nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

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

analysing repeated measurements with possibly missing observations by modelling marginal distributions suppose that subjects are observed repeatedly over a common set of time points with possibly time dependent covariates and possibly missing observations at each time point we model the marginal distribution of the response variable and the effect of the covariates on that distribution using a class of quasi likelihood models studied in mccullagh and nelder no parametric model of dependence of the repeated observations of the subject is assumed for large samples the quasi likelihood estimates of the time specific regression coefficients over the set of predetermined time points are shown to be approximately jointly normal this coupled with various inference procedures provides a global picture about the effects of the covariates on the response variable over the entire study period a lack of fit test for testing the adequacy of the assumed quasi likelihood model is also provided all the methods considered here are illustrated with real life examples

analytic structure of two 1d transport equations with nonlocal fluxes abstract we replace the flux term in burger s equation by two simple alternates that contain contributions depending globally on the solution in one case the term is in the form of a hyperbolic equation where the characteristic speed is nonlocal and in the other the term is in conservation form in both cases the nonanalytic is due to the presence of the hilbert transform the equations have a loose analogy to the motion of vortex sheets in particular they both form singularities in finite time in the absence of viscous effects our motivation then is to study the influence of viscosity in one case viscosity does not prevent singularity formation in the other we can prove solutions exist for all time and determine the likely weak solution as viscosity vanishes an interesting aspect of our work is that singularity formation can be viewed as the motion of singularities in the complex physical plane that reach the real axis in finite time in one case the singularity is a pole and causes the solution to blow up when it reaches the real axis in the other numerical solutions and an asymptotic analysis suggest that the weak solution contains a square root singularity that reaches the real axis in finite time and then propagates along it we hope our results will spur further interest in the role of singularities in the complex spatial plane in solutions to transport equations

singularity formation for prandtl s equations abstract we consider prandtl s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method we classify van dommelen and shen s singularity as a cubic root singularity we introduce a class of initial data uniformly bounded in h 1 which have a dipole singularity in the complex plane these data lead to a solution blow up whose time can be made arbitrarily short within the class this is numerical evidence of the ill posedness of the prandtl equations in h 1 the presence of a small viscosity in the streamwise direction changes the behavior of the singularities they stabilize at a distance from the real axis which depends on the amount of viscosity we show that the van dommelen and shen singularity and the singularity predicted by e and engquist in w e b engquist blowup of the solutions to the unsteady prandtl s equations comm pure appl math 50 1997 1287 1293 have different complex structures

generalized estimating equation model for binary outcomes with missing covariates summary this paper presents an approach to handling missing covariates in the generalized estimating equation gee model for binary outcomes when the probability of missingness depends on the observed outcomes and covariates the proposed method is to replace the missing quantities in the estimating function with consistent estimates in special cases the proposed model reduces to a weighted gee model for the completely observed units where the weight is the inverse of the probability of missingness our method can be viewed as an extension of the mean score method by reilly and pepe 1995 biometrika 82 299 314 to the gee context under certain regularity conditions the estimates of the regression coefficients obtained by the proposed method are consistent and asymptotically normally distributed the finite sample properties of the estimates are illustrated via computer simulations an application to the study of dementia among stroke patients is presented

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

a modified pseudolikelihood approach for analysis of longitudinal data we consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters in general inconsistent estimators of the covariance variance correlation parameters will be produced when the working correlation matrix is misspecified which may result in great loss of efficiency of the mean parameter estimators albeit the consistency is preserved we consider using different working correlation models for the variance and the mean parameters in particular we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified the designated working correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters simulation studies indicate that the proposed algorithm performs very well we also applied different estimation procedures to a data set from a clinical trial for illustration

on analytic properties of periodic solutions for equation mathcal h u_x u u p 0 the paper focuses on 2l periodic solutions ul x of the nonlocal equation p 2 is integer is the hilbert transform we give numerical evidence for the existence of a continuous family of these solutions parametrized by l for p 3 4 5 the features of the solutions ul x are discussed in particular we offer strong numerical arguments that ul x can be continued analytically from the real axis to some strip in the complex plane ul x ul z the singularities which arise under analytical continuation of these solutions into the complex plane are considered it is proved theorem 1 that ul z cannot be any power of a meromorphic function the formula which describes the asymptotic behavior of ul z in a neighborhood of the singularity is given it agrees with the numerical results computations also allow us to locate the closest to the real axis singularity of ul z and estimate the width of the analyticity strip

energy minimization using sobolev gradients application to phase separation and ordering a common problem in physics and engineering is the calculation of the minima of energy functionals the theory of sobolev gradients provides an efficient method for seeking the critical points of such a functional we apply the method to functionals describing coarse grained ginzburg landau models commonly used in pattern formation and ordering processes

3d euler equations and ideal mhd mapped to regular systems probing the finite time blowup hypothesis abstract we prove by an explicit construction that solutions to incompressible 3d euler equations defined in the periodic cube 0 l 3 can be mapped bijectively to a new system of equations whose solutions are globally regular we establish that the usual beale kato majda criterion for finite time singularity or blowup of a solution to the 3d euler system is equivalent to a condition on the corresponding regular solution of the new system in the hypothetical case of euler finite time singularity we provide an explicit formula for the blowup time in terms of the regular solution of the new system the new system is amenable to being integrated numerically using similar methods as in euler equations we propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite time singularity problem of euler equations based on the conservation of quantities directly related to energy and circulation the method of mapping to a regular system can be extended to any fluid equation that admits a beale kato majda type of theorem e g 3d navier stokes 2d and 3d magnetohydrodynamics and 1d inviscid burgers we discuss briefly the case of 2d ideal magnetohydrodynamics in order to illustrate the usefulness of the mapping we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1d inviscid burgers equation

second order expansion of mean squared error matrix of generalized least squares estimator with estimated parameters summary the second order expansion of the mean squared error matrix of the generalized least squares estimators for the regression parameters is obtained when the samples and the estimators for the structural parameters contained in the variance covariance matrix are independent in this paper we treat the second order expansion of the mean squared error matrix of the generalized least squares estimator fj in the statistical linear model y x u we consider the case where the variance covariance matrix v 00 of u is specified by finite parameters 00 and where a consistent estimator 0 of 00 is available which is independent of the observation y williams 1975 gave a sufficient condition concerning the order of the weak consistency of fj and commented on the asymptotic efficiency of fj but he did not obtain a result for the latter fuller rao 1978 dealt with the case where the observations fell into k groups with constant error variance for each group they obtained the asymptotic distribution of an estimator fj which used a weight matrix when the number of groups was infinite assuming the numbers of elements nij in the groups formed a fixed sequence our model corresponds to fixing k and letting ni increase

a numerical approach to blow up issues for dispersive perturbations of burgers equation abstract we provide a detailed numerical study of various issues pertaining to the dynamics of the burgers equation perturbed by a weak dispersive term blow up in finite time versus global existence nature of the blow up existence for long times and the decomposition of the initial data into solitary waves plus radiation we numerically construct solitary waves for fractional korteweg de vries equations

model problems for the multigrid optimization of systems governed by differential equations we discuss a multigrid approach to the optimization of systems governed by differential equations such optimization problems appear in many applications and are of a different nature than systems of equations our approach uses an optimization based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids our goal is not to argue for a particular optimization based multigrid algorithm but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms furthermore using several model problems we give evidence both theoretical and numerical that the optimization setting is well suited to multigrid algorithms some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation in addition we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization based multigrid algorithm

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

a case study of methods of series summation kelvin helmholtz instability of finite amplitude we compute the singularities of the solution of the birkhoff rott equation that governs the evolution of a planar periodic vortex sheet our approach uses the taylor series obtained by meiron et al j fluid mech 114 1982 283 for a flat sheet subject initially to a sinusoidal disturbance of amplitude a the series is then summed by using various generalisations of the pade method we find approximate values for the location and type of the principal singularity as a ranges from zero to infinity finally the results are used as a basis to guide the choice of methods of summing series arising from problems in fluid mechanics

a remark on the blowing up of solutions to the cauchy problem for nonlinear schrodinger equations on considere des solutions de iu t u u p 1 u u o u 0 ou x appartient a un domaine lisse cr n et on demontre que sous de bonnes conditions sur p n et u 0 h 2 h 0 1 u t l 2 eclate en un temps fini

on optimal reconstruction of constitutive relations abstract in this investigation we develop and validate a computational method for reconstructing constitutive relations based on measurement data applicable to problems arising in nonequilibrium thermodynamics and continuum mechanics this parameter estimation problem is solved as pde constrained optimization using a gradient based technique in the optimize then discretize framework the principal challenge is that the control variable i e the relation characterizing the constitutive property is not a function of the independent variables in the problem but of the state dependent variable the proposed method allows one to reconstruct a smooth constitutive relation defined over a broad range of the dependent variable it relies on three main ingredients a computationally friendly expression for the cost functional gradient sobolev gradients used in lieu of discontinuous l 2 gradients and a systematic technique for shifting the identifiability region the performance of this approach is illustrated by the reconstruction of the temperature dependence of the thermal conductivity in a one dimensional model problem

functional association models for multivariate survival processes we consider multivariate temporal processes that are continuously observed within overlapping time windows the intended application is in censored multistate and multivariate survival settings where point processes are continuously observed these data differ from other functional data like longitudinal data which are discretely observed at irregular times existing functional approaches to survival processes use intensity models which require smoothing and depend critically on the choice of smoothing parameters similarly to discretely observed data in this article we study functional mean and association regression models for the point processes with unspecified time varying coefficients the continuous observation scheme is exploited the coefficients may be estimated nonparametrically by extending generalized estimating equations to continuously observed data the estimators automatically converge at the parametric rate without smoothing unlike with discretely observed data uniform consistenc

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

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

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

generalized linear models with varying dispersion summary generalized linear models are further generalized to include a linear predictor for the dispersion as well as for the mean it is shown how the convenient structure of generalized linear models can be carried over to this more general setting by considering the mean and dispersion structure separately mean and dispersion submodels are formulated for this the dependent variable for the dispersion submodel being the deviance components of the mean submodel the fact that both submodels are essentially generalized linear models themselves is used to derive simple expressions for the likelihood equations and for asymptotic tests estimation algorithms are proposed which have good convergence properties the results apply mainly to the normal inverse gaussian and gamma distributions but can be extended to discrete distributions by using quasi likelihoods the methods developed are applied to a well known data set

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

singularity formation for complex solutions of the 3d incompressible euler equations abstract moore s approximation method first formulated for vortex sheets is generalized and applied to axi symmetric flow with swirl and with smooth initial data the approximation preserves the forward cascade of energy but neglects any backflow of energy it splits the euler equations into two sets of equations one for u u r z t containing all non negative wavenumbers in z and the second for u u the equations for u are exactly the euler equations but with complex initial data traveling waves solutions u u r z i t with imaginary wave speed are found numerically for this problem the asymptotic properties of the resulting fourier coefficients show a singularity forming in finite time at which the velocity blows up

skin friction and pressure the footprints of turbulence abstract the problems of exact state reconstruction and approximate state estimation based on wall information in a wall bounded incompressible unsteady flow are addressed it is shown that if in an arbitrarily small neighborhood of time t precise measurements are made of the two components of wall skin friction and the wall pressure all terms in the taylor series expansions of the unsteady flow state near the wall at time t may be determined in the linear setting this determination may be made based on skin friction measurements alone combining this fact with the analyticity of solutions of the nonlinear navier stokes equation and the unique continuation theorem for analytic functions in theory complete reconstruction of a fully developed turbulent flow in a channel at any reynolds number at time t is possible given only information about the unsteady flow available at the wall in a neighborhood of time t without knowledge of the initial conditions of the flow thus skin friction and pressure measurements on the wall in a neighborhood of time t provide a unique footprint of the entire unsteady turbulent flow state no other flow can have the same footprint indeed higher order terms are shown to uniformly improve the correlation of truncated taylor series expansions with the dns of a turbulent flow near the wall however such series extrapolations amplify measurement noise as they require differentiation in both space and time of the measurements and the radius of convergence of the taylor series expansions is less than 10 wall units the so called linear stochastic estimation technique in which the polynomials forming the basis of the series expansion are replaced by well behaved functions such as pod modes on the entire flow domain also demonstrates very poor convergence in light of these limitations on direct extrapolations from measurements in the practical setting an adjoint based algorithm is presented and numerically tested for estimating the state of an entire turbulent channel flow system based on a time history of noisy measurements at the wall this algorithm effectively uses the unsteady nonlinear navier stokes equation itself as a filter to find the flow solution that is most consistent with the available measurements

sharp bounds on enstrophy growth in the viscous burgers equation we use the cole hopf transformation and the laplace method for the heat equation to justify the numerical results on enstrophy growth in the viscous burgers equation on the unit circle we show that the maximum enstrophy achieved in the time evolution is scaled as e 3 2 where e is the large initial enstrophy whereas the time needed for reaching the maximal enstrophy is scaled as e 1 2 these bounds are sharp for initial conditions given by odd c 3 functions that are convex on half period

on singularity formation of a 3d model for incompressible navier stokes equations we investigate the singularity formation of a 3d model that was recently proposed by hou and lei 2009 in 15 for axisymmetric 3d incompressible navier stokes equations with swirl the main difference between the 3d model of hou and lei and the reformulated 3d navier stokes equations is that the convection term is neglected in the 3d model this model shares many properties of the 3d incompressible navier stokes equations one of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3d inviscid model for a class of initial boundary value problems with smooth initial data of finite energy we also prove the global regularity of the 3d inviscid model for a class of small smooth initial data

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

parametric variance function estimation for nonnormal repeated measurement data zeger and liang 1986 biometrics 42 121 130 proposed a procedure for analyzing nonnormal longitudinal data in the context of the generalized linear model this procedure is extended to model variance heterogeneity allowing the observations to come from distributions with different scale parameters loss of efficiency is evaluated when heterogeneity of scale factor is ignored

multiple imputation methods for the missing covariates in generalized estimating equation this paper discusses the missing covariates problem in the generalized estimating equation gee model estimates by various multiple imputation techniques mi are examined and compared to the sample average imputation method sa through simulations and an example the simulation results show that under the correct model specification the mi estimators have negligible bias and have fairly similar efficiencies as the sa estimator a practical advantage of the mi estimates is that the standard errors can be more easily computed than the sa estimates

using second order generalized estimating equations to model heterogeneous intraclass correlation in cluster randomized trials in cluster randomized trials it is commonly assumed that the magnitude of the correlation among subjects within a cluster is constant across clusters however the correlation may in fact be heterogeneous and depend on cluster characteristics accurate modeling of the correlation has the potential to improve inference we use second order generalized estimating equations to model heterogeneous correlation in cluster randomized trials using simulation studies we show that accurate modeling of heterogeneous correlation can improve inference when the correlation is high or varies by cluster size we apply the methods to a cluster randomized trial of an intervention to promote breast cancer screening

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

nonlinear dynamics and wall touch up in unstably stratified multilayer flows in horizontal channels under the action of electric fields this study considers the nonlinear dynamics of stratified immiscible fluids when an electric field acts perpendicular to the direction of gravity a particular setup is investigated in detail namely two stratified fluids inside a horizontal channel of infinite extent the fluids are taken to be perfect dielectrics and a constant horizontal field is imposed along the channel the sharp interface separating the two fluids may or may not support surface tension and the rayleigh taylor instability is typically present when the heavier fluid is on top a novel system of partial differential equations that describe the interfacial position and the leading order horizontal velocity in the fluid layers is studied analytically and computationally the system is valid in the asymptotic limit of one layer being asymptotically thin compared to the second fluid layer and as a result nonlocal electrostatic terms arise due to the multiscale nature of the physical setup the initial value problem on spatially perio

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

modeling liquid association in 2002 ker chau li introduced the liquid association measure to characterize three way interactions between genes and developed a computationally efficient estimator that can be used to screen gene expression microarray data for such interactions that study and others published since then have established the biological validity of the method and clearly demonstrated it to be a useful tool for the analysis of genomic data sets to build on this work we have sought a parametric family of multivariate distributions with the flexibility to model the full range of trivariate dependencies encompassed by liquid association such a model could situate liquid association within a formal inferential theory in this article we describe such a family of distributions a trivariate conditional normal model having gaussian univariate marginal distributions and in fact including the trivariate gaussian family as a special case perhaps the most interesting feature of the distribution is that the parameterization naturally parses the three way dependence structure into a number of distinct interpretable components one of these components is very closely aligned to liquid association and is developed as a measure we call modified liquid association we develop two methods for estimating this quantity and propose statistical tests for the existence of this type of dependence we evaluate these inferential methods in a set of simulations and illustrate their use in the analysis of publicly available experimental data

numerical study of fractional nonlinear schrodinger equations using a fourier spectral method we provide a detailed numerical investigation of dispersive schrodinger type equations involving a fractional laplacian in an one dimensional case by an appropriate choice of the dispersive exponent both mass and energy sub and supercritical regimes can be identified this allows us to study the possibility of finite time blow up versus global existence the nature of the blow up the stability and instability of nonlinear ground states and the long time dynamics of solutions the latter is also studied in a semiclassical setting moreover we numerically construct ground state solutions of the fractional nonlinear schrodinger equation

on maximum enstrophy growth in a hydrodynamic system abstract enstrophy e plays an important role in the study of regularity of solutions to the three dimensional navier stokes equation the best estimates for its growth available to date do not rule out the possibility of enstrophy becoming unbounded in finite time which would indicate loss of regularity of solutions it is therefore interesting to investigate sharpness of such finite time bounds for enstrophy growth we consider this question in the context of burgers equation which is used as a toy model the problem of saturation of finite time estimates for the enstrophy growth is stated as a pde constrained optimization problem max e t e 0 subject to e 0 e 0 where the control variable represents the initial condition which is solved numerically using an adjoint based gradient method for a wide range of time windows t and initial enstrophies e 0 we show that this optimization problem admits a discrete family of maximizers parameterized by the wavenumber m whose members are rescaled copies of the fundamental maximizer corresponding to m 1 it is found that the maximum enstrophy growth in finite time scales with the initial enstrophy as e 0 where 3 2 the exponent is smaller than 3 predicted by analytic means therefore suggesting possible lack of sharpness of analytical estimates

stability of computational methods for constrained dynamics systems many methods have been proposed for numerically integrating the differential algebraic systems arising from the euler lagrange equations for constrained motion these are based on various problem formulations and discretizations we offer a critical evaluation of these methods from the standpoint of stability nconsidering a linear model we first give conditions under which the differential algebraic problem is well conditioned this involves the concept of an essential underlying ode we review a variety of reformulations which have been proposed in the literature and show that most of them preserve the well conditioning of the original problem then we consider stiff and nonstiff discretizations of such reformulated models in some cases the same implicit discretization may behave in a very different way when applied to different problem formulations acting as a stiff integrator on some formulations and as a nonstiff integrator on others we present the approach of projected invariants as a method for yielding problem reformulations which are desirable in this sense

complex space singularities of 2d euler flow in lagrangian coordinates we show that for 2d space periodic incompressible flow the solution can be evaluated numerically in lagrangian coordinates with the same accuracy that is achieved in standard eulerian spectral methods this allows the determination of complex space lagrangian singularities lagrangian singularities are found to be closer to the real domain than eulerian singularities and seem to correspond to fluid particles which escape to complex infinity by the current time various mathematical conjectures regarding eulerian lagrangian singularities are presented

focusing and multi focusing solutions of the nonlinear schro dinger equation abstract a method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic schrodinger equations in two and three dimensions and describe their universal properties the same method is applied to simulate the multi focusing phenomena produced by simple models of saturating nonlinearities

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

calculation of complex singular solutions to the 3d incompressible euler equations abstract this paper presents numerical computations of complex singular solutions to the 3d incompressible euler equations the euler solutions found here consist of a complex valued velocity field u that contains all positive wavenumbers u satisfies the usual euler equations but with complex initial data the real valued velocity u u u where u u is an approximate singular solution to the euler equations under moore s approximation the method for computing singular solutions is an extension of that in caflisch 1993 25 for axisymmetric flow with swirl but with several improvements that prevent the extreme magnification of round off error which affected previous computations this enables the first clean analysis of the singular surface in three dimensional complex space we find singularities in the velocity field of the form u 1 for near 3 2 and u log where 0 denotes the singularity surface the logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations

estimates on enstrophy palinstrophy and invariant measures for 2 d turbulence we construct semi integral curves which bound the projection of the global attractor of the 2 d navier stokes equations in the plane spanned by enstrophy and palinstrophy of particular interest are certain regions of the plane where palinstrophy dominates enstrophy previous work shows that if solutions on the global attractor spend a significant amount of time in such a region then there is a cascade of enstrophy to smaller length scales one of the main features of 2 d turbulence theory the semi integral curves divide the plane into regions having limited ranges for the direction of the flow this allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy we also derive a sharp universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero

effects of variance function misspecification in analysis of longitudinal data the approach of generalized estimating equations gee is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within subject correlations the variance is often assumed to be a known function of the mean this article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses our numerical studies indicate that 1 correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified 2 misspecification of the variance function impacts much more on estimators for within cluster covariates than for cluster level covariates and 3 if the variance function is misspecified correct choice of the correlation structure may not necessarily improve estimation efficiency we illustrate impacts of different variance functions using a real data set from cow growth

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

computational synergetics and mathematical innovation it is demonstrated how the computer used in a heuristic mode has greatly augmented our understanding of the mathematics of nonlinear dynamical process examples are given of recent work in soliton mathematics mainly but also nonlinear iterative mappings and low order nonlinear ordinary differential equation systems the role of good graphics in enhancing the discovery and retention of new mathematical properties of equations is illustrated several examples are given of the present frontier of research in two space dimensional nonlinear evolutionary problems

estimating equations for association structures this paper investigates generalized estimating equations for association parameters which are frequently of interest in family studies with emphasis on covariance estimation separate link functions are used to connect the mean the scale and the correlation to linear predictors involving possibly different sets of covariates and separate estimating equations are proposed for the three sets of parameters simulations show that the robust sandwich variance estimator and the jackknife variance estimator for the correlation parameters are generally close to the empirical variance for the sample size of 50 clusters the results contradict ziegler et al and kastner and ziegler where the sandwich estimator obtained from the software mareg was shown to be unsuitable for practical usage the problem appears to arise because the mareg variance estimator does not account for variability in estimation of the scale parameters but may be valid with fixed scale we also find that the formula for the approximate jackknife variance estimator in ziegler et al is deficient resulting in systematic deviations from the fully iterated jackknife variance estimator a general jackknife formula is provided and performs well in numerical studies data from a study on the genetics of alcoholism is used to illustrate the importance of reliable variance estimation in biomedical applications

a computational framework for the regularization of adjoint analysis in multiscale pde systems this paper examines the regularization opportunities available in the adjoint analysis and optimization of multiscale pde systems regularization may be introduced into such optimization problems by modifying the form of the evolution equation and the forms of the norms and inner products used to frame the adjoint analysis typically l2 brackets are used in the definition of the cost functional the adjoint operator and the cost functional gradient if instead we adopt the more general sobolev brackets the various fields involved in the adjoint analysis may be made smoother and therefore easier to resolve numerically the present paper identifies several relationships which illustrate how the different regularization options fit together to form a general framework the regularization strategies proposed are exemplified using a id kuramoto sivashinsky forecasting problem and computational examples are provided which exhibit their utility a multiscale preconditioning algorithm is also proposed that noticeably accelerates convergence of the optimization procedure application of the proposed regularization strategies to more complex optimization problems of physical and engineering relevance is also discussed

descent methods for inverse problems of fundamental importance in modelling is the problem of recovering coe cient functions in a difierential equation from appropriate measurements on the solution thisisinvestigatedbymeansofsteepestdescenttechniquesusingcertainnew banach space gradients

convergence of an energy preserving scheme for the zakharov equations in one space dimension an energy preserving linearly implicit finite difference scheme is presented for approximating solutions to the periodic cauchy problem for the one dimensional zakharov system of two nonlinear partial differential equations first order convergence estimates are obtained in a standard energy norm in terms of the initial errors and the usual discretization errors

deciphering singularities by discrete methods we consider the problem of estimating numerically the parameters of singularities of solutions of differential equations we propose a novel approach which is based on discretizing the governing equation and timestepping in the complex domain some applications to ordinary and partial differential equations are discussed

stability of periodic solutions of index 2 differential algebraic systems this paper deals with periodic index 2 differential algebraic equations and the question whether a periodic solution is stable in the sense of lyapunov as the main result a stability criterion is proved this criterion is formulated in terms of the original data so that it may be used in practical computations

as flat as possible how does one determine a surface which is as flat as possible such as those created by soap film surfaces what does it mean to be as flat as possible in this paper we address this question from two distinct points of view one local and one global in nature continuing with this theme we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible this elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students build geometric and analytic understanding of solutions of certain elliptic and parabolic partial differential equations

how floquet theory applies to index 1 differential algebraic equations local stability of periodic solutions is established by means of a floquet theory for index 1 differential algebraic equations linear differential algebraic equations with periodic coefficients are considered in detail and a natural notion of the monodromy matrix is obtained that generalizes the well known theory for regular ordinary differential equations

linear mixed models with heterogeneous within cluster variances this paper describes an extension of linear mixed models to allow for heterogeneous within cluster variances in the analysis of clustered data unbiased estimating equations based on quasilikelihood pseudolikelihood and method of moments are introduced and are shown to give consistent estimators of the regression coefficients variance components and heterogeneity parameter under regularity conditions cluster specific random effects and variances are predicted by the posterior modes the method is illustrated through an analysis of menstrual diary data and its properties are evaluated in a simulation study

adjoint based optimization of pde systems with alternative gradients in this work we investigate a technique for accelerating convergence of adjoint based optimization of pde systems based on a nonlinear change of variables in the control space this change of variables is accomplished in the differentiate then discretize approach by constructing the descent directions in a control space not equipped with the hilbert structure we show how such descent directions can be computed in general lebesgue and besov spaces and argue that in the besov space case determination of descent directions can be interpreted as nonlinear wavelet filtering of the adjoint field the freedom involved in choosing parameters characterizing the spaces in which the steepest descent directions are constructed can be leveraged to accelerate convergence of iterations our computational examples involving state estimation problems for the 1d kuramoto sivashinsky and 3d navier stokes equations indeed show significantly improved performance of the proposed method as compared to the standard approaches

optical tomography reconstruction algorithm with the finite element method an optimal approach with regularization tools optical tomography is mathematically treated as a non linear inverse problem where the optical properties of the probed medium are recovered through the minimization of the errors between the experimental measurements and their predictions with a numerical model at the locations of the detectors according to the ill posed behavior of the inverse problem some regularization tools must be performed and the tikhonov penalization type is the most commonly used in optical tomography applications this paper introduces an optimized approach for optical tomography reconstruction with the finite element method an integral form of the cost function is used to take into account the surfaces of the detectors and make the reconstruction compatible with all finite element formulations continuous and discontinuous through a gradient based algorithm where the adjoint method is used to compute the gradient of the cost function an alternative inner product is employed for preconditioning the reconstruction algorithm moreover appropriate re parameterization of the optical properties is performed these regularization strategies are compared with the classical tikhonov penalization one it is shown that both the re parameterization and the use of the sobolev cost function gradient are efficient for solving such an ill posed inverse problem

efficient semiparametric mean association estimation for longitudinal binary responses semiparametric methods for longitudinal data with association within subjects have recently received considerable attention however existing methods for semiparametric longitudinal binary regression modeling i mainly concern mean structures with association parameters treated as nuisance ii generally require a correct specification of the covariance structure for misspecified covariance structure may lead to inefficient mean parameter estimates and iii usually run into computation and estimation problems when the time points are irregularly and possibly subject specific in this article we propose a semiparametric logistic regression model which simultaneously takes into account both the mean and response association structures via conditional log odds ratio for multivariate longitudinal binary outcomes our main interest lies in efficient estimation of both the marginal and association parameters the estimators of the parameters are obtained via the profile kernel approach we evaluate the proposed methodology through simulation studies and apply it to a real dataset both theoretical and empirical results demonstrate that the proposed method yields highly efficient estimators and performs satisfactorily copyright 2012 john wiley sons ltd

multimaterial topology optimization by volume constrained allen cahn system and regularized projected steepest descent method abstract a new computational algorithm is introduced in the present study to solve multimaterial topology optimization problems it is based on the penalization of the objective functional by the multiphase volume constrained ginzburg landau energy functional the update procedure is based on the gradient flow of the objective functional by a fractional step projected steepest descent method in the first step the new design is found based on the projected steepest descent method to ensure the reduction in the objective functional simultaneously satisfying the control constraints in the second step regularization step an h 1 regularity of the solution is ensured while keeping the feasibility of solution with respect to the set of control constraints the presented algorithm could be accounted as a constrained h 1 optimization algorithm which according to our knowledge has not been reported to solve such kind of problems yet the success and efficiency of the presented method are shown through several test problems numerical results show that the presented algorithm ends with a near 0 1 topology and its computational cost scales sub linearly by the number of phases for the sake of reader convenience and the ease of further extension the matlab implementation of the presented algorithm is included in the appendix

a simple illustration of a weak spectral cascade the textbook first encounter with nonlinearity in a partial differential equation pde is the first order wave equation u t u u x 0 often referred to as the inviscid burgers equation this equation is familiar to many in the theoretical contexts of characteristics wavebreaking or shock propagation another canonical behavior contained within this simplest of pdes is the spectral cascade surprisingly buried in a little known 1964 article by g w platzman is an elegant example of an exact fourier series solution associated with a purely sinusoidal initial condition this fourier representation valid prior to wavebreaking is generalized to arbitrary continuous initial conditions on both the periodic and infinite domains for the specific example of platzman s original problem the fourier coefficients decay exponentially with increasing wavenumber and the decay rate flattens to zero precisely at the time of wavebreaking it is demonstrated that two simplified descriptions a downscale truncat

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

numerical study of shock formation in the dispersionless kadomtsev petviashvili equation and dispersive regularizations abstract the formation of singularities in solutions to the dispersionless kadomtsev petviashvili dkp equation is studied numerically for different classes of initial data the asymptotic behavior of the fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity the approach is first tested in detail in 1 1 dimensions for the known case of the hopf equation where it is shown that the break up of the solution can be identified with prescribed accuracy for dissipative regularizations of this shock formation as the burgers equation and for dispersive regularizations as the korteweg de vries equation the fourier coefficients indicate as expected global regularity of the solutions the kadomtsev petviashvili kp equation can be seen as a dispersive regularization of the dkp equation the behavior of kp solutions for small dispersion parameter 1 near a break up of corresponding dkp solutions is studied it is found that the difference between kp and dkp solutions for the same initial data at the critical point scales roughly as 2 7 as for the korteweg de vries equation

on the stability behaviour of systems obtained by index reduction index reduction may destabilize the asymptotic stability of stationary solutions we analyze in some detail how index reduction via an artificial lagrangian multiplier as well as the baumgarte method behaves

on complex singularities of the 2d euler equation at short times abstract we present a study of complex singularities of a two parameter family of solutions for the two dimensional euler equation with periodic boundary conditions and initial conditions 0 z 1 z 2 f p cos p z f q cos q z in the short time asymptotic regime as has been shown numerically in pauls et al w pauls t matsumoto u frisch j bec nature of complex singularities for the 2d euler equation physica d 219 2006 40 59 the type of the singularities depends on the angle between the modes p and q thus the fourier coefficients of the solutions decrease as g k c k e k with the exponent depending on here we show for the two particular cases of going to zero and to that the type of the singularities can be determined very accurately being characterised by 5 2 and 3 respectively in these two cases we are also able to determine the subdominant corrections furthermore we find that the geometry of the singularities in these two cases is completely different the singular manifold being located over different points in the real domain

interplay between the beale kato majda theorem and the analyticity strip method to investigate numerically the incompressible euler singularity problem decrease to zero in order to sustain a finite time singularity consistent with the bkm theorem a simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the bkm theorem with the analyticity strip method it is shown that a finite time blowup can exist only if t vanishes sufficiently fast at the singularity time in particular if a power law is assumed for t then its exponent must be greater than some critical value thus providing a new test that is applied to our 4096 3 taylor green numerical simulation our main conclusion is that the numerical results are not inconsistent with a singularity but that higher resolution studies are needed to extend the time interval on which a well resolved power law behavior of t takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay

the effect of estimating weights in weighted least squares abstract in weighted least squares it is typical that the weights are unknown and must be estimated most packages provide standard errors assuming that the weights are known this is fine for sufficiently large sample sizes but what about for small to moderate sample sizes the investigation of this article into the effect of estimating weights proceeds under the assumption typical in practice that one has a parametric model for the variance function in this context generalized least squares consists of a an initial estimate of the regression parameter b a method for estimating the variance function and c the number of iterations in reweighted least squares by means of expansions for the covariance it is shown that each of a c can matter in problems of small to moderate size a few conclusions may be of practical interest first estimated standard errors assuming that the weights are known can be too small in practice the investigation indicates that a simple bootstrap operation resul

the cauchy lagrangian method for numerical analysis of euler flow a novel semi lagrangian method is introduced to solve numerically the euler equation for ideal incompressible flow in arbitrary space dimension it exploits the time analyticity of fluid particle trajectories and requires in principle only limited spatial smoothness of the initial data efficient generation of high order time taylor coefficients is made possible by a recurrence relation that follows from the cauchy invariants formulation of the euler equation zheligovsky and frisch 2014 44 truncated time taylor series of very high order allow the use of time steps vastly exceeding the courant friedrichs lewy limit without compromising the accuracy of the solution tests performed on the two dimensional euler equation indicate that the cauchy lagrangian method is more and occasionally much more efficient and less prone to instability than eulerian runge kutta methods and less prone to rapid growth of rounding errors than the high order eulerian time taylor algorithm we also develop tools of analysis adapted to the cauchy lagrangian method such as the monitoring of the radius of convergence of the time taylor series certain other fluid equations can be handled similarly