homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

weak type 1 1 inequalities and regularity properties of adjoint and normalized adjoint solutions to linear nondivergence form operators with vmo coefficients

boundary regularity on the parabolic monge ampere equation abstract this paper establishes boundary estimates for solutions to the parabolic monge ampere equation u t det d 2 u 1 n f x t

multiplicity considerations in subgroup analysis this paper deals with the general topic of subgroup analysis in late stage clinical trials with emphasis on multiplicity considerations the discussion begins with multiplicity issues arising in the context of exploratory subgroup analysis including principled approaches to subgroup search that are applied as part of subgroup exploration exercises as well as in adaptive biomarker driven designs key considerations in confirmatory subgroup analysis based on one or more pre specified patient populations are reviewed including a survey of multiplicity adjustment methods recommended in multi population phase iii clinical trials guidelines for interpretation of significant findings in several patient populations are introduced to facilitate the decision making process and achieve consistent labeling across development programs

opportunities and challenges of combined effect measures based on prioritized outcomes many authors have proposed different approaches to combine multiple endpoints in a univariate outcome measure in the literature in case of binary or time to event variables composite endpoints which combine several event types within a single event or time to first event analysis are often used to assess the overall treatment effect a main drawback of this approach is that the interpretation of the composite effect can be difficult as a negative effect in one component can be masked by a positive effect in another recently some authors proposed more general approaches based on a priority ranking of outcomes which moreover allow to combine outcome variables of different scale levels these new combined effect measures assign a higher impact to more important endpoints which is meant to simplify the interpretation of results whereas statistical tests and models for binary and time to event variables are well understood the latter methods have not been investigated in detail so far in this paper we will investigate the statistical properties of prioritized combined outcome measures we will perform a systematical comparison to standard composite measures such as the all cause hazard ratio in case of time to event variables or the absolute rate difference in case of binary variables to derive recommendations for different clinical trial scenarios we will discuss extensions and modifications of the new effect measures which simplify the clinical interpretation moreover we propose a new method on how to combine the classical composite approach with a priority ranking of outcomes using a multiple testing strategy based on the closed test procedure

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

the regularity theory for the parabolic double obstacle problem in this paper we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator the result in this paper are generalizations of the theory for the elliptic problem in lee et al calc var partial differ equ 58 3 104 2019 and lee and park the regularity theory for the double obstacle problem for fully nonlinear operator 2018 to parabolic case and also the theory for the parabolic single obstacle problem in caffarelli et al j am math soc 17 4 827 869 2004 to double obstacle case new difficulties in the theory which are generated by the characteristic of parabolic pdes and the existence of the upper obstacle are discussed in detail furthermore the thickness assumptions to have the regularity of the free boundary are carefully considered

aleksandrov bakelman pucci maximum principles for a class of uniformly elliptic and parabolic integro pde abstract we prove generalized aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes of hamilton jacobi bellman isaacs types whose pde parts are either uniformly elliptic or uniformly parabolic the proofs of these results are based on the classical aleksandrov bakelman pucci maximum principles for the elliptic and parabolic pdes and an iteration procedure using solutions of pucci extremal equations we also provide proofs of nonlocal versions of the classical aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

constructing multiple test procedures for partially ordered hypothesis sets a popular method to control multiplicity in confirmatory clinical trials is to use a so called hierarchical or fixed sequence test procedure this requires that the null hypotheses are ordered a priori for example in order of clinical importance the procedure tests the hypotheses in this order using level tests until one is not rejected it then stops so that no subsequent hypotheses are eligible for rejection this procedure strongly controls the familywise error rate fwe that is to say the probability that any true hypotheses are rejected this paper describes a simple generalization of this approach in which the null hypotheses are partially ordered it is convenient to display the partial ordering in a directed acyclic graph dag we consider sequentially rejective procedures based on the partial ordering in which a hypothesis is tested only when all preceding hypotheses have been tested and rejected in general such procedures do not control the fwe but it is shown that when certain intersection hypotheses are added strong control of the fwe is obtained the purpose of the method is to construct inference strategies for confirmatory clinical trials that better reflect the trial objectives copyright 2007 john wiley sons ltd

discontinuous galerkin spectral element approximations on moving meshes we derive and evaluate high order space arbitrary lagrangian eulerian ale methods to compute conservation laws on moving meshes to the same time order as on a static mesh we use a discontinuous galerkin spectral element method dgsem in space and one of a family of explicit time integrators such as adams bashforth or low storage explicit runge kutta the approximations preserve the discrete metric identities and the discrete geometric conservation law dgcl by construction we present time step refinement studies with moving meshes to validate the approximations the test problems include propagation of an electromagnetic gaussian plane wave a cylindrical pressure wave propagating in a subsonic flow and a vortex convecting in a uniform inviscid subsonic flow each problem is computed on a time deforming mesh with three methods used to calculate the mesh velocities from exact differentiation from the integration of an acceleration equation and from numerical differentiation of the mesh position

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

weak harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications abstract the weak harnack inequality for l p viscosity supersolutions of fully nonlinear second order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved it is shown that holder continuity of l p viscosity solutions is derived from the weak harnack inequality for l p viscosity supersolutions the local maximum principle for l p viscosity subsolutions and the harnack inequality for l p viscosity solutions are also obtained several further remarks are presented when equations have superlinear growth in the first space derivatives

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

a gatekeeping procedure to test a primary and a secondary endpoint in a group sequential design with multiple interim looks glimm et al 2010 and tamhane et al 2010 studied the problem of testing a primary and a secondary endpoint subject to a gatekeeping constraint using a group sequential design gsd with k 2 looks in this article we greatly extend the previous results to multiple k 2 looks if the familywise error rate fwer is to be controlled at a preassigned level then it is clear that the primary boundary must be of level we show under what conditions one level primary boundary is uniformly more powerful than another based on this result we recommend the choice of the o brien and fleming 1979 boundary over the pocock 1977 boundary for the primary endpoint for the secondary endpoint the choice of the boundary is more complicated since under certain conditions the secondary boundary can be refined to have a nominal level while still controlling the fwer at level thus boosting the secondary power we carry out secondary power comparisons via simulation between different choices of primary secondary boundary combinations the methodology is applied to the data from the rales study pitt et al 1999 wittes et al 2001 an r library package gsrsb to implement the proposed methodology is made available on cran

mixtures of multiple testing procedures for gatekeeping applications in clinical trials this paper proposes a general framework for constructing gatekeeping procedures for clinical trials with hierarchical objectives such problems frequently exhibit complex structures including multiple families of hypotheses and logical restrictions the proposed framework is based on combining multiple procedures across families it enables the construction of powerful and flexible gatekeeping procedures that account for general logical restrictions among the hypotheses of interest a clinical trial in patients with schizophrenia is used to illustrate the approach for parallel gatekeeping whereas another clinical trial in patients with hypertension is used to illustrate the approach for gatekeeping with general logical restrictions

elliptic equations on convex domains with nonhomogeneous dirichlet boundary conditions abstract in this paper we will study the differentiability on the boundary of solutions of elliptic non divergence differential equations on convex domains the results are divided into two cases i at the boundary points where the blow up of the domain is not the half space if the boundary function is differentiable then the solution is differentiable ii at the boundary points where the blow up of the domain is the half space the differentiability of the solution needs an extra dini condition for the boundary function counterexample is given to show that our results are optimal

approximation integration and differentiation of time functions using a set of orthogonal hybrid functions hf and their application to solution of first order differential equations abstract differential equations of different types and orders are of utmost importance for mathematical modeling of control system problems state variable method uses the concept of expressing n number of first order differential equations in vector matrix form to model and analyze synthesize control systems the present work proposes a new set of orthogonal hybrid functions hf which evolved from synthesis of sample and hold functions shf and triangular functions tf this hf set is used to approximate a time function in a piecewise linear manner with the mean integral square error mise much less than block pulse function based approximation which always provides staircase solutions the operational matrices for integration and differentiation in hf domain are also derived and employed for solving non homogeneous and homogeneous differential equations of the first order as well as state equations the results are compared with exact solutions the 4th order runge kutta method and its further improved versions proposed by simos 6 the presented hf domain theory is well supported by a few illustrations

powerful short cuts for multiple testing procedures with special reference to gatekeeping strategies in this paper we present a general testing principle for a class of multiple testing problems based on weighted hypotheses under moderate conditions this principle leads to powerful consonant multiple testing procedures furthermore short cut versions can be derived which simplify substantially the implementation and interpretation of the related test procedures it is shown that many well known multiple test procedures turn out to be special cases of this general principle important examples include gatekeeping procedures which are often applied in clinical trials when primary and secondary objectives are investigated and multiple test procedures based on hypotheses which are completely ordered by importance we illustrate the methodology with two real clinical studies

variable stepsize chebyshev type methods for the integration of second order i v p s panovsky and richardson a family of implicit chebyshev methods for the numerical integration of second order differential equations j comput appl math 23 1988 35 51 presented a method based on chebyshev approximations for numerically solving the problem y f x y being the steplength constant coleman and booth analysis of a family of chebyshev methods for y f x y j comput appl math 44 1992 95 114 made an analysis of the above method and suggested the convenience to design a variable steplength implementation as far as we know this goal has not been achieved until now later on we extended the above method this journal 2003 and obtained a scheme for numerically solving the equation y 2gy g 2 w 2 f x y the question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper

parabolic equations in time dependent reifenberg domains in this paper we define time dependent parabolic reifenberg domains and study lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains the basic assumption is that the principal coefficients are of parabolic bmo space with small parabolic bmo seminorms it is shown that lp estimates hold for time dependent parabolic d reifenberg domains

fast closed testing for exchangeable local tests summarymultiple hypothesis testing problems arise naturally in science this note introduces a new fast closed testing method for multiple testing which controls the familywise error rate controlling the familywise error rate is state of the art in many important application areas and is preferred over false discovery rate control for many reasons including that it leads to stronger reproducibility the closure principle rejects an individual hypothesis if all global nulls of subsets containing it are rejected using some test statistics it takes exponential time in the worst case when the tests are symmetric and monotone the proposed method is an exact algorithm for computing the closure is quadratic in the number of tests and is linear in the number of discoveries our framework generalizes most examples of closed testing such as holm s method and the bonferroni method as a special case of the method we propose the simes and higher criticism fusion test which is powerful both for detecting a few strong signals and for detecting many moderate signals

a note on testing families of hypotheses using graphical procedures regulatory guidelines for drug development suggest a strong control of the familywise error rate when multiple hypotheses are simultaneously tested in confirmatory clinical trials accordingly a variety of multiple test procedures exist for pharmaceutical trial applications which are typically defined on a single structured family of null hypotheses for some confirmatory clinical trials with multiple objectives however it is advantageous to address the arising multiplicity problems via grouped families of hypotheses graphical test procedures have been developed to construct visualize and perform multiple test procedures that are tailored to either a single or multiple families of structured hypotheses of interest this note complements the existing literature by introducing a general algorithm to calculate adjusted p values for any sequentially rejective graphical test procedure where the test procedures within the individual families of hypotheses are not necessarily graphical

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

on the aleksandrov bakel man pucci estimate for some elliptic and parabolic nonlinear operators in this work we prove the aleksandrov bakel man pucci estimate for possibly degenerate nonlinear elliptic and parabolic equations of the form rm div left f left nabla u x right right f left x right quad rm in omega subset mathbb r n and u t x t rm div left f left nabla u x t right right f left x t right quad rm in q subset mathbb r n 1 for f a fancyscript c 1 monotone field under some suitable conditions examples of applications such as the p laplacian and the mean curvature flow are considered as well as extensions of the general results to equations that are not in divergence form such as the m curvature flow

a new theoretical approach to runge kutta methods runge kutta rk methods are treated here as linear multistage methods within the author s concept of a methods as a consequence the traditional difference in treatment of rk methods and linear multistep methods vanishes in particular it is not necessary to consider elementary differentials it is shown how the order of a rk method depends on the error constants of its stages and the conditions nonlinear equations for order p are generated from a very simple recursion

a study of rosenbrock type methods of high order this paper deals with the solution of nonlinear stiff ordinary differential equations the methods derived here are of rosenbrock type this has the advantage that they area stable or stiffly stable and nevertheless do not require the solution of nonlinear systems of equations we derive methods of orders 5 and 6 which require one evaluation of the jacobian and onelu decomposition per step we have written programs for these methods which use richardson extrapolation for the step size control and give numerical results

on a class of fully nonlinear flows in kahler geometry in this paper we study a class of fully nonlinear metric flow on kahler manifolds which includes the j flow as a special case we provide a sufficient and necessary condition for the long time convergence of the flow generalizing the result of songweinkove as a consequence under the given condition we solved the corresponding euler equation which is fully nonlinear of monge ampere type as an application we also discuss a complex monge ampere type equation including terms of mixed degrees which was first posed by chen

higher order convergence rates in theory of homogenization ii oscillatory initial data abstract we establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic cauchy problems accompanied with rapidly oscillating initial data such result is new even for linear problems here we construct higher order initial layer and interior correctors which describe the oscillatory behavior near the initial and interior time zone of the domain to construct higher order correctors we develop a regularity theory in macroscopic scales and prove an exponential decay estimate for initial layer correctors the higher order expansion requires an iteration process successively correcting the initial layer then the interior this leads to a more complicated asymptotic expansion as compared to the non oscillating data case and this complexity is present even in the linear case a notable observation for fully nonlinear operators is that even if the given operator is space time periodic the interior correctors become aperiodic in the time variable as we proceed with the iteration process moreover each interior corrector of higher order is paired with a space time periodic version and the difference between the two decays exponentially fast with time

testing each hypothesis marginally at alpha while still controlling fwer how and when this paper proposes a multiple testing procedure that allows one to reject each individual hypothesis at a prespecified level while still controlling the familywise error rate at in the strong sense typically rejecting a hypothesis when its marginal p value is in a multiple hypothesis testing setting will lead to an inflation of familywise error rate however this inflation can be avoided if a particular consistency criterion is prespecified and incorporated in the testing algorithm the criterion is equivalent to requiring that all p values be smaller than or equal to a particular threshold in the one sided hypothesis testing setting extensions to the two sided hypothesis testing setting and extensions to situations where the criterion can be chosen per user s preference are also presented copyright 2012 john wiley sons ltd

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

key multiplicity issues in clinical drug development much progress has been made over the past decade with the development of novel methods for addressing increasingly more complex multiplicity problems arising in confirmatory phase iii clinical trials this includes traditional problems with a single source of multiplicity for example analysis of multiple endpoints or dose placebo contrasts in addition more advanced problems with several sources of multiplicity have attracted attention in clinical drug development these problems include two or more families of objectives such as multiple endpoints evaluated at multiple dose levels or in multiple patient populations this paper provides a review of concepts that play a central role in defining and solving multiplicity problems error rate definitions and introduces main classes of multiple testing procedures widely used in clinical trials nonparametric semiparametric and parametric procedures the paper also presents recent advances in multiplicity research including gatekeeping procedures for clinical trials with multiple sets of objectives the concepts and methods introduced in the paper are illustrated using several case studies on the basis of real clinical trials software implementation of commonly used multiple testing and gatekeeping procedures is discussed copyright 2012 john wiley sons ltd

testing for efficacy in primary and secondary endpoints by partitioning decision paths testing for efficacy in multiple endpoints has emerged as an important statistical problem the food and drug administration fda will issue a guidance on multiple endpoints in the near future when there are primary and secondary endpoints efficacy in the secondary endpoint is relevant only if efficacy in the primary endpoint has been shown thus there are defined paths to decision making current approaches to this problem are based on closed testing testing all possible intersection hypotheses and collating the results for decision making to follow predefined paths strategic choices of test statistics and critical values must be made as the number of doses and endpoints increase such strategic choices become increasingly difficult partition testing is an alternative to closed testing it provides insight into confidence sets for stepwise tests and can be more powerful than closed testing it also can simplify problem formulation when decision making follows specific paths for the primary se

beyond conventional runge kutta methods in numerical integration of odes and daes by use of structures and local models there are two parts in this paper in the first part we consider an overdetermined system of differential algebraic equations daes we are particularly concerned with hamiltonian and lagrangian systems with holonomic constraints the main motivation is in finding methods based on gauss coefficients preserving not only the constraints symmetry symplecticness and variational nature of trajectories of holonomically constrained hamiltonian and lagrangian systems but also having optimal order of convergence the new class of s s gauss lobatto specialized partitioned additive runge kutta spark methods uses greatly the structure of the daes and possesses all desired properties in the second part we propose a unified approach for the solution of ordinary differential equations odes mixing analytical solutions and numerical approximations the basic idea is to consider local models which can be solved efficiently for example analytically and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction in order to preserve also symmetry we define the new class of symmetrized runge kutta methods with local model srklm

free boundary regularity close to initial state for parabolic obstacle problem here partial derivative p denotes the parabolic boundary h is a parabolic operator with certain properties q is the upper half of the unit cylinder in rn 1 and the equation is satisfied in t

explicit solutions of the g heat equation for a class of initial conditions abstract we obtain the viscosity solution of the g heat equation with the initial condition x x n for each integer n 1 using the method of g brownian motion

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

a multiple comparison procedure for hypotheses with gatekeeping structure we develop gatekeeping procedures that focus on comparing multiple treatments with a control when there are multiple endpoints our procedures utilize estimated correlations among individual test statistics without parametric assumptions we make comparisons with other gatekeeping procedures with respect to properties of the trade off in statistical power between families of hypotheses we introduce a reward function to facilitate these comparisons we illustrate our methods by simulation and an analysis of data from a randomized multi armed clinical trial copyright 2013 oxford university press

a general multistage procedure for k out of n gatekeeping we generalize a multistage procedure for parallel gatekeeping to what we refer to as k out of n gatekeeping in which at least k out of n hypotheses 1 k n in a gatekeeper family must be rejected in order to test the hypotheses in the following family this gatekeeping restriction arises in certain types of clinical trials for example in rheumatoid arthritis trials it is required that efficacy be shown on at least three of the four primary endpoints we provide a unified theory of multistage procedures for arbitrary k with k 1 corresponding to parallel gatekeeping and k n to serial gatekeeping the theory provides an insight into the construction of truncated separable multistage procedures using the closure method explicit formulae for calculating the adjusted p values are given the proposed procedure is simpler to apply for this particular problem using a stepwise algorithm than the mixture procedure and the graphical procedure with memory using entangled graphs

worn stones with flat sides all time regularity of the interface we study the all time regularity of the free boundary problem associated to the deformation of a compact weakly convex surface in 3 with a flat side by its gaussian curvature we show that under certain necessary regularity and non degeneracy initial conditions the interface separating the flat from the strictly convex side remains smooth on 0 t tc up to the vanishing time tc of the flat side

numerical methods for ordinary differential equations in the 20th century abstract numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons the first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of bashforth and adams for linear multistep methods and runge for runge kutta methods other reasons which of course apply to numerical analysis in general are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics this survey paper follows many of the main strands in the developments of these methods both for general problems stiff systems and for many of the special problem types that have been gaining in significance as the century draws to an end

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

on c 1 c 2 c2 and weak type 1 1 estimates for linear elliptic operators part ii we extend and improve the results in dong and kim commun partial differ equ 42 3 417 435 2017 showing that weak solutions to full elliptic equations in divergence form with zero dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have dini mean oscillation and the lower order coefficients verify certain conditions similar results are obtained for non divergence form equations we extend the weak type 1 1 estimates in dong and kim commun partial differ equ 42 3 417 435 2017 and escauriaza duke math j 74 1 177 201 1994 up to the boundary and derive a harnack inequality for non negative adjoint solutions to non divergence form elliptic equations when the leading coefficients have dini mean oscillation

solvability of uniformly elliptic fully nonlinear pde we get existence uniqueness and non uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the hamilton jacobi bellman isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness some of our results are new even for equations in divergence form

holder estimates for non local parabolic equations with critical drift abstract in this paper we extend previous results on the regularity of solutions of integro differential parabolic equations the kernels are non necessarily symmetric which could be interpreted as a non local drift with the same order as the diffusion we provide a growth lemma and a harnack inequality which can be used to prove higher regularity estimates

traditional multiplicity adjustment methods in clinical trials this tutorial discusses important statistical problems arising in clinical trials with multiple clinical objectives based on different clinical variables evaluation of several doses or regiments of a new treatment analysis of multiple patient subgroups etc simultaneous assessment of several objectives in a single trial gives rise to multiplicity if unaddressed problems of multiplicity can undermine integrity of statistical inferences the tutorial reviews key concepts in multiple hypothesis testing and introduces main classes of methods for addressing multiplicity in a clinical trial setting general guidelines for the development of relevant and efficient multiple testing procedures are presented on the basis of application specific clinical and statistical information case studies with common multiplicity problems are used to motivate and illustrate the statistical methods presented in the tutorial and software implementation of the multiplicity adjustment methods is discussed copyright 2013 john wiley sons ltd

hypersurfaces with mean curvature given by an ambient sobolev function we consider n hypersurfaces j with interior ej whose mean curvature are given by the trace of an ambient sobolev function uj w 1 p r h j uj ej on j 0 1 where ej denotes the inner normal of j we investigate 0 1 when j weakly as varifolds and prove that is an integral n varifold with bounded first variation which still satisfies 0 1 for uj u ej e p has to satisfy

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

twice differentiability of solutions to fully nonlinear parabolic equations near the boundary abstract in this paper we prove h 2 regularity for viscosity solutions to non convex fully nonlinear parabolic equations near the boundary this constitutes the parabolic counterpart of a similar c 2 regularity result due to silvestre and sirakov proved in 17 for solutions to non convex fully nonlinear elliptic equations

solving directly special fourth order ordinary differential equations using runge kutta type method in this paper an explicit runge kutta method for solving directly fourth order ordinary differential equations odes is constructed and denoted as rkfd we present a relevant colored tree theory and the associated b series theory for the order conditions based on the order conditions a three stage fourth order rkfd method and a three stage fifth order rkfd method are constructed numerical illustrations are presented to show the efficiency of the new rkfd methods by comparing them with other existing runge kutta nystrom rkn and runge kutta rk methods in the scientific literature after converting the problem into a system of second order odes and a system of first order odes respectively

compatible simultaneous lower confidence bounds for the holm procedure and other bonferroni based closed tests we consider the problem of simultaneously testing multiple one sided null hypotheses single step procedures such as the bonferroni test are characterized by the fact that the rejection or non rejection of a null hypothesis does not take the decision for any other hypothesis into account for stepwise test procedures such as the holm procedure the rejection or non rejection of a null hypothesis may depend on the decision of other hypotheses it is well known that stepwise test procedures are by construction more powerful than their single step counterparts this power advantage however comes only at the cost of increased difficulties in constructing compatible simultaneous confidence intervals for the parameters of interest for example such simultaneous confidence intervals are easily obtained for the bonferroni method but surprisingly hard to derive for the holm procedure in this paper we discuss the inherent problems and show that ad hoc solutions used in practice typically do not control the pre specified simultaneous confidence level instead we derive simultaneous confidence intervals that are compatible with a certain class of closed test procedures using weighted bonferroni tests for each intersection hypothesis the class of multiple test procedures covered in this paper includes gatekeeping procedures based on bonferroni adjustments fixed sequence procedures the simple weighted or unweighted bonferroni procedure by holm and the fallback procedure we illustrate the results with a numerical example

numerical solution of third order linear differential equations using generalized one shot operational matrices in orthogonal hybrid function domain abstract the present work proposes a new set of orthogonal hybrid functions hf which evolved from synthesis of orthogonal sample and hold functions shf and triangular functions tf the hf set is used to approximate a time function in a piecewise linear manner for such approximation the mean integral square error mise is much less than block pulse function based approximation which always provides staircase solutions and it is found that hf based approximation is a strong contender of approximations based upon orthogonal polynomials like legendre polynomials the respective mise s and elapsed times for mise computations for these three kinds of approximations are also studied in detail the one shot operational matrices for integration of different orders in hf domain are also derived these matrices are employed for more accurate multiple integration an example is treated to determine efficiency of these matrices finally these matrices are employed for solving third order non homogeneous differential equations followed by a numerical example the results obtained via matlab are compared with the exact solution as well as the results obtained via 4th order runge kutta method

the l p dirichlet problem and nondivergence harmonic measure we consider the dirichlet problem lu 0 in d u g on d for two second order elliptic operators l k u n i j 1 a i j k x ij u x k 0 1 in a bounded lipschitz domain d r n the coefficients a i j k belong to the space of bounded mean oscillation bmo with a suitable small bmo modulus we assume that l 0 is regular in l p d d for some p 1 p that is nu lp c g lp for all continuous boundary data g here is the surface measure on d and nu is the nontangential maximal operator the aim of this paper is to establish sufficient conditions on the difference of the coefficients e i j x a i j 1 x a i j 0 x that will assure the perturbed operator l 1 to be regular in l q d d for some q 1 q

on the viscosity solutions of a stochastic differential utility problem we prove existence uniqueness and gradient estimates of stochastic differential utility as a solution of the cauchy problem for the following equation in r 3 xxu u yu tu f u where f is lipschitz continuous we also characterize the solution in the vanishing viscosity sense r 2002 elsevier science usa all rights reserved

a group sequential holm procedure with multiple primary endpoints we propose a group sequential holm procedure when there are multiple primary endpoints this method addresses multiplicities arising from multiple primary endpoints and from multiple analyses in a group sequential design it has been shown to be a closed testing procedure and preserves the familywise error rate in the strong sense when multiple endpoints are the only concern without an interim analysis the method simplifies to the weighted holm procedure the proposed method is more powerful than the parallel group sequential method and avoids the need to anticipate the testing order as in the fixed sequence testing scheme

a new partition testing strategy for multiple endpoints to evaluate efficacy in multiple endpoints in confirmatory clinical trials is a challenging problem in multiple hypotheses testing the difficulty comes from the different importance of each endpoint and their underlying correlation current approaches to this problem which test the efficacy in certain dose endpoint combinations and collate the results are based on closed testing or partition testing despite their different formulations all current approaches test their dose endpoint combinations as intersection hypotheses and apply various union intersection tests likelihood ratio test is seldom used owing to the extensive computation and lack of consistent inferences in this article we first generalize the formulation of multiple endpoints problem to include the cases of alternative primary endpoints and co primary endpoints then we propose a new partition testing approach that is based on consonance adjusted likelihood ratio test the new procedure provides consistent inferences and yet it is still conservative and does not rely on the estimation of endpoint correlation or independence assumptions that might be challenged by regulatory agencies

time domain goal oriented adaptivity using pseudo dual error representations abstract goal oriented adaptive algorithms produce optimal grids to solve challenging engineering problems recently a novel error representation using unconventional pseudo dual problems for goal oriented adaptivity in the context of frequency domain wave propagation problems has been developed in this paper we extend this error representation to the case of time domain problems we express the entire problem in weak form in order to derive the adjoint formulation and apply goal oriented adaptivity one dimensional 1d numerical results show that upper bounds for the new error representation are sharper than the classical ones therefore this new error representation can be used to design more efficient goal oriented adaptive methodologies

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

a fourth order runge kutta method based on bdf type chebyshev approximations in this paper we consider a new fourth order method of bdf type for solving stiff initial value problems based on the interval approximation of the true solution by truncated chebyshev series it is shown that the method may be formulated in an equivalent way as a runge kutta method having stage order four the method thus obtained have good properties relatives to stability including an unbounded stability domain and large a value concerning a a stability a strategy for changing the step size based on a pair of methods in a similar way to the embedding pair in the runge kutta schemes is presented the numerical examples reveals that this method is very promising when it is used for solving stiff initial value problems

lateral boundary differentiability of solutions of parabolic equations in nondivergence form abstract the lateral boundary differentiability is shown for solutions of parabolic differential equations in nondivergence form under the assumptions that the parabolic boundary satisfies the exterior dini condition and is punctually c 1 differentiable one sided in t direction the classical barrier technique the maximum principle the interior harnack inequality and an iteration procedure are the main analytical tools

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order

differentiability at lateral boundary for fully nonlinear parabolic equations abstract for fully nonlinear uniformly parabolic equations the first derivatives regularity of viscosity solutions at lateral boundary is studied under new dini type conditions for the boundary which is called reifenberg dini conditions and is weaker than usual dini conditions

graphical approaches using a bonferroni mixture of weighted simes tests graphical approaches to multiple testing procedures are very flexible and easy to communicate with non statisticians the availability of the r package gmcp further propelled the application of graphical approaches in randomized clinical trials bretz et al biometrical journal 2011 53 894 913 introduced a class of nonparametric testing procedures based on a bonferroni mixture of weighted simes tests for intersection hypotheses such approaches are extremely useful when the conditions for the simes test are known to hold for hypotheses within certain subsets but may not hold for hypotheses across subsets we describe the calculation of adjusted p values for such approaches which is currently not available in the gmcp package we also optimize the generation of the weights for each intersection hypothesis in the closure of a graph based multiple testing procedure which can dramatically reduce the computing time for simulation based power calculations we show the validity of the simes test for comparing several treatments with a control performing noninferiority and superiority tests or testing the treatment effect in an overall and a subpopulation for the normal binary count and time to event data the proposed method is illustrated using an example for designing a confirmatory clinical trial copyright 2016 john wiley sons ltd

a global approach to fully nonlinear parabolic problems we consider the general initial boundary value problem 1 u t f x t u du du f x t x t qt 0 t 2 g x t u du g x t x t st 0 t 3 u x 0 h x x where is a bounded open set in rn with sufficiently smooth boundary the problem 1 3 is first reduced to the analogous problem in the space w 4 0 p qt with zero initial condition and f w 2 0 p qt g w 3 1 p 0 p st the resulting problem is then reduced to the problem au 0 where the operator a w 4 0 p qt w 4 0 p qt satisfies condition s this reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in sobolev spaces the local and global solvability of the operator equation au 0 are achieved via topological methods developed by i v skrypnik further applications are also given involving relevant coercive problems as well as galerkin approximations

testing strategies in multi dose experiments including active control inferential test strategies for multi arm trials are adapted or proposed for the special situation when more than one dose of a test treatment placebo and active control s are compared this includes between doses dose placebo and dose active control comparisons the procedures refer to situations when detailed comparisons make sense only if the sensitivity of the trial has been shown for example if a dose response relationship or a difference between active control and placebo has been established split strategies hierarchical assuming an order restriction among doses or linked procedures are introduced in linked procedures equivalence to the active control will be established only if the dose is also shown to be effective as compared to placebo all the inferential procedures control the experimentwise error rate in the strong sense for the respective sets of null hypotheses considered

a graphical approach to sequentially rejective multiple test procedures for clinical trials with multiple treatment arms or endpoints a variety of sequentially rejective weighted bonferroni type tests have been proposed such as gatekeeping procedures fixed sequence tests and fallback procedures they allow to map the difference in importance as well as the relationship between the various research questions onto an adequate multiple test procedure since these procedures rely on the closed test principle they usually require the explicit specification of a large number of intersection hypotheses tests the underlying test strategy may therefore be difficult to communicate we propose a simple iterative graphical approach to construct and perform such bonferroni type tests the resulting multiple test procedures are represented by directed weighted graphs where each node corresponds to an elementary hypothesis together with a simple algorithm to generate such graphs while sequentially testing the individual hypotheses the approach is illustrated with the visualization of several common gatekeeping strategies a case study is used to illustrate how the methods from this article can be used to tailor a multiple test procedure to given study objectives