a priori l 2 error estimates for finite element methods for nonlinear diffusion equations with memory this paper studies finite element approximations to the solutions of the nonlinear diffusion equations with memory an elliptic projection with memory associated with our equations is defined and then used in the derivations of optimal error estimates for semidiscrete and crank nicolson finite element approximations

extension of hereditary symmetry operators two models of candidates for hereditary symmetry operators are proposed and thus many nonlinear systems of evolution equations possessing infinitely many commuting symmetries may be generated some concrete structures of hereditary symmetry operators are carefully analysed on the basis of the resulting general conditions and several corresponding nonlinear systems are explicitly given as illustrative examples

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

necessary conditions for existence of non degenerate hamiltonian structures the necessary criteria are pointed out for the exisence of hamiltonian and bi hamiltonian non degenerate structures for a nonlinear system of partial differential equations of first order the results are formulated in terms of the new invariants of the intrinsic geometry introduced in this paper connected with the nijenhuis and haantjes tensors of a 1 1 tensor field

on the structure of 2 1 dimensional commutative and noncommutative integrable equations we develop the symbolic representation method to derive the hierarchies of 2 1 dimensional integrable equations from the scalar lax operators and to study their properties globally the method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in c or a noncommutative associative algebra we prove that these hierarchies are indeed quasi local in the commutative case as conjectured by mikhailov and yamilov j phys a 31 6707 1998 we propose a ring extension in the noncommutative case based on the symbolic representation as examples we give noncommutative versions of kadomtsev petviashvili kp modified kadomtsev petviashvili mkp and boussinesq equations

on the inefficiency of the adaptive design for monitoring clinical trials summary adaptive designs which allow the sample size to be modified based on sequentially computed observed treatment differences have been advocated recently for monitoring clinical trials although such methods have a great deal of appeal on the surface we show that such methods are inefficient and that one can improve uniformly on such adaptive designs using standard group sequential tests based on the sequentially computed likelihood ratio test statistic

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

theory of tensor invariants of integrable hamiltonian systems i incompatible poisson structures this paper develops a new theory of tensor invariants of a completely integrable non degenerate hamiltonian system on a smooth manifoldmn the central objects in this theory are supplementary invariant poisson structurespc which are incompatable with the original poisson structurep1 for this hamiltonian system a complete classification of invariant poisson structures is derived in a neighbourhood of an invariant toroidal domain this classification resolves the well known inverse problem that was brought into prominence by magri s 1978 paper deveoted to the theory of compatible poisson structures applications connected with the kam theory with the kepler problem with the basic integrable problem of celestial mechanics and with the harmonic oscillator are pointed out a cohomology is defined for dynamical systems on smooth manifolds the physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of poisson structures are introduced to study the diversity of pairs of poisson structures incompatible in magri s sense it is proved that if a dynamical systemv preserves two strongly dynamically compatible poisson structuresp1 andp2 in a general position then this system is completely integrable such a systemv generates a hierarchy of integrable dynamical systems which in general are not hamiltonian neither with respect top1 nor with respect top2 necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible poisson structures

lax wendroff methods for hyperbolic history value problems this paper is concerned with lax wendroff methods for a class of hyperbolic history value problems these problems have the feature that globally in time smooth solutions exist if the data are sufficiently small and that solutions develop singularities for large data we prove second order convergence of the lax wendroff method for smooth solutions and investigate numerically the dependence on the initial data we demonstrate the occurrence of shock type singularities and compare the results to the quasilinear wave equation without volterra term

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

an alternative approach to solve the mixed akns equations abstract the algebraic geometric solutions of the mixed akns equations are investigated through a finite dimensional lie poisson hamiltonian system which is generated by the nonlinearization of the adjoint equation related to the akns spectral problem first each mixed akns equation can be decomposed into two compatible lie poisson hamiltonian flows then the separated variables on the coadjoint orbit are introduced to study these lie poisson hamiltonian systems further based on the hamilton jacobi theory the relationship between the action angle coordinates and the jacobi inversion problem is established in the end using riemann jacobi inversion the algebraic geometric solutions of the first three mixed akns equations are obtained

hamiltonian theory over noncommutative rings and integrability in multidimensions a generalization of the adler gel fand dikii scheme is used to generate bi hamiltonian structures in two spatial dimensions in order to implement this scheme a hamiltonian theory is built over a noncommutative ring namely the ring of formal pseudodifferential operators bi hamiltonian structures generated in this way can be used for the kadomtsev petviashvili equation as well as other integrable equations in 2 1

schouten tensor and bi hamiltonian systems of hydrodynamic type the necessary conditions are derived for existence of a bi hamiltonian structure for a given hydrodynamic type system one of the conditions is the vanishing of its haantjes tensor a theorem is proved on the canonical forms of the generic bi hamiltonian systems the schouten 2 1 tensor sijk is connected with any hamiltonian system of hydrodynamic type the complete symmetry of the 3 0 tensor sijk is demonstrated the necessary conditions for existence of a single nondegenerate hamiltonian structure are obtained in terms of the special differential k forms p1 pk u1 uk

on a class of quasilinear partial integrodifferential equations with singular kernels abstract we prove local and global existence theorems for a model equation in nonlinear viscoelasticity in contrast to previous studies we allow the memory function to have a singularity we approximate the equation by equations with regular kernels and use energy estimates to prove convergence of the approximate solutions

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

an inversion formula for recursion operator of kp hierarchy an inversion formula is established for the recursion operator of the kp hierarchy the formula explains in part how operators y i are cancelled each other in certain combination of i y j that is essential for generating the kp hierarchy

the hamiltonian structure of the 2 1 dimensional ablowitz kaup newell segur hierarchy by considering hamiltonian theory over a suitable noncommutative ring the nonlinear evolution equations of the ablowitz kaup newell segur 2 1 hierarchy are incorporated into a hamiltonian framework and a modified lenard scheme

on the theory of recursion operator the general structure and properties of recursion operators for hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered weak and strong recursion operators are introduced the conditions which determine weak and strong recursion operators are found

adaptive and nonadaptive group sequential tests summary methods have been proposed for redesigning a clinical trial at an interim stage in order to increase power in order to preserve the type i error rate methods for unplanned design change have to be defined in terms of nonsufficient statistics and this calls into question their efficiency and the credibility of conclusions reached we evaluate schemes for adaptive redesign extending the theoretical arguments for use of sufficient statistics of tsiatis mehta 2003 and assessing the possible benefits of preplanned adaptive designs by numerical computation of optimal tests these optimal adaptive designs are concrete examples of optimal sequentially planned sequential tests proposed by schmitz 1993 we conclude that the flexibility of unplanned adaptive designs comes at a price and we recommend that the appropriate power for a study should be determined as thoroughly as possible at the outset then standard error spending tests possibly with unevenly spaced analyses provide efficient designs but it is still possible to fall back on flexible methods for redesign should study objective change unexpectedly once the trial is under way

the partitioning principle a powerful tool in multiple decision theory a first general principle and nowadays state of the art for the construction of powerful multiple test procedures controlling a multiple level a is the so called closure principle in this article we introduce another powerful tool for the construction of multiple decision procedures especially for the construction of multiple test procedures and selection procedures this tool is based on a partition of the parameter space and will be called partitioning principle pp in the first part of the paper we review basic concepts of multiple hypotheses testing and discuss a slight generalization of the current theory in the second part we present various variants of the pp for the construction of multiple test procedures these are a general pp gpp a weak pp wpp and a strong pp spp it will be shown that depending on the underlying decision problem a pp may lead to more powerful test procedures than a formal application of the closure principle fcp moreover the more complex spp may be more powerful than the wpp based on a duality between testing and selecting pps can also be applied for the construction of more powerful selection procedures in the third part of the paper fcp wpp and spp are applied and compared in some examples

adaptive randomized trial designs that cannot be dominated by any standard design at the same total sample size summary prior work has shown that certain types of adaptive designs can always be dominated by a suitably chosen standard group sequential design this applies to adaptive designs with rules 10 for modifying the total sample size a natural question is whether analogous results hold for other types of adaptive designs we focus on adaptive enrichment designs which involve preplanned rules for modifying enrollment criteria based on accrued data in a randomized trial such designs often involve multiple hypotheses e g one for the total population and one for a predefined subpopulation such as those with high disease se verity at baseline we fix the total 15 sample size and consider overall power defined as the proba bility of rejecting at least one false null hypothesis we present adaptive enrichment designs whose overall power at two alternatives cannot simultaneously be matched by any standard design in some scenarios there is a substantial gap between the overall power achieved by these adaptive designs and that of any standard design we also prove that such gains in overall power come at a cost to attain overall power 20 above what is achievable by certain standard designs it is necessary to increase power to reject some hypotheses and reduce power to reject others we conclude by showing the class of adaptive enrichment designs allows certain power tradeoffs that are not available when restricting to standard designs we illustrate our results in the context o f planning a hypothetical randomized trial of a new antidepressant using data distributions fro m kirsch et al 2008 25

hamiltonian structure for the modulation equations of a sine gordon wavetrain on etudie l equation sinus gordon e 2 t 2 x 2 u sin u 0 ou e est un petit parametre on construit une solution asymptotique sous la forme d un train d onde n phase variant lentement

block diagonalizability problem for hydrodynamic type systems the necessary and sufficient conditions are derived for the reducibility of a hydrodynamic type system into a block diagonal form with k mutually interacting blocks applications to the perturbations of the benney system and to the hamiltonian systems of partial differential equations are presented the algebraic identities connecting the nijenhuis tensors nb a u v and na u v and haantjes tensors hb a u v and ha u v are discovered

optimizing randomized trial designs to distinguish which subpopulations benefit from treatment it is a challenge to evaluate experimental treatments where it is suspected that the treatment effect may only be strong for certain subpopulations such as those having a high initial severity of disease or those having a particular gene variant standard randomized controlled trials can have low power in such situations they also are not optimized to distinguish which subpopulations benefit from a treatment with the goal of overcoming these limitations we consider randomized trial designs in which the criteria for patient enrollment may be changed in a preplanned manner based on interim analyses since such designs allow data dependent changes to the population enrolled care must be taken to ensure strong control of the familywise type i error rate our main contribution is a general method for constructing randomized trial designs that allow changes to the population enrolled based on interim data using a prespecified decision rule for which the asymptotic familywise type i error rate is strongly controlled at a specified level as a demonstration of our method we prove new sharp results for a simple two stage enrichment design we then compare this design to fixed designs focusing on each design s ability to determine the overall and subpopulation specific treatment effects

applications and numerical analysis of partial differential volterra equations a brief survey this article contains a concise survey of the numerical analysis of volterra equations and leads up to some recent results on a posteriori error estimation for finite element approximations

bi hamiltonian structure in 2 d field theory we exhibit the bi hamiltonian structure of the equations of associativity witten dijkgraaf verlinde verlinde dubrovin equations in 2 d topological field theory which reduce to a single equation of monge ampere type f ttt f xxt 2 f xxx f xtt in the case of three primary fields the first hamiltonian structure of this equation is based on its representation as a 3 component system of hydrodynamic type and the second hamiltonian structure follows from its formulation in terms of a variational principle with a degenerate lagrangian

on the decoupling problem of general quasilinear first order systems in two independent variables abstract the paper deals with the decoupling problem of general quasilinear first order systems in two independent variables we consider either the case of homogeneous and autonomous systems or the one of nonhomogeneous and or nonautonomous systems necessary and sufficient conditions for the partial or full decoupling of the systems at hand are provided the conditions involve the properties of eigenvalues and eigenvectors of the coefficient matrix and provide the differential constraints whose integration leads to the decoupling transformation some applications of physical interest are also given

hyperbolicity and change of type in the flow of viscoelastic fluids the equations governing the flow of viscoelastic liquids are classified according to the symbol of their differential operators propagation of singularities is discussed and conditions for a change of type are investigated the vorticity equation for steady flow can change type when a critical condition involving speed and stresses is satisfied this leads to a partitioning of the field of flow into subcritical and supercritical regions as in the problem of transonic flow

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

hamiltonian structure of real monge ampere equations the variational principle for the real homogeneous monge ampere equation in two dimensions is shown to contain three arbitrary functions of four variables there exist two different specializations of this variational principle where the lagrangian is degenerate and furthermore contains an arbitrary function of two variables the hamiltonian formulation of these degenerate lagrangian systems requires the use of dirac s theory of constraints as in the case of most completely integrable systems the constraints are second class and dirac brackets directly yield the hamiltonian operators thus the real homogeneous monge ampere equation in two dimensions admits two classes of infinitely many hamiltonian operators namely a family of local as well as another family non local hamiltonian operators and symplectic 2 forms which depend on arbitrary functions of two variables the simplest non local hamiltonian operator corresponds to the kac moody algebra of vector fields and functions on the unit circle hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair in the case of real monge ampere equations with constant right hand side this compatible pair is the only pair of hamiltonian operators that survives then the complete integrability of all these real monge ampere equations follows by magri s theorem some of the remarkable properties we have obtained for the hamiltonian structure of the real homogeneous monge ampere equation in two dimensions turn out to be generic to the real homogeneous monge ampere equation and the geodesic flow for the complex homogeneous monge ampere equation in arbitrary number of dimensions hence among all integrable nonlinear evolution equations in one space and one time dimension the real homogeneous monge ampere equation is distinguished as one that retains its character as an integrable system in multiple dimensions

decoupling problem for systems of quasi linear pde s the necessary and sufficient conditions for the decoupling of a quasi linear system of partial differential equations into k non interacting subsystems are derived several necessary conditions for the decoupling are found and applied to the benney system

yang baxter equations and intermediate long wave hierarchies in the framework of the gn manifold approach the algebraic structures of two representations of the ilw hierarchy are investigated it is shown that this hierarchy can be obtained by different realizations of two abstract structures strictly related with the yang baxter equations a new reduction theorem for the second representation of ilw is also formulated

a unified theory of two stage adaptive designs adaptation of a clinical trial occurs when there is a change from the planned course of action in design sample size or method of analysis there are many proposed adaptations in the literature but no comprehensive mathematical theory to support them the standard neyman pearson theory is not suitable for the setting and its use may result in invalid inference this article provides a general formulation of adaptation under which its validity can be broadly and rigorously established the basic theory on distribution stochastic independence point estimation confidence intervals hypothesis testing and asymptotics is developed various examples are given to illustrate applications of the theory the goal is to provide a solid theoretical foundation to aid the development of adaptive methods for clinical investigations