a l interpretation geometrique de ces equations de riccati matricielles et des contraintes associees nous deduisons un procede de linearisation et les equations se reduisent a celles de zakharov mikhailov et shabat 7 10 un procede inductif est utilise pour resoudre explicitement une succession de transformations de backlund par une transformation purement algebrique des solutions de la premiere etape la formule de superposition non lineaire qui en resulte permet de demontrer un theoreme de permutabilite et d analyser l apparition de singularites

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

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

the hamiltonian structure of the 2 1 dimensional ablowitz kaup newell segur hierarchy by considering hamiltonian theory over a suitable noncommutative ring the nonlinear evolution equations of the ablowitz kaup newell segur 2 1 hierarchy are incorporated into a hamiltonian framework and a modified lenard scheme

bi hamiltonian structures of the coupled akns hierarchy and the coupled yajima oikawa hierarchy the hamiltonian theory for the two component akns hierarchy and yajima oikawa hierarchy is considered from the viewpoint of reduction we show that the second hamiltonian structures of the former is a dirac reduction of the sl 3 current algebra while the latter is related to the classical w 3 4 algebra

group theoretical symmetries and generalized backlund transformations for integrable systems a notion of symmetry for 1 1 dimensional integrable systems is presented which is consistent with their group theoretic description it is shown how a group symmetry may be used together with a dynamical reduction to produce new generalizations of the backlund transformation for the korteweg de vries equation to its sl n c generalization an additional application to the relativistic invariance of the leznov saveliev systems is given

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

linear integral equations and multicomponent nonlinear integrable systems i abstract a systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations pde s is developed the method starts from a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k the spectral parameter special k dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term in this way the direct linearization is obtained of multicomponent generalizations of various nonlinear pde s such as the nonlinear schrodinger equation the derivative nonlinear schrodinger equations the isotropic heisenberg spin chain equation the complex sine gordon equation and the massive thirring model equations in the present paper i we present the general framework to derive finite matrix pde s and we also discuss backlund transformations and multicomponent lattice versions in a subsequent paper ii we treat a variety of examples of multicomponent pde s and discuss miura transformations and gauge equivalences

the matrix nonlinear schrodinger equation in dimension 2 in this paper we study the existence of global solutions to the cauchy problem for the matrix nonlinear schrodinger equation mnls in 2 space dimensions a sharp condition for the global existence is obtained for this equation this condition is in terms of an exact stationary solution of a semilinear elliptic equation in the scalar case the mnls reduces to the well known cubic nonlinear schrodinger equation for which existence of solutions has been studied by many authors

prediction using partly conditional time varying coefficients regression models summary data collected longitudinally in time provide the opportunity to develop predictive models of future observations given current data for an individual such models may be of particular value in defining individuals at high risk and thereby in suggesting subgroups for targeting of prevention intervention research efforts in this paper we propose a method for estimating predictive functions the method uses an extension of the marginal regression analysis methods of liang and zeger 1986 biometrika73 13 22 and is implemented using simple estimating equations a key feature of the models is that regression coefficients are modelled as smooth functions of the times both at and for prediction data from a study of obesity in childhood and early adulthood is used to demonstrate the methodology criteria for defining individuals to be at high risk can be defined on the basis of estimated predictive functions we suggest methods for evaluating the diagnostic accuracy sensitivity and specificity of such rules using cross validation the method holds promise as a robust and technically easy way of evaluating information about future prognosis that may be gleaned from a patient s current and past clinical status

cross validation the jackknife and the bootstrap excess error estimation in forward logistic regression abstract given a prediction rule based on a set of patients what is the probability of incorrectly predicting the outcome of a new patient call this probability the true error an optimistic estimate is the apparent error or the proportion of incorrect predictions on the original set of patients and it is the goal of this article to study estimates of the excess error or the difference between the true and apparent errors i consider three estimates of the excess error cross validation the jackknife and the bootstrap using simulations and real data the three estimates for a specific prediction rule are compared when the prediction rule is allowed to be complicated overfitting becomes a real danger and excess error estimation becomes important the prediction rule chosen here is moderately complicated involving a variable selection procedure based on forward logistic regression

the algebraic curve of classical superstrings on ads5 s5 we investigate the monodromy of the lax connection for classical iib superstrings on ads5 s5 for any solution of the equations of motion we derive a spectral curve of degree 4 4 the curve consists purely of conserved quantities all gauge degrees of freedom have been eliminated in this form the most relevant quantities of the solution such as its energy can be expressed through certain holomorphic integrals on the curve this allows for a classification of finite gap solutions analogous to the general solution of strings in flat space the role of fermions in the context of the algebraic curve is clarified finally we derive a set of integral equations which reformulates the algebraic curve as a riemann hilbert problem they agree with the planar one loop open image in new window supersymmetric gauge theory proving the complete agreement of spectra in this approximation

aks hierarchy and bi hamiltonian geometry of gelfand zakharevich type a bi hamiltonian system is a system of differential equations which can be written in hamiltonian form in two distinct ways the applications of gelfand zakharevich bi hamiltonian structure which is an extension of a poisson nijenhuis structure on phase space has been extensively explored by falqui magri and pedroni in the context of separation of variables it is well known that the integrable hamiltonian systems defined by the adler kostant symes aks scheme contains bi hamiltonian structure in this paper we unveil the connection between adler kostant symes formalism applied to loop algebra and the gelfand zakharevich bi hamiltonian structure by superposition the results of fordy and kulish in the aks scheme we also study the commuting flows of the aks hierachy and its connection to the zakharov shabat hierarchy

quantum integrability and quantum groups abstract this report presents a brief summary of the theory of quantum integrable systems first the quantum inverse scattering method is used to give an algebraic formulation of the bethe ansatz method second the yang baxter relation a sufficient condition for integrability is introduced and its relation to quantum groups is discussed third some recent results on quantum integrable systems with long range interactions are explained to present the basic ideas and concepts in an introductory way specific models are chosen the results however apply to a wider class of quantum integrable models

riemann equations in bidifferential calculus we consider equations that formally resemble a matrix riemann or hopf equation in the framework of bidifferential calculus with different choices of a first order bidifferential calculus we obtain a variety of equations including a semi discrete and a fully discrete version of the matrix riemann equation a corresponding universal solution generating method then either yields a continuous or discrete cole hopf transformation or leaves us with the problem of solving riemann equations hence an application of the hodograph method if the bidifferential calculus extends to second order solutions of a system of riemann equations are also solutions of an equation that arises on the universal level of bidifferential calculus as an integrability condition depending on the choice of bidifferential calculus the latter can represent a number of prominent integrable equations like self dual yang mills as well as matrix versions of the two dimensional toda lattice hirota s bilinear difference equation 2 1 dimensional nonlinear schrodinger nls kadomtsev petviashvili kp equation and davey stewartson equations for all of them a recent non isospectral binary darboux transformation result in bidifferential calculus applies which can be specialized to generate solutions of the associated riemann equations for the latter we clarify the relation between these specialized binary darboux transformations and the aforementioned solution generating method from arbitrary size matrix versions of the riemann equations associated with an integrable equation possessing a bidifferential calculus formulation multi soliton type solutions of the latter can be generated this includes breaking multi soliton type solutions of the self dual yang mills and the 2 1 dimensional nls equation which are parametrized by solutions of riemann equations

t duality in 2d integrable models the non conformal analogue of abelian t duality transformations relating pairs of axial and vector integrable models from the non abelian affine toda family is constructed and studied in detail

reductions of n wave interactions related to low rank simple lie algebras i z2 reductions the analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem we show how the second order reductions of the n wave interactions related to low rank simple lie algebras g can be embedded also in the weyl group of g this allows us to display along with the well known ones a number of new types of integrable n wave systems some of the reduced systems find applications to nonlinear optics

twisted hierarchies associated with the generalized sine gordon equation twisted u and twisted u k hierarchies are soliton hierarchies introduced by terng to find higher flows of the generalized sine gordon equation twisted o j j o j o j hierarchies are among the most important classes of twisted hierarchies in this paper we derive explicit interesting first and higher flows of twisted o j j o j o j hierarchies justify that the one dimensional systems of twisted o j j o j o j hierarchies for j iq n q 1 q n 1 called the generalized sinh gordon equations are the gauss codazzi equations for n dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo euclidean 2n 1 dimensional space r2q 12n 1 with index 2q 1 furthermore a unified treatment of the inverse scattering theory for twisted o j j o j o j hierarchies is provided

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

quasiperiodic solutions for matrix nonlinear schrodinger equations the adler kostant symes theorem yields isospectral hamiltonian flows on the dual erm g of a lie subalgebra erm g of a loop algebra erm g a general approach relating the method of integration of krichever novikov and dubrovin to such flows is used to obtain finite gap solutions of matrix nonlinear schrodinger equations in terms of quotients of functions

integrable extension of nonlinear sigma model we propose an integrable extension of nonlinear sigma model on the target space of hermitian symmetric space hss starting from a discussion of soliton solutions of o 3 model and an integrally extended version of it we construct general theory defined on arbitrary hss by using the coadjoint orbit method it is based on the exploitation of a covariantized canonical structure on hss this term results in an additional first order derivative term in the equation of motion which accommodates the zero curvature representation infinite conservation laws of nonlocal charges in this model are derived

integrable dynamical systems with quadratic and cubic nonlinearities related to graded lie algebras practical implementation of two important methods of constructing integrable dynamical systems the lax pair method and the adler konstant symes theorem is studied general considerations are illustrated by several examples that contain multiwave interaction systems and interacting garnier systems as particular cases

existence of solutions for anti periodic boundary value problems abstract by using schauder s fixed point theorem and the lower and upper solutions method we discuss the anti periodic boundary value problem for a class of second order differential equations some sufficient conditions for the existence of solutions are obtained

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

vertex operators and soliton solutions of affine toda model with u 2 symmetry the symmetry structure of the non abelian affine toda model based on the coset sl 3 sl 2 u 1 is studied it is shown that the model possess non abelian noether symmetry closing into a q deformed sl 2 u 1 algebra specific two vertex soliton solutions are constructed

dynamic generalized linear models and bayesian forecasting abstract dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and posterior distributions for the exponential family parameters this leads to the calculation of closed standard form predictive distributions for forecasting and model criticism the structure of the models depends on the time evolution of underlying state variables and the feedback of observational information to these variables is achieved using linear bayesian prediction methods data analytic aspects of the models concerning scale parameters and outliers are discussed and some applications are provided dynamic bayesian models are developed for application in nonlinear non normal time series and regression problems providing dynamic extensions of standard generalized linear models a key feature of the analysis is the use of conjugate prior and

four lie algebras associated with r6 and their applications the first part in the paper reads that a three dimensional lie algebra is first introduced whose corresponding loop algebra is constructed for which isospectral problems are established by employing zero curvature equations a modified kaup newell mkn soliton hierarchy of evolution equations is obtained the corresponding hereditary operator and hamiltonian structure are worked out respectively then two types of enlarging semisimple lie algebras isomorphic to the linear space r6 are followed to construct one of them is a complex lie algebra their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mkn hierarchy the hereditary operators hamiltonian structures of the hierarchies are produced again respectively the exact computing formulas of the constant appearing in the trace identity and the variational identity are derived under the semisimple algebras the second part of this pap

nonautonomous integrable nonlinear schrodinger equations with generalized external potentials inhomogeneous nonlinear schrodinger equations are constructed systematically using the covariance with respect to the darboux transformation the darboux invariants and the spectral parameter are functions of time and space which result in equations describing nonautonomous solitons moving in generalized external potentials all the known inhomogeneous nonlinear schrodinger equations as well as new ones of a more generalized form can be derived from the constructed equation one and two soliton solutions are explicitly constructed using the darboux transformation

integrable evolution equations on associative algebras this paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra which includes matrix clifford and group algebra valued systems a variety of new examples of integrable systems possessing higher order symmetries are presented symmetry reductions lead to an associative algebra valued version of the painleve transcendent equations the basic theory of hamiltonian structures for associative algebra valued systems is developed and the bihamiltonian structures for several examples are found

hierarchies of evolution equations associated with certain symmetric spaces hierarchies of evolution equations associated with certain symmetric spaces are presented main examples of such symmetric spaces are hermitian symmetric spaces and the obtained evolution equations include generalized nonlinear schrodinger equations those results are obtained by extending the lie algebra in the ablowitz kaup newell segur scheme from sl 2 c to more general one

integrable systems and invariant curve flows in centro equiaffine symplectic geometry abstract in this paper the theory for curves in centro equiaffine symplectic geometry is established integrable systems satisfied by the curvatures of curves under inextensible motions in centro equiaffine symplectic geometry are identified it is shown that certain non stretching invariant curve flows in centro equiaffine symplectic geometry are closely related to the matrix kdv equations and their extension

generalized heisenberg ferromagnet models via hermitian symmetric spaces we study the generalized heisenberg ferromagnet model using a duality with the hermitian symmetric space nonlinear schrodinger equation we explain the mathematical structure of the hermiticity of the space in connection with the soliton surface theory and obtain various spin configurations in n dimension using this correspondence

dressing method for the vector sine gordon equation and its soliton interactions in this paper we develop the dressing method to study the exact solutions for the vector sine gordon equation the explicit formulas for one kink and one breather are derived the method can be used to construct multi soliton solutions two soliton interactions are also studied the formulas for position shift of the kink and position and phase shifts of the breather are given these quantities only depend on the pole positions of the dressing matrices

anti periodic boundary value problems of second order impulsive differential equations this paper discusses anti periodic boundary value problems of second order impulsive differential equations by using the method of upper and lower solutions coupled with the monotone iterative technique new existence results of coupled solutions and uniqueness of problems are obtained

a new family of integrable models in 2 1 dimensions associated with hermitian symmetric spaces in a series of papers fordy and his collaborators have studied families of integrable models in 1 1 dimensions associated with hermitian symmetric spaces these models were also generalized to 2 1 dimensions however the generalization used is not unique and in this paper a different generalization is considered resulting in new families of integrable models in 2 1 dimensions

darboux transformation and exact multisolitons of cpn nonlinear sigma model abstract exact multisolitons of c p n sigma model are obtained by using darboux transformation in terms of quasideterminant darboux matrix the multisoliton solutions are shown to be expressed in terms of quasideterminants of gelfand and retakh 1991 10 the method of quasideterminant darboux matrix has been shown to be related with the dressing method that has already been used to compute iteratively exact multisolitons of the c p n sigma model by means of nonlinear superposition principle of backlund transformations explicit expressions of one two and three soliton solutions of c p 1 model have been computed by using properties of quasideterminants

anti periodic solutions of lienard equations with state dependent impulses abstract subject to a priori bounds lienard equations with state dependent impulsive forcing are shown to admit a unique absolutely continuous anti periodic solution with first derivative of bounded variation on finite intervals the point wise convergence of a sequence of iterates to the solution is obtained along with a bound for the rate of convergence the results are applied to josephson s and van der pol s equations

a discretization of the matrix nonlinear schrodinger equation in this paper we shall show that a class of solutions to the discrete coupled matrix nonlinear schrodinger equation dcmnlse is gauge equivalent to the discrete equation of the schrodinger flow of maps into the grassmannian and the realizing gauge transformation is only the discretization of a classical gauge transformation between the matrix nonlinear schrodinger equation mnlse and the schrodinger flow of maps into the grassmannian in other words from the viewpoint of gauge equivalence a class of solutions of the dcmnlse is a correct discretization of the mnlse

matrix nonlinear schrodinger equations and moment maps into loop algebras it is shown how darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear schrodinger equations are realized as isospectral hamiltonian flows the parametrization follows from a moment map embedding of the symplectic vector space reduced by suitable group actions into the dual g of the algebra g of positive frequency loops in a lie algebra g the resulting phase space is identified with a poisson subspace of g consisting of elements that are rational in the loop parameter reduced coordinates associated to the various hermitian symmetric lie algebras g k corresponding to the classical lie algebras are obtained

existence results for discontinuous functional differential systems in this paper we study solvability of first order discontinuous functional differential systems equipped with discontinuous functional initial conditions a concrete example is presented and solved to illustrate the obtained results

a powerful approach to generate new integrable systems in this paper an effective algorithm to generate integrable systems is given as a result many new integrable equations are derived in a systematic way

vector nonlinear schrodinger equation on the half line we investigate the manakov model or more generally the vector nonlinear schrodinger equation on the half line using a backlund transformation method two classes of integrable boundary conditions are derived mixed neumann dirichlet and robin boundary conditions integrability is shown by constructing a generating function for the conserved quantities we apply a nonlinear mirror image technique to construct the inverse scattering method with these boundary conditions the important feature in the reconstruction formula for the fields is the symmetry property of the scattering data emerging from the presence of the boundary particular attention is paid to the discrete spectrum an interesting phenomenon of transmission between the components of a vector soliton interacting with the boundary is demonstrated this is specific to the vector nature of the model and is absent in the scalar case for one soliton solutions we show that the boundary can be used to make certain components of the incoming soliton vanishingly small this is reminiscent of the phenomenon of light polarization by reflection

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

quantum nonlinear wave interaction system abstract the quantum spectral transform method is applied to a lie model quantum three wave interaction system the eigenvectors of the integrals of motion are constructed using the algebraic bethe ansatz different reductions and generalizations of the three wave system are presented

homogeneous kahler manifolds paving the way towards new supersymmetric sigma models homogeneous kahler manifolds give rise to a broad class of supersymmetric sigma models containing as a rather special subclass the more familiar supersymmetric sigma models based on hermitian symmetric spaces in this article all homogeneous kahler manifolds with semisimple symmetry groupg are constructed and are classified in terms of dynkin diagrams explicit expressions for the complex structure and the kahler structure are given in terms of the lie algebra g ofg it is shown that for compactg one can always find an einstein kahler structure which is unique up to a constant multiple and for which the kahler potential takes a simple form

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

some series of infinitely many symmetry generators in symmetric space chiral models this paper shows that starting from any conserved current generated by some given infinitesimal symmetry generator one may use finite dual transformations to induce infinitely many infinitesimal symmetry generators thus besides starting from ordinary isotopic and space time translation this paper also discovers the infinitesimal generators for backlund transformation for dual symmetry itself and other general cases and then uses them to generate infinitely many local or nonlocal currents respectively

sit nls solitons in hermitian symmetric spaces soliton solutions of multi component sit nls systems associated with hermitian symmetric spaces are obtained using the backlund transformation projectors of special form are introduced and solved to satisfy specific conditions to obtain one soliton solutions the non abelian permutability theorem is developed and used to obtain two soliton solutions

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

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures