an approximate analytic solution method for the biharmonic problem a method is presented for the solution of the biharmonic equation through the use of a combination of polynomial and eigenfunction solutions results are presented comparing the method with an exact solution excellent agreement is obtained for economical solution representations calculations and comparison with established results of the displacement of an elastic plate under a uniform applied load are presented further results are presented of the calculation of zero reynolds number flow in a driven cavity

infinite systems for a biharmonic problem in a rectangle this paper addresses a general analytical method based upon mellin s transforms technique of investigating asymptotic behaviour of infinite systems of linear algebraic equations occurring in some basic two dimensional biharmonic problems in a rectangular domain the object of this paper is to prove the advantages of the asymptotic analysis when studying the concrete problems of an equilibrium of an elastic rectangle creeping flow of viscous fluid set up in a rectangular cavity by tangential velocities applied along its walls and bending of a clamped thin rectangular plate by a normal load at one surface the method is illustrated by several numerical examples the rapidity of convergence and the accuracy of results are studied

infinite systems for a biharmonic problem in a rectangle discussion of non uniqueness meleshko gomilko referred to below as mg developed a systematic technique based on mellin transforms for analysing the infinite systems that occur in various biharmonic problems for the rectangular domain by estimating the asymptotic behaviour of the solution sequences the convergence of the series is considerably improved even at the corner points however mg overlooked the existence of solutions of the homogeneous systems the consequent non uniqueness and the need to satisfy solvability conditions in order for the solutions to exist fortunately their main results remain valid and so the brief discussion here is aimed at elucidating the structures of the systems

a fast algorithm for simulating vesicle flows in three dimensions vesicles are locally inextensible fluid membranes that can sustain bending in this paper we extend the study of veerapaneni et al s k veerapaneni d gueyffier g biros d zorin a numerical method for simulating the dynamics of 3d axisymmetric vesicles suspended in viscous flows journal of computational physics 228 19 2009 7233 7249 to general non axisymmetric vesicle flows in three dimensions although the main components of the algorithm are similar in spirit to the axisymmetric case spectral approximation in space semi implicit time stepping scheme important new elements need to be introduced for a full 3d method in particular spatial quantities are discretized using spherical harmonics and quadrature rules for singular surface integrals need to be adapted to this case an algorithm for surface reparameterization is needed to ensure stability of the time stepping scheme and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs to characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi implicit time stepping scheme we perform a spectral analysis of the evolution operator on the unit sphere by introducing these algorithmic components we obtain a time stepping scheme that circumvents the stability constraint on the time step and achieves spectral accuracy in space we present results to analyze the cost and convergence rates of the overall scheme to illustrate the applicability of the new method we consider a few vesicle flow interaction problems a single vesicle in relaxation sedimentation shear flows and many vesicle flows

the evolution of piston driven pipe flow thermal transpiration the steady state transition from and to the uniform entry and exit flow profiles is well described at large aspect ratio in terms of the stream function by the pipe eigenfunctions but these latter are unsuited to oscillatory motion or the time evolution of the symmetric piston driven pipe flow for which an appropriate solution has a combination of a fourier series along the finite pipe and a fourier bessel series in the transverse direction a non uniqueness requires the identification of a solvability condition and care is needed in demonstrating its satisfaction an additional result is that the solution must be constructed to satisfy the normal flow conditions identically application is made to thermal transpiration recently explained by the revised navier stokes equations and boundary conditions

an approximate generalized linear model with random effects for informative missing data this paper develops a class of models to deal with missing data from longitudinal studies we assume that separate models for the primary response and missingness e g number of missed visits are linked by a common random parameter such models have been developed in the econometrics heckman 1979 econometrica 47 153 161 and biostatistics wu and carroll 1988 biometrics 44 175 188 literature for a gaussian primary response we allow the primary response conditional on the random parameter to follow a generalized linear model and approximate the generalized linear model by conditioning on the data that describes missingness the resultant approximation is a mixed generalized linear model with possibly heterogeneous random effects an example is given to illustrate the approximate approach and simulations are performed to critique the adequacy of the approximation for repeated binary data

sinc galerkin method for solving biharmonic problems there are many techniques available to numerically solve the biharmonic equation in this paper we show that the sinc galerkin method is a very effective tool in numerically solving this equation hermite interpolation is used to treat the nonhomogeneous boundary conditions our method is tested on examples and comparisons with other methods are made it is shown that the sinc galerkin method yields good results even when singularities occur at the boundaries

a new method for free vibration and buckling analysis of rectangular orthotropic plates a new method for free vibration and buckling analysis of rectangular orthotropic plates is presented the proposed method differs significantly from previously published methods in that it allows obtaining practically exact results for free vibration and buckling of orthotropic plates by enhancing the method of superposition the boundary value problem is essentially reduced to an infinite system of linear algebraic equations of particular significance of this research is that the existence of bounded solution of the infinite system after elimination of the unknowns has been captured in an elegant but quantitative way to converge upon any natural frequency or critical buckling load of the orthotropic plate to any desired accuracy the theory is applied to the particular case of orthotropic plates with clamped edges but it can be extended to other boundary conditions buckling analysis is carried out as a degenerate case of free vibration at zero frequency comparisons of computed results from the present theory with those available in the literature are made wherever possible and some noteworthy conclusions are drawn

fluid mechanics in rectangular cavities analytical model and numerics abstract we present an analytical model of stokes creeping flows in rectangular cavities in the model the fluid normal velocity at the wall is zero while the fluid tangential velocity at the wall differs from the wall velocity numerical work is used to validate some of the ideas generated from this realizable model

a phase field model for vesicle substrate adhesion in this paper a phase field model is developed for vesicle adhesion involving complex substrate and vesicle geometries the model takes into account an adhesion potential that depends on the distance of vesicle to the substrate a variational problem is solved in a 3d computational domain by minimizing the contribution of bending elastic energy and the adhesion energy under the constraints of total surface area and volume described via a phase function an adaptive finite element method is used to efficiently compute the numerical solutions of the model the computational results are validated through comparison of several axisymmetric shapes with the sharp interface ode solution moreover we compute shapes for non axisymmetric situations to support the observation that concave substrates favor adhesion

general solutions of the stokes flow equations abstract the papkovich neuber solution of the stokes flow equations for a viscous incompressible flow at very low reynolds numbers is derived using three alternative approaches in each case the class of solutions which is valid varies depending on the assumptions made on the domain considered the technique is illustrated by several examples the solution is identical to that of incompressible linear elasticity upon appropriate modification

on handling non homogeneous corner data in confined steady stokes flow although the method of eigenfunction expansions is a powerful tool to solve problems of steady stokes flow in closed containers of simple geometry there are difficulties in its use when the data at the corners of the containers are non homogeneous a method is suggested to overcome this limitation and explicit formulae are derived for the cases of two dimensional flow in a rectangular container and three dimensional flow in a cylindrical container of circular section as an application of the method we demonstrate the improved resolution of the field near the corner in the case of slow natural convection in a square container

small sample inference for incomplete longitudinal data with truncation and censoring in tumor xenograft models in cancer drug development demonstrating activity in xenograft models where mice are grafted with human cancer cells is an important step in bringing a promising compound to humans a key outcome variable is the tumor volume measured in a given period of time for groups of mice given different doses of a single or combination anticancer regimen however a mouse may die before the end of a study or may be sacrificed when its tumor volume quadruples and its tumor may be suppressed for some time and then grow back thus incomplete repeated measurements arise the incompleteness or missingness is also caused by drastic tumor shrinkage 0 01 cm3 or random truncation because of the small sample sizes in these models asymptotic inferences are usually not appropriate we propose two parametric test procedures based on the em algorithm and the bayesian method to compare treatment effects among different groups while accounting for informative censoring a real xenograft study on a new antitumor agent temozolomide combined with irinotecan is analyzed using the proposed methods

method for efficient shape parametrization of fluid membranes and vesicles in this paper we describe a method for the parametrization of the shapes adopted by fluid membranes and vesicles the method is based upon a boundary value approach to geometry description in which smooth surfaces are produced as the solution to an elliptic partial differential equations shape parameters are introduced through the boundary conditions which control the shape of the vesicle models in combination with a model for the surface energy and a method for numerical minimization it is shown how the method can accurately approximate the shapes of both axisymmetric and nonaxisymmetric vesicles over a wide range of control parameters the particular value of the method lies in its ability to parametrize complicated shapes efficiently a feature that becomes especially valuable when seeking shapes of minimal energy using direct optimization techniques

equivalence of two approaches to integrable hierarchies of kdv type the equivalence between the approaches of drinfeld sokolov and feigin frenkel to the mkdv and kdv hierarchies is established a new derivation of the mkdv equations in the zero curvature form is given connection with the baker akhiezer function and the tau function is also discussed

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

simulating the deformation of vesicle membranes under elastic bending energy in three dimensions in this paper we study the three dimensional deformation of a vesicle membrane under the elastic bending energy with prescribed bulk volume and surface area both static and dynamic deformations are considered a newly developed energetic variational formulation is employed to give an effective eulerian description efficient time and spatial discretizations are considered and implemented numerical experiments illustrate some fascinating phenomena that are of interests in real applications

a finite element method to compute three dimensional equilibrium configurations of fluid membranes we introduce a finite element method to compute equilibrium configurations of fluid membranes identified as stationary points of a curvature dependent bending energy functional under certain geometric constraints the reparameterization symmetries in the problem pose a challenge in designing parametric finite element methods and existing methods commonly resort to lagrange multipliers or penalty parameters in contrast we exploit these symmetries by representing solution surfaces as normal offsets of given reference surfaces and entirely bypass the need for artificial constraints we then resort to a galerkin finite element method to compute discrete c 1 approximations of the normal offset coordinate the variational framework presented is suitable for computing deformations of three dimensional membranes subject to a broad range of external interactions we provide a systematic algorithm for computing large deformations wherein solutions at subsequent load steps are identified as perturbations of previously computed ones we discuss the numerical implementation of the method in detail and demonstrate its optimal convergence properties using examples we discuss applications of the method to studying adhesive interactions of fluid membranes with rigid substrates and to investigate the influence of membrane tension in tether formation

numerical observation of nonaxisymmetric vesicles in fluid membranes by means of surface evolver a software package of brute force energy minimization over a triangulated surface developed by the geometry center of the university of minnesota exp math 1 141 1992 we have numerically searched the nonaxisymmetric shapes under the helfrich spontaneous curvature sc energy model we show that there are abundant mechanically stable nonaxisymmetric vesicles in the sc model including both regular shapes with intrinsic geometric symmetry and complex irregular ones we report in this paper a catalog of interesting shapes including a corniculate shape with six corns a quadriconcave shape a shape resembling sickle cells a shape resembling acanthocytes and two tubelike shapes most of these shapes can be related to experimental observations in red blood cells and other experiments in fluid membrane s1063 651x 98 06510 6

steady stokes flow in a rectangular cavity this paper addresses a general analytical method for investigating the two dimensional creeping flow field set up in a rectangular cavity by velocities applied along its top and bottom walls in any arbitrary manner the main goal is to prove the advantages of that approach when studying a local stream function behaviour near the corner points the values of amplitudes of the so called corner eddies were established based on the solution of the finite system of linear algebraic equations the method is illustrated by several numerical examples the rapidity of convergence and the accuracy of results are investigated the streamline patterns for some typical distribution of velocities are shown graphically

generalized norton simon models of tumour growth this paper considers the analysis of serial data on the growth of tumours in laboratory rodents i propose a model a generalization of the tumour growth model of norton and simon that leads to a rich family of growth and decay curves the model assumes that unperturbed growth follows the generalized logistic form it accommodates time varying treatment effects through an effective dose function i fit two such models to data on a human prostate tumour growing in nude mice and compare the fitted curves and dose functions with a non parametric curve and dose function estimated from a cubic spline model all three models account for both random animal effects and autocorrelation monte carlo results suggest that a maximum likelihood estimates of growth parameters are biased although not severely and b standard errors are conservative in small samples but become increasingly accurate in larger samples

legendre galerkin spectral element method for the biharmonic equations and its applications abstract a spectral element method based on the legendre galerkin approximation is presented to solve the two dimensional biharmonic equations rigorous error analysis is carried out to establish the convergence of the method by constructing appropriate basis functions which satisfy the boundary conditions of the differential equations the discrete variational formulation is reduced to linear system with sparse and symmetric matrices which can be efficiently solved by a fast schur complement method accuracy test is provided to confirm the convergence rate of the theoretical results finally the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero reynolds number flow in a driven cavity the results compare well with established results

steady stokes flow in a finite cylinder this article addresses an analytical approach for solving the problem of steady three dimensional creeping flow in a finite cylindrical container with arbitrary velocities prescribed over its surface to illustrate the advantages of the developed approach the flow in the container with a constant tangential velocity applied at one of the flat end walls is considered it is shown that the local behaviour of the velocity field near the cylinder rim with discontinuity in velocities is similar to that of the goodier taylor velocity field in a two dimensional wedge the kinematics of the corner eddy structures is illustrated by several numerical examples it is shown that the structure of the corner eddies is more complicated compared with the well known moffatt corner eddies the streamlines near the quiet rim of the cylinder having their spiral shapes

infinite systems for a biharmonic problem in a rectangle further discussion davis 2003 discussed non uniqueness and solvability conditions of two types of infinite systems that occur in various biharmonic boundary value problems for a rectangular domain studied by meleshko gomilko 1997 this paper further elucidates the property of the non uniqueness of the solution of infinite linear systems in a historical retrospective it is shown that in spite of the existence of non unique solutions of homogeneous infinite systems the physical quantities involved are always uniquely determined knowledge of asymptotic behaviour of unknowns is important when solving infinite systems by the method of reduction

sine gordon equation and representations of affine kac moody algebra sl2 abstract we give a representation theoretic interpretation of the sine gordon equation using the action of affine kac moody algebra sl 2 on the weyl algebra in this setup functions become functions in non commuting variables the skew casimir operators that we introduce here give rise to a hierarchy of partial differential equations that includes the sine gordon equation and two copies of the korteweg de vries equation

a numerical method for simulating the dynamics of 3d axisymmetric vesicles suspended in viscous flows we extend shravan k veerapaneni denis gueyffier denis zorin george biros a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d journal of computational physics 228 7 2009 2334 2353 to the case of three dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous flows although the main components of the algorithm are similar in spirit to the 2d case spectral approximation in space semi implicit time stepping scheme the main differences are that the bending and viscous force require new analysis the linearization for the semi implicit schemes must be rederived a fully implicit scheme must be used for the toroidal topology to eliminate a cfl type restriction and a novel numerical scheme for the evaluation of the 3d stokes single layer potential on an axisymmetric surface is necessary to speed up the calculations by introducing these novel components we obtain a time scheme that experimentally is unconditionally stable has low cost per time step and is third order accurate in time we present numerical results to analyze the cost and convergence rates of the scheme to verify the solver we compare it to a constrained variational approach to compute equilibrium shapes that does not involve interactions with a viscous fluid to illustrate the applicability of method we consider a few vesicle flow interaction problems the sedimentation of a vesicle interactions of one and three vesicles with a background poiseuille flow

a comparison of least squares and conditional maximum likelihood estimators under volume endpoint censoring in tumor growth experiments measurements in tumor growth experiments are stopped once the tumor volume exceeds a preset threshold a mechanism we term volume endpoint censoring we argue that this type of censoring is informative further least squares ls parameter estimates are shown to suffer a bias in a general parametric model for tumor growth with an independent and identically distributed measurement error both theoretically and in simulation experiments in a linear growth model the magnitude of bias in the ls growth rate estimate increases with the growth rate and the standard deviation of measurement error we propose a conditional maximum likelihood estimation procedure which is shown both theoretically and in simulation experiments to yield approximately unbiased parameter estimates in linear and quadratic growth models both ls and maximum likelihood estimators have similar variance characteristics in simulation studies these properties appear to extend to the case of moderately dependent measurement error the methodology is illustrated by application to a tumor growth study for an ovarian cancer cell line

analysis of multi arm tumor growth trials in xenograft animals using phase change adaptive piecewise quadratic models xenograft trials allow tumor growth in human cell lines to be monitored over time in a mouse model we consider the problem of inferring the effect of treatment combinations on tumor growth a piecewise quadratic model with flexible phase change locations is proposed to model the effect of change in therapy over time each piece represents a growth phase with phase changes in response to change in treatment piecewise slopes represent phase specific log linear growth rates and curvature parameters represent departure from linear growth trial data are analyzed in two stages i subject specific curve fitting ii analysis of slope and curvature estimates across subjects a least squares approach with penalty for phase change point location is proposed for curve fitting in simulation studies the method is shown to give consistent estimates of slope and curvature parameters under independent and ar 1 measurement error the piecewise quadratic model is shown to give excellent fit median r 2 0 98 to growth data from a six armed xenograft trial on a lung carcinoma cell line

tau functions and virasoro symmetries for drinfeld sokolov hierarchies abstract for each drinfeld sokolov integrable hierarchy associated to affine kac moody algebra we obtain a uniform construction of tau function by using tau symmetric hamiltonian densities moreover we represent its virasoro symmetries as linear nonlinear actions on the tau function the relations between the tau function constructed in this paper and those defined for particular cases of drinfeld sokolov hierarchies in the literature are clarified we also show that whenever the affine kac moody algebra is simply laced or twisted the tau function of the drinfeld sokolov hierarchy coincides with the solution of the corresponding kac wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra

drinfeld sokolov hierarchies and diagram automorphisms of affine kac moody algebras for a diagram automorphism of an affine kac moody algebra such that the folded diagram is still an affine dynkin diagram we show that the associated drinfeld sokolov hierarchy also admits an induced automorphism then we show how to obtain the drinfeld sokolov hierarchy associated to the affine kac moody algebra that corresponds to the folded dynkin diagram from the invariant sub hierarchy of the original drinfeld sokolov hierarchy

bihamiltonian cohomologies and integrable hierarchies ii the tau structures starting from a so called flat exact semisimple bihamiltonian structure of hydrodynamic type we arrive at a frobenius manifold structure and a tau structure for the associated principal hierarchy we then classify the deformations of the principal hierarchy which possess tau structures