a new description of force free magnetic fields the moses curl eigenfunctions are used to describe linear force free magnetic fields enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space this change of viewpoint suggests an orderly approach to the exploration and classification of the properties of such fields which is briefly sketched the simplest force free fields defined on zero one and two dimensional sets on the transform sphere are exhibited and the possibility of fields with fractal support sets suggested connections with other mathematical descriptions are pointed out as well as several promising directions for further exploration all results apply equally well to the description of the trkalian subset of beltrami fields in fluid dynamics

subgroup identification based on differential effect search a recursive partitioning method for establishing response to treatment in patient subpopulations we propose a novel recursive partitioning method for identifying subgroups of subjects with enhanced treatment effects based on a differential effect search algorithm the idea is to build a collection of subgroups by recursively partitioning a database into two subgroups at each parent group such that the treatment effect within one of the two subgroups is maximized compared with the other subgroup the process of data splitting continues until a predefined stopping condition has been satisfied the method is similar to interaction tree approaches that allow incorporation of a treatment by split interaction in the splitting criterion however unlike other tree based methods this method searches only within specific regions of the covariate space and generates multiple subgroups of potential interest we develop this method and provide guidance on key topics of interest that include generating multiple promising subgroups using different splitting criteria choosing optimal values of complexity parameters via cross validation and addressing type i error rate inflation inherent in data mining applications using a resampling based method we evaluate the operating characteristics of the procedure using a simulation study and illustrate the method with a clinical trial example copyright 2011 john wiley sons ltd

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

improvements of jackknife confidence limit methods summary edgeworth expansion is applied to studentized parameter estimates when the standard error has been computed by a jackknife method adjustments to the usual jackknife confidence limit formulae are obtained this approach is contrasted with a bootstrap approach in numerical illustrations for estimation of a ratio jackknife methods are nonparametric methods for estimating the bias and standard error of an estimate t approximate confidence limits for the estimand can be found by using a large sample normal approximation for t somewhat curiously little is known about possible improvements to the normal approximation in this context although improvements based on edgeworth expansions are familiar in other problems in this paper we show that edgeworth expansion methods can be applied in the jackknife context numerical results for a particular application raise the possibility that the resulting improvements of jackknife methods can be matched by a suitable use of bootstrap methods efron 1982

the hopf bifurcation with symmetry for the navier stokes equations in l p ω n with application to plane poiseuille flow the existence of periodic solutions of the navier stokes equations in function spaces based upon lp nis proved the paper has three parts a a proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of lp n b the evolution equation is restricted to a space of time periodic functions and a fredholm integral equation on this space is formed the lyapunov schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry c the theory is applied to the bifurcation of periodic solutions from planar poiseuille flow in the presence of symmetry so 2 x o 2 x s1 yielding new results for this classic problem the o 2 invariance is in the spanwise direction with the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction there are however points of degeneracy on the neutral surface a numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi periodic solutions

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

semilinear evolution equations in banach spaces with application to parabolic partial differential equations a theory for a class of semilinear evolution equations in banach spaces is developed which when applied to certain parabolic partial differential equations with nonlinear terms in divergence form gives strong solutions even for nondifferentiable data

peristaltic transport of a heat conducting viscous fluid as an application of abstract differential equations and semigroup of operators abstract a recent abstract result of rankin trans amer math soc to appear is applied to the peristaltic transport of a heat conducting fluid in a flexible tube the oberbeck boussinesq equations are used as the governing equations a pressure drop over a wavelength of the tube is prescribed to ensure uniqueness and newton s cooling law for the temperature is imposed on the boundary with the notion of a weak solution introduced local existence uniqueness regularity and continuation of solutions are considered in particular w r 2 regularity of the solutions is established

calibrating confidence coefficients abstract two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed the first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level this is achieved through a combination of computer simulation and density estimation the asymptotic validity of the procedure is proved for a number of common situations a classical example is the nonparametric estimation of the variance of a population using the normal theory interval here it is shown that the estimated coverage probability consistently estimates the true coverage probability if the population distribution possesses a finite sixth moment the second approach is more traditional it is a procedure for modifying an interval to yield improved coverage properties given a confidence interval its estimated coverage probability obtained in the first approach is used to alter the nominal level of the interval

nonstationary peristaltic transport of a heat conducting fluid abstract we study the peristaltic transport of a heat conducting fluid in a flexible lube with a prescribed pressure drop over a wavelength of the lube and newton s cooling law for the temperature and a prescribed velocity on the boundary using the oberbeck boussinesq equations as the governing equations and introducing a generalized solution we prove existence uniqueness and stability theorems for the nonstationary problem and develop and justify the stokes and long wave asymptotic expansions

complex trkalian fields and solutions to euler s equations for the ideal fluid we consider solutions to the complex trkalian equation vec nabla times vc vc where vc is a 3 component vector function with each component in the complex field and may be expressed in the form vc e ig vec nabla f with g real and f complex we find there are precisely two classes of solutions one where g is a cartesian variable and one where g is the spherical radial coordinate we consider these flows to be the simplest of all exact 3 d solutions to the euler s equation for the ideal incompressible fluid the novel approach we use in solving for these classes of solutions to these 3 dimensional vector pdes involves differential geometric techniques one may employ the method to generate solutions to other classes of vector pdes

identification of subgroups with differential treatment effects for longitudinal and multiresponse variables we describe and evaluate a regression tree algorithm for finding subgroups with differential treatments effects in randomized trials with multivariate outcomes the data may contain missing values in the outcomes and covariates and the treatment variable is not limited to two levels simulation results show that the regression tree models have unbiased variable selection and the estimates of subgroup treatment effects are approximately unbiased a bootstrap calibration technique is proposed for constructing confidence intervals for the treatment effects the method is illustrated with data from a longitudinal study comparing two diabetes drugs and a mammography screening trial comparing two treatments and a control copyright 2016 john wiley sons ltd

weak solutions for a class of parabolic volterra integrodifferential equations abstract we define weak solutions for a class of volterra integrodifferential equations of the form u t au t 0t a t s g s u s ds f t u t t 0 u 0 0 the operator a is the negative infinitesimal generator of an analytic semigroup in a banach space x the operator g t u is related to a by a special form g t u a 1 2 q t u where q t u is an appropriate lower order operator we show the existence and uniqueness of weak solutions and their continuability to infinity under suitable conditions using our results we study the asymptotic behavior as time goes to infinity of strong solutions of a second order initial boundary value problem

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

a note on the l_1 consistency of variable kernel estimates 1 introduction most consistent nonparametric density estimates have a built in smoothing parameter numerous schemes have been proposed see e g references found in rudemo 1982 or devroye and penrod 1984 for selecting the smoothing parameter as a function of the data only a process called automatization and for introducing locally adaptable smoothing parameters in this note we give conditions which insure that estimators of the form

the beltrami spectrum for incompressible fluid flows recently v yakhot s orszag and their co workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate beltrami flows a mathematical framework is developed here to study organized beltrami hierarchies in a systematic fashion this framework is applied to several important classes of examples with universal beltrami hierarchies an analysis of the persistence of such beltrami hierarchies is also presented for general solutions of the navier stokes equations

some notes on periodic beltrami fields in cartesian geometry some mathematical facts about beltrami fields which are time independent solutions of the three dimensional incompressible euler equations with nontrivial helicity are assembled the linearization about an arbitrary fixed beltrami field is studied in a hamiltonian framework a factorization of this linearization is introduced and used to characterize null spaces for beltrami fields with ergodic streamlines an interesting property of unstable modes is noticed and its consequences concerning the nature of potential instabilities are discussed

convection of microstructure and related problems we study how two flows that vary spatially in two widely separated scales evolve under the dynamics of euler s equations we use a multiple scale approach which we motivate by studying some simpler model problems

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

subgroups from regression trees with adjustment for prognostic effects and postselection inference identification of subgroups with differential treatment effects in randomized trials is attracting much attention many methods use regression tree algorithms this article addresses 2 important questions arising from the subgroups how to ensure that treatment effects in subgroups are not confounded with effects of prognostic variables and how to determine the statistical significance of treatment effects in the subgroups we address the first question by selectively including linear prognostic effects in the subgroups in a regression tree model the second question is more difficult because it falls within the subject of postselection inference we use a bootstrap technique to calibrate normal theory t intervals so that their expected coverage probability averaged over all the subgroups in a fitted model approximates the desired confidence level it can also provide simultaneous confidence intervals for all subgroups the first solution is implemented in the guide algorithm and is applicable to data with missing covariate values 2 or more treatment arms and outcomes subject to right censoring bootstrap calibration is applicable to any subgroup identification method it is not restricted to regression tree models two real examples are used for illustration a diabetes trial where the outcomes are completely observed but some covariate values are missing and a breast cancer trial where the outcome is right censored

a new method for testing separate families of hypotheses abstract theoretical and empirical evidence are reported showing that the tests given in cox 1961 and williams 1970a b for the problem of testing separate families of hypotheses suffer from fundamental difficulties neither test is guaranteed to be asymptotically level a even for very smooth problems a new method proposed here as an alternative works adaptively by repeated application of the parametric bootstrap idea over slowly shrinking confidence regions theoretical properties of the method are derived and its finite sample behavior is studied in a number of examples

subgroup analysis via recursive partitioning subgroup analysis is an integral part of comparative analysis where assessing the treatment effect on a response is of central interest its goal is to determine the heterogeneity of the treatment effect across subpopulations in this paper we adapt the idea of recursive partitioning and introduce an interaction tree it procedure to conduct subgroup analysis the it procedure automatically facilitates a number of objectively defined subgroups in some of which the treatment effect is found prominent while in others the treatment has a negligible or even negative effect the standard cart breiman et al 1984 methodology is inherited to construct the tree structure also in order to extract factors that contribute to the heterogeneity of the treatment effect variable importance measure is made available via random forests of the interaction trees both simulated experiments and analysis of census wage data are presented for illustration

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

a regression tree approach to identifying subgroups with differential treatment effects in the fight against hard to treat diseases such as cancer it is often difficult to discover new treatments that benefit all subjects for regulatory agency approval it is more practical to identify subgroups of subjects for whom the treatment has an enhanced effect regression trees are natural for this task because they partition the data space we briefly review existing regression tree algorithms then we introduce three new ones that are practically free of selection bias and are applicable to data from randomized trials with two or more treatments censored response variables and missing values in the predictor variables the algorithms extend the guide approach by using three key ideas i treatment as a linear predictor ii chi squared tests to detect residual patterns and lack of fit and iii proportional hazards modeling via poisson regression importance scores with thresholds for identifying influential variables are obtained as by products a bootstrap technique is used to construct confidence intervals for the treatment effects in each node the methods are compared using real and simulated data