confidence intervals for an effect size measure based on the mann whitney statistic part 1 general issues and tail area based methods for two random variables x and y pr y x pr y x is advocated as a general measure of effect size to characterize the degree of separation of their distributions it is estimated by u mn a generalization of the mann whitney u statistic derived by dividing u by the product of the two sample sizes it is equivalent to the area under the receiver operating characteristic curve it is readily visualized in terms of two gaussian distributions with appropriately separated peaks the effect of discretization of a continuous variable is explored tail area based confidence interval methods are developed which can be applied to very small samples or extreme outcomes copyright 2005 john wiley sons ltd

exact nonparametric inference when doses are measured with random errors studies that estimate the effects of exposure to a possibly harmful agent often compare exposed subjects who received varied doses with matched controls who received zero dose if the doses are measured with error then one may wish to use the fallible doses to estimate a linear relationship between the unobserved true dose and the observed response if one is willing to assume that the dose errors for exposed subjects are symmetrically distributed about 0 that is the dose errors are pure errors and not say systematic underreporting of exposure then the presence of zero dose controls is all that is needed to obtain exact distribution free confidence intervals and tests and consistent point estimates the method is simpler for matched pairs than for matched sets with two or more matched subjects and it is illustrated using two studies one of each kind with matched pairs as in this first example the method uses wilcoxon s signed rank test as the basis for inference when there are several zero dos

generalized drinfel d sokolov hierarchies a general approach is adopted to the construction of integrable hierarchies of partial differential equations a series of hierarchies associated to untwisted kac moody algebras and conjugacy classes of the weyl group of the underlying finite lie algebra is obtained the generalized kdv hierarchies of v g drinfel d and v v sokolov are obtained as the special case for the coxeter element various examples of the general formalism are treated in some detail including the fractional kdv hierarchies

a stochastic version of center manifold theory random dynamical systems arise naturally if the influence of white or real noise on the parameters of a nonlinear determinstic dynamical system is studied in this situation lyapunov exponents attached to the linearized flow replace the real parts of the eigenvalues and describe the stability behavior of the linear system if at least one of them vanishes then it is possible to prove the existence of a stochastic analogue of the deterministic center manifold the asymptotic behavior of the entire system can then be derived from the lower dimensional system restricted to this stochastic center manifold a dynamical characterization of the stochastic center manifold is given and approximation results are proved

exemplary data set sample size calculation for wilcoxon mann whitney tests zhao rahardja and qu consider sample size calculation for wilcoxon mann whitney wmw tests for data with ties and present a straightforward formula we observe that the exemplary data set approach usually applied in more complex situations has a close relationship to the zhao rahardja qu method for wmw sample size estimation and they are asymptotically equivalent therefore the exemplary data set approach can be used to easily obtain estimates similar to those that the closed formula gives we illustrate application of both methods for a wmw sample size estimation example and also extend the simulation study presented by zhao et al we find that the zhao rahardja qu formula and by extension the exemplary data set method can give estimates just as accurate as those obtained using either the kolassa approach via nquery advisor or the o brien castelloe approach via sas 9 2 proc power for 1 1 and 1 2 allocation ratios however the latter two methods can be more accurate for a ratio of 1 4 or 1 19 finally we note the general utility of the exemplary data set approach for sample size estimation even in other situations where closed form sample size formulae exist

sit nls integrable equations associated with hermitian symmetric spaces we study multi component sit nls systems associated with hermitian symmetric spaces we introduce generalized polarizations and express them in terms of the variable of hermitian symmetric spaces the equations of motion are expressed using electromagnetic fields and generalized polarizations the backlund transformation of the sit nls system is introduced which relates two solutions of sit nls systems

sample size estimation for the van elteren test a stratified wilcoxon mann whitney test the van elteren test is a type of stratified wilcoxon mann whitney test for comparing two treatments accounting for strata in this paper we study sample size estimation methods for the asymptotic version of the van elteren test assuming that the stratum fractions ratios of each stratum size to the total sample size and the treatment fractions ratios of each treatment size to the stratum size are known in the study design in particular we develop three large sample sample size estimation methods and present a real data example to illustrate the necessary information in the study design phase in order to apply the methods simulation studies are conducted to compare the performance of the methods and recommendations are made for method choice finally sample size estimation for the van elteren test when the stratum fractions are unknown is also discussed

the prolongation structures of quasi polynomial flows we look closely at the process of finding a wahlquist estabrook prolongation structure for a given system of nonlinear evolution equation s there are two main steps in this calculation the first to reduce the problem to the investigation of a finitely generated free lie algebra with constraints the second to find a finite dimensional linear representation of these generators we discuss some of the difficulties that arise in this calculation for quasi polynomial flows defined later we give an algorithm for the first step we do not totally solve the problems of the second step but do give an algebraic framework and a number of techniques that are quite generally applicable we illustrate these methods with many examples several of which are new

local hamiltonian structures of multicomponent kdv equations the author discusses the behaviour of hamiltonian structures for a generalised akns hierarchy under componentwise reductions to equations of kdv and mkdv type identifying those cases where a new local structure arises

a generalized formula for integrable classes of surfaces in lie algebras we discuss relations between the approach of fokas and gelfand to immersions on lie algebras and the theory of soliton surfaces of sym we show that many results concerning immersions on lie algebras can be reduced to or interpreted within the soliton surfaces approach we present also some new results including a generalization of the fokas gelfand formula for integrable classes of surfaces in lie algebras and in particular in pseudo euclidean n dim spaces the generalized formula is used to formulate a method of constructing integrable classes of surfaces as an example we discuss the class of linear weingarten surfaces defined by the linear relationship between gaussian and mean curvatures we construct explicitly a one parameter family of linear weingarten surfaces parallel equidistant to a given pseudospherical surface

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

integrable counterparts of the d kaup newell soliton hierarchy two integrable counterparts of the d kaup newell soliton hierarchy are constructed from a matrix spectral problem associated with the three dimensional special orthogonal lie algebra so 3 r an application of the trace identity presents hamiltonian or quasi hamiltonian structures of the resulting counterpart soliton hierarchies thereby showing their liouville integrability i e the existence of infinitely many commuting symmetries and conserved densities the involved hamiltonian and quasi hamiltonian properties are shown by computer algebra systems

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

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

determining the appropriate sample size for nonparametric tests for location shift for the case in which two independent samples arc to be compared using a nonparametric test for location shift we propose a bootstrap technique for estimating the sample sizes required to achieve a specified power the estimator called boot uses information from a small pilot experiment for the special case of the wilcoxon test a simulation study is conducted to compare boot to two other sample size estimators one method called anpv is based on the assumption that the underlying distribution is normal with a variance estimated from the pilot data the other method called noether adapts the sample size formula of noether for use with a location shift alternative the boot and noether sample size estimators are particularly appropriate for this nonparametric setting because they do not require assumptions about the shape of the underlying continuous probability distribution the simulation study shows that a sample size estimates can have large uncertainty b boot is at least as accurate as an

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

probabilistic index an intuitive non parametric approach to measuring the size of treatment effects effect sizes es tell the magnitude of the difference between treatments and ideally should tell clinicians how likely their patients will benefit from the treatment currently used es are expressed in statistical rather than in clinically useful terms and may not give clinicians the appropriate information we restrict our discussion to studies with two groups one with n patients receiving a new treatment and the other with m patients receiving the usual or no treatment the standardized mean difference e g cohen s d is a well known index for continuous outcomes there is some intuitive value to d but measuring improvement in standard deviations sd is a statistical concept that may not help a clinician how much improvement is a half sd a more intuitive and simple to calculate es is the probability that the response of a patient given the new treatment x is better than the one for a randomly chosen patient given the old or no treatment y i e p x y larger values meaning better outcomes this probability has an immediate identity with the area under the curve auc measure in procedures for receiver operator characteristic roc curve comparing responses to two treatments it also can be easily calculated from the mann whitney u wilcoxon or kendall statistics we describe the characteristics of an ideal es we propose p x y as an alternative index summarize its correspondence with well known non parametric statistics compare it to the standardized mean difference index and illustrate with clinical data copyright 2005 john wiley sons ltd

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

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

power and sample size calculations for discrete bounded outcome scores we consider power and sample size calculations for randomized trials with a bounded outcome score bos as primary response adjusted for a priori chosen covariates we define bos to be a random variable restricted to a finite interval typically a bos has a j or u shaped distribution hindering traditional parametric methods of analysis when no adjustment for covariates is needed a non parametric test could be chosen however there is still a problem with calculating the power since the common location shift alternative does not hold in general for a bos in this paper we consider a parametric approach and assume that the observed bos is a coarsened version of a true bos which has a logit normal distribution in each treatment group allowing correction for covariates a two step procedure is used to calculate the power firstly the power function is defined conditionally on the covariate values secondly the marginal power is obtained by averaging the conditional power with respect to an assumed distribution for the covariates using monte carlo integration a simulation study evaluates the performance of our method which is also applied to the ecass 1 stroke study

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

two integrable hamiltonian hierarchies in sl 2 r and so 3 r with three potentials by introducing two specific matrix spectral problems associated with sl 2 r and so 3 r matrix lie algebras we generate two integrable hamiltonian hierarchies with three potentials the computation and analysis on their hamiltonian structures by means of the trace identity show that the resulting hierarchies are liouville integrable namely that each hierarchy consists of commuting hamiltonian soliton equations

an integrable su 3 spin chain abstract an integrable su 3 soin chain equation is constructed an explicit form of the lax pair representation is given and it is shown to be gauge equivalent to a set of nonlinear schrodinger type equations for three complex variables

completeness of the eigenfunctions for the caudrey beals coifman system the direct and the inverse scattering problem for the first order linear systems of the type l x i d dx q x j x 0 j h q x gj which generalizes the zakharov shabat system and the systems studied by caudrey beals and coifman cbc is analyzed herein here j is a regular complex constant element of the cartan subalgebra h g of the simple lie algebra g and the potential q x vanishes fast enough for x taking values in the image gj of adj the cbc results are generalized and the fundamental analytic solution m x for any choice of the irreducible finite dimensional representation v of g is constructed four pairwise equivalent minimal sets of scattering data for l invariant with respect to the choice of the representation of g are extracted from the asymptotics of m x for x from m x the resolvent of l is constructed in the adjoint representation vad and the completeness relation is proven for the eigenfunctions of l in vad it is also proven that the discrete spectrum

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

additive noise induced turing transitions in spatial systems with application to neural fields and the swift hohenberg equation this work studies the spatio temporal dynamics of a generic integral differential equation subject to additive random fluctuations it introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for fokker planck equations and studies analytically the systems stability near turing bifurcations in addition two types of fluctuation are studied namely fluctuations uncorrelated in space and time and global fluctuations which are constant in space but uncorrelated in time we show that the global fluctuations shift the turing bifurcation threshold this shift is proportional to the fluctuation variance applications to a neural field equation and the swift hohenberg equation reveal the shift of the bifurcation to larger control parameters which represents a stabilization of the system all analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral differential equation to gain some insight into experimental manifestations the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively

estimating the power of the two sample wilcoxon test for location shift summary traditional methods for calculating the power of a statistical test for location shift require knowledge of the shape of the underlying probability distribution the distribution shape however may be unknown this paper describes a bootstrap method for using observed data or pilot data to approximate the power no assumptions need be made about the shape of the underlying continuous probability distribution simulation evidence shows that when applied to the wilcoxon two sample test for location shift the suggested method is reliable the evidence also shows that it is more accurate than a benchmark traditional approach the bootstrap method is applied to a real data example the analysis demonstrates how the method can be used to determine sample sizes and how to choose the more powerful of two alternative tests for location shift

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

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

calculation of power and sample size with bounded outcome scores the two sample wilcoxon rank sum test is the most popular non parametric test for the comparison of two samples when the underlying distributions are not normal although the underlying distributions need not be known in detail to calculate the null distribution of the test statistic parametric assumptions are often made to determine the power of the test or the sample size we encountered difficulties with this approach in the planning of a recent clinical trial in stroke patients it is shown that for power and sample size estimation it can be dangerous to apply the classical formulae routinely especially with outcome scores having a u shaped or a j shaped distribution as an example we have taken the barthel index a quality of life outcome measure in stroke patients further we have investigated alternative methods by means of monte carlo simulation the distributional characteristics of the estimated powers were compared our findings suggest more appropriate computer software is necessary for the calculation of power and sample size when efficacy is measured by a non parametric method

on differential systems describing surfaces of constant curvature abstract the geometric notion of a differential system describing surfaces of constant nonzero gaussian curvature is introduced the nonlinear schrodinger equation nls with 1 and 1 is shown to describe a family of spherical surfaces s s and pseudospherical surfaces p s s respectively the schrodinger flow of maps into s 2 the hf model and its generalized version the landau lifschitz equation are shown to describe spherical surfaces the schrodinger flow of maps into h 2 the m hf model provides another example of a system describing pseudo spherical surfaces new differential systems describing surfaces of nonzero constant gaussian curvature are obtained furthermore we give a characterization of evolution systems which describe surfaces of nonzero constant gaussian curvature in particular we determine all differential systems of type u t v xx h 11 u v u x h 12 u v v x h 13 u v v t u xx h 21 u v u x h 22 u v v x h 23 u v which describe pseudospherical or spherical surfaces as an application we obtain four parameter family of such systems for a complex valued function q u iv given by iq t q xx i q 2 q x i q x q 2 q q 0 where 0 if 0 particular cases of this family obtained by the vanishing of the parameters are the linear equations the nls equation the derivative nonlinear schrodinger equation dnls and the mixed nls dnls equation

gauge transformation of the third kind for u 1 invariant coupled schrodinger equations we consider a large class of canonical coupled nonlinear schrodinger equations invariant over the action of the u 1 group the most general nonlinearity is taken into account through a matrix that without loss of generality can be separated into the sum of a hermitian matrix and an anti hermitian matrix the u 1 symmetry implies the existence of a set of continuity equations for the conserved densities where the corresponding currents have in general a nonlinear structure for this class of coupled schrodinger equations we introduce a nonlinear gauge transformation which changes the nonlinear matrix into another one purely hermitian consequently the currents are transformed in the standard bilinear form generalization to noncanonical systems is also discussed some examples are presented to illustrate the applicability of the method

quasideterminant solutions of the generalized heisenberg magnet model in this paper we present the darboux transformation for the generalized heisenberg magnet ghm model based on the general linear lie group gl n and construct multi soliton solutions in terms of quasideterminants further we relate the quasideterminant multi soliton solutions obtained by means of darboux transformation with those obtained by the dressing method we also discuss the model based on the lie group su n and obtain explicit soliton solutions of the model based on su 2

a u statistics based approach to sample size planning of two arm trials with discrete outcome criterion aiming to establish either superiority or noninferiority in current practice the most frequently applied approach to the handling of ties in the mann whitney wilcoxon mww test is based on the conditional distribution of the sum of mid ranks given the observed pattern of ties starting from this conditional version of the testing procedure a sample size formula was derived and investigated by zhao et al stat med 2008 in contrast the approach we pursue here is a nonconditional one exploiting explicit representations for the variances of and the covariance between the two u statistics estimators involved in the mann whitney form of the test statistic the accuracy of both ways of approximating the sample sizes required for attaining a prespecified level of power in the mww test for superiority with arbitrarily tied data is comparatively evaluated by means of simulation the key qualitative conclusions to be drawn from these numerical comparisons are as follows with the sample sizes calculated by means of the respective formula both versions of the test maintain the level and the prespecified power with about the same degree of accuracy despite the equivalence in terms of accuracy the sample size estimates obtained by means of the new formula are in many cases markedly lower than that calculated for the conditional test perhaps a still more important advantage of the nonconditional approach based on u statistics is that it can be also adopted for noninferiority trials copyright 2016 john wiley sons ltd

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

confidence intervals for an effect size measure based on the mann whitney statistic part 2 asymptotic methods and evaluation several asymptotic confidence interval methods for u mn the mann whitney u statistic divided by the product of the two sample sizes are developed and evaluated alongside published methods novel methods derived from gaussian and beta models perform well though overall a modification of the hanley mcneil approach method 5 in this evaluation performed better and is recommended for calculating confidence intervals for u mn for continuously distributed data

integrable decompositions of a symmetric matrix kaup newell equation and a symmetric matrix derivative nonlinear schrodinger equation it is shown that the reduction conditions in the spectral problems of soliton equations can be replaced by more equations of eigenfunctions based on this the method of nonlinearization of spectral problem is extended to soliton equations with reduction conditions integrable decompositions of a symmetric matrix kaup newell equation and a symmetric matrix derivative nonlinear schrodinger equation are constructed

lie algebras and korteweg de vries equations abstract for each metrizable lie algebra g an infinite commuting hierarchy of bi hamiltonian integrable extensions of the korteweg de vries kdv hierarchy is found the classical kdv hierarchy itself corresponds to the case g 0 the construction is based on the hamiltonian formalism on the dual space to a certain one parameter family of differential lie algebras produced from g

power and sample size calculations for the wilcoxon mann whitney test in the presence of death censored observations we consider a clinical trial of a potentially lethal disease in which patients are randomly assigned to two treatment groups and are followed for a fixed period of time a continuous endpoint is measured at the end of follow up for some patients however death or severe disease progression may preclude measurement of the endpoint a statistical analysis that includes only patients with endpoint measurements may be biased an alternative analysis includes all randomized patients with rank scores assigned to the patients who are available for the endpoint measurement on the basis of the magnitude of their responses and with worst rank scores assigned to those patients whose death precluded the measurement of the continuous endpoint the worst rank scores are worse than all observed rank scores the treatment effect is then evaluated using the wilcoxon mann whitney test in this paper we derive closed form formulae for the power and sample size of the wilcoxon mann whitney test when missing measurements of the continuous endpoints because of death are replaced by worst rank scores we distinguish two approaches for assigning the worst rank scores in the tied worst rank approach all deaths are weighted equally and the worst rank scores are set to a single value that is worse than all measured responses in the untied worst rank approach the worst rank scores further rank patients according to their time of death so that an earlier death is considered worse than a later death which in turn is worse than all measured responses in addition we propose four methods for the implementation of the sample size formulae for a trial with expected early death we conduct monte carlo simulation studies to evaluate the accuracy of our power and sample size formulae and to compare the four sample size estimation methods

resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations we consider a general method for the macroscale modelling of nonlinear nonautomomous partial differential equations or difference equations our method is able to resolve the complex fluctuations which arise in nonliner nonautonomous systems these fluctuations are not accounted for by popular multiscale averaging procedures and homogenization our method is illustrated with the example of a forced burgers equation coarse grained macroscale spatial discretisations of nonlinear nonautonomous partial differential difference equations are given novel support by centre manifold theory dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements the crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements the techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential difference equations such as the forced burgers equation adopted here as an illustrative example

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

bifurcations in fluctuating systems the center manifold approach bifurcations in fluctuating dynamical systems are studied using the ideas of center manifold reduction the method provides not only a systematic procedure for the reduction of the system to a small number of variables but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points the joint probability density factorizes into a stationary gaussian densityp v u in the fast variablesv and a time dependent densityp u f in the slow variablesu describing the dynamics on the center manifoldv v0 u p u t obeys a reduced fokker planck equation that can be written in a normal form by means of local nonlinear transformations both additive and multiplicative white noise are considered as is colored noise the results extend and formalize haken s concept of adiabatic elimination of fast variables

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

conservation laws in higher order nonlinear schrodinger equations conservation laws of the nonlinear schr odinger equation are studied in the presence of higher order optical effects including the third order dispersion and the self steepening in a context of group theory we derive general expressions for infinitely many conserved currents and charges of a coupled higher order nonlinear schr odinger equation the first few currents and associated charges are also presented explicitly due to the higher order effects the conservation laws of the nonlinear schr odinger equation are violated in general the differences between the types of the conserved currents for the hirota and the sasa satsuma equations imply that the higher order terms determine the inherent types of conserved quantities for each integrable case of the higher order nonlinear schr odinger equation

curve motions from the integrable equations having variable spectral parameters this paper reports on a study of the motion of moving space curves induced from integrable equations having variable spectral parameters we explain the geometric structure of the curves using the associated linear equations of integrable equations a generalized form of the hasimoto transformation is introduced this form relates the curvature and torsion of the curves to the variables of integrable equations some explicit curve motions including the solitonic and kelvin type curves are calculated using the sym tafel formula

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

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

the use of bootstrap methods for estimating sample size and analysing health related quality of life outcomes health related quality of life hrqol measures are increasingly used in trials as primary outcome measures investigators are now asking statisticians for advice on how to plan and analyse studies using such outcomes hrqol outcomes like the sf 36 are usual measured on an ordinal scale although most investigators assume that there exists an underlying continuous latent variable and that the actual measured outcomes the ordered categories reflect contiguous intervals along this continuum the ordinal scaling of hrqol measures means they tend to generate data that have discrete bounded and skewed distributions thus standard methods of analysis that assume normality and constant variance may not be appropriate for this reason conventional statistical advice would suggest non parametric methods be used to analyse hrqol data the bootstrap is one such computer intensive non parametric method for estimating sample sizes and analysing data we describe three methods of estimating sample sizes for two group cross sectional comparisons of hrqol outcomes we then compared the power of the three methods for a two group cross sectional study design using bootstrap simulation the results showed that under the location shift alternative hypothesis conventional methods of sample size estimation performed well particularly whitehead s method whitehead s method is recommended if the hrqol outcome has a limited number of discrete values 7 and or the expected proportion of cases at either of the bounds is high if a pilot data set is readily available then bootstrap simulation will provide a more accurate and reliable estimate than conventional methods finally we used the bootstrap for hypothesis testing and the estimation of standard errors and confidence intervals for parameters in an example data set we then compared and contrasted the bootstrap with standard methods of analysing hrqol outcomes in the data set studied with the sf 36 outcome the use of the bootstrap for estimating sample sizes and analysing hrqol data produces results similar to conventional statistical methods these results suggest that bootstrap methods are not more appropriate for analysing hrqol outcome data than standard methods

design sensitivity in observational studies summary outside the field of statistics the literature on observational studies offers advice about research designs or strategies for judging whether or not an association is causal such as multiple operationalism or a dose response relationship these useful suggestions are typically informal and qualitative a quantitative measure design sensitivity is proposed for measuring the contribution such strategies make in distinguishing causal effects from hidden biases several common strategies are then evaluated in terms of their contribution to design sensitivity a related method for computing the power of a sensitivity analysis is also developed

sample size determination for testing whether an identified treatment is best laska and meisner 1989 biometrics 45 1139 1151 dealt with the problem of testing whether an identified treatment belonging to a set of k 1 treatments is better than each of the other k treatments they calculated sample size tables for k 2 when using multiple t tests or wilcoxon mann whitney tests both under normality assumptions in this paper we provide sample size formulas as well as tables for sample size determination for k or 2 when t tests under normality or wilcoxon mann whitney tests under general distribution assumptions are used

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

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

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

power and sample size evaluation for the cochran mantel haenszel mean score wilcoxon rank sum test and the cochran armitage test for trend the power of a chi square test and thus the required sample size are a function of the noncentrality parameter that can be obtained as the limiting expectation of the test statistic under an alternative hypothesis specification herein we apply this principle to derive simple expressions for two tests that are commonly applied to discrete ordinal data the wilcoxon rank sum test for the equality of distributions in two groups is algebraically equivalent to the mann whitney test the kruskal wallis test applies to multiple groups these tests are equivalent to a cochran mantel haenszel mean score test using rank scores for a set of c discrete categories although various authors have assessed the power function of the wilcoxon and mann whitney tests herein it is shown that the power of these tests with discrete observations that is with tied ranks is readily provided by the power function of the corresponding cochran mantel haenszel mean scores test for two and r 2 groups these expressions yield results virtually identical to those derived previously for rank scores and also apply to other score functions the cochran armitage test for trend assesses whether there is an monotonically increasing or decreasing trend in the proportions with a positive outcome or response over the c ordered categories of an ordinal independent variable for example dose herein it is shown that the power of the test is a function of the slope of the response probabilities over the ordinal scores assigned to the groups that yields simple expressions for the power of the test copyright 2011 john wiley sons ltd

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

slow manifold and averaging for slow fast stochastic differential system abstract we consider stochastic dynamical systems with multiple time scales an intermediate reduced model is obtained and explored for a slow fast system when the fast mode is driven by white noise first and for later comparison one approximation to the reduced stochastic system on the exponentially attracting stochastic slow manifold is derived to errors of order o second because the noise only drives the fast modes averaging derives an autonomous deterministic system with errors of order o then a martingale argument accounts for fluctuations about the averaged system to form an intermediate reduced model with errors of order o the autonomous deterministic system now driven by white noise this intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold these results not only connect averaging with the stochastic slow manifold they also provide a martingale method for improving averaged models of stochastic systems

on the recovering of a coupled nonlinear schrodinger potential we establish a priori conditions for a gel fand levitan gl integral using some results of the fredholm theory as consequence we obtain a recovering formula for the potential of the coupled nonlinear schrodinger equations the remarkable fact is that the recovering formula is given in terms of the solutions of a classical gl integral equation

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

sample size determination for some common nonparametric tests abstract the article discusses the problem of determining the number of observations required by some common nonparametric tests so that the tests have power at least 1 against alternatives that differ sufficiently from the hypothesis being tested it is shown that the number of observations depends on certain simple probabilities a method is suggested for fixing the value of the appropriate probability when determining sample size

sample size calculation for the wilcoxon mann whitney test adjusting for ties in this paper we study sample size calculation methods for the asymptotic wilcoxon mann whitney test for data with or without ties the existing methods are applicable either to data with ties or to data without ties but not to both cases while the existing methods developed for data without ties perform well the methods developed for data with ties have limitations in that they are either applicable to proportional odds alternatives or have computational difficulties we propose a new method which has a closed form formula and therefore is very easy to calculate in addition the new method can be applied to both data with or without ties simulations have demonstrated that the new sample size formula performs very well as the corresponding actual powers are close to the nominal powers

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

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

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

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

bicomplex formulation and moyal deformation of 2 1 dimensional fordy kulish systems using bicomplex formalism we construct generalizations of fordy kulish systems of matrix nonlinear schrodinger equations on two dimensional space time in two respects firstly we obtain corresponding equations in three space time dimensions secondly a moyal deformation is applied to the space time coordinates and the ordinary product of functions replaced by the moyal product in a suitable way both generalizations preserve the existence of an infinite set of conservation laws

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

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

a note on sample size determination for a nonparametric test of location sample size determination methods are considered for hypothesis testing about location parameters of two continuous distributions using independent random samples and the mann whitney wilcoxon mww test for a specified shift size and power these methods are based on the so called noether s formula derived from a normal approximation to the power of the mww test and a pilot sample from each of the distributions compared with alternate methods including the one using bootstrap similar to that of hamilton and collings 1991 the new methods are shown to be at least as good in terms of accuracy and substantially more efficient in terms of speed and variability the simpler method using linearly smoothed empirical cdf s of the pilot samples and requiring no bootstrapping is recommended for practical use extensions and adaptations for obtaining an upper confidence bound on the sample size and to general linear rank tests are indicated

on sample size of the kruskal wallis test with application to a mouse peritoneal cavity study summary as the nonparametric generalization of the one way analysis of variance model the kruskal wallis test applies when the goal is to test the difference between multiple samples and the underlying population distributions are nonnormal or unknown although the kruskal wallis test has been widely used for data analysis power and sample size methods for this test have been investigated to a much lesser extent this article proposes new power and sample size calculation methods for the kruskal wallis test based on the pilot study in either a completely nonparametric model or a semiparametric location model no assumption is made on the shape of the underlying population distributions simulation results show that in terms of sample size calculation for the kruskal wallis test the proposed methods are more reliable and preferable to some more traditional methods a mouse peritoneal cavity study is used to demonstrate the application of the methods

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

optimal sample size planning for the wilcoxon mann whitney test there are many different proposed procedures for sample size planning for the wilcoxon mann whitney test at given type i and type ii error rates and respectively most methods assume very specific models or types of data to simplify calculations eg ordered categorical or metric data location shift alternatives etc we present a unified approach that covers metric data with and without ties count data ordered categorical data and even dichotomous data for that we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following synthetic data approach we evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities then well known relations for the ranks of the data are used for the calculations in addition to computing the necessary sample size n for a fixed allocation proportion t n1 n where n1 is the sample size in the first group and n n1 n2 is the total sample size we provide an interval for the optimal allocation rate t which minimizes the total sample size n it turns out that for certain distributions a balanced design is optimal we give a characterization of such distributions furthermore we show that the optimal choice of t depends on the ratio of the two variances which determine the variance of the wilcoxon mann whitney statistic under the alternative this is different from an optimal sample size allocation in case of the normal distribution model

on the low dimensional modelling of stratonovich stochastic differential equations abstract we develop further ideas on how to construct low dimensional models of stochastic dynamical systems the aim is to derive a consistent and accurate model from the originally high dimensional system this is done with the support of centre manifold theory and techniques aspects of several previous approaches are combined and extended adiabatic elimination has previously been used but centre manifold techniques simplify the noise by removing memory effects and with less algebraic effort than normal forms analysis of associated fokker plank equations replace nonlinearly generated noise processes by their long term equivalent white noise the ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations