the classification of novikov algebras in low dimensions novikov algebras were introduced in connection with the poisson brackets of hydrodynamic type and hamiltonian operators in the formal variational calculus for further our understanding and physical applications we give a classification of novikov algebras in dimensions two and three in this paper

novikov poisson algebras abstract we classify novikov poisson algebras whose novikov algebras are simple with an idempotent element moreover a class of simple novikov algebras without idempotent elements is constructed through novikov poisson algebras certain new simple lie superalgebras induced by novikov poisson algebras are introduced

on fermionic novikov algebras novikov algebras were introduced in connection with the poisson brackets of hydrodynamic type and hamiltonian operators in formal variational calculus they are a class of left symmetric algebras with commutative right multiplication operators which can be viewed as bosonic fermionic novikov algebras are a class of left symmetric algebras with anti commutative right multiplication operators they correspond to a certain hamiltonian superoperator in a supervariable in this paper we commence a study on fermionic novikov algebras from the algebraic point of view we will show that any fermionic novikov algebra in dimension 3 must be bosonic moreover we give the classification of real fermionic novikov algebras on four dimensional nilpotent lie algebras and some examples in higher dimensions as a corollary we obtain kinds of four dimensional real fermionic novikov algebras which are not bosonic all of these examples will serve as a guide for further development including the application in physics

corrected confidence intervals after sequential testing with applications to survival analysis corrected confidence intervals are developed for an unknown parameter for data from a sequential experiment the corrections are obtained from asymptotic expansions for the distribution of the signed root transformation and they have a simple form the model does not require independent and identically distributed random variables or smooth stopping boundaries inference about the failure rate of an exponential distribution when data are subject to random censoring censoring by real time and optional stopping is used to illustrate the main results the accuracy of the method is confirmed through simulation studies

higher order pdes and symmetric stable processes we describe a connection between the semigroup of a symmetric stable process with rational index and higher order partial differential equations as an application we obtain a variational formula for the eigenvalues associated with the process killed upon leaving a bounded open set d the variational formula is more user friendly than the classical rayleigh ritz formula we illustrate this by obtaining upper bounds on the eigenvalues in terms of dirichlet eigenvalues of the laplacian on d these results generalize some work of banuelos and kulczycki on the cauchy process along the way we prove an operator inequality for the operators associated with the transition densities of brownian motion and the brownian motion killed upon leaving d

the realization of non transitive novikov algebras novikov algebras were introduced in connection with hydrodynamic type poisson brackets and hamiltonian operators in the formal variational calculus we have given a kind of realization of transitive novikov algebras through the novikov algebras given by s gelfand and their compatible infinitesimal deformations in bai and meng 2001 j phys a math gen 34 3363 72 as a further and continuous study we extend this realization theory to the non transitive novikov algebras in the paper in two and three dimensions we find that all non transitive novikov algebras also can be realized as the novikov algebras given by s gelfand and their compatible infinitesimal deformations moreover they have simpler formulae

correction to higher order pdes and symmetric stable processes probability theory and related fields 129 495 536 2004 since 0 as m but there is a hidden dependence on m in the s and one dimensional examples show 1 1 is false this was pointed out to us by the anonymous referee of the article 2 in fact the conclusion of theorem 1 5 is false as one can show via a one dimensional counterexample the only place in we used theorem 1 5 in the paper was in the proof of theorem 1 3 we outline a corrected proof of theorem 1 3 that does not appeal to theorem 1 5

point and interval estimation following a sequential clinical trial summary the usual methodology employed in analysis after a sequential clinical trial is based on orderings of the possible samples resulting from the design however this approach lacks flexibility for use in wider applications in this paper two estimation techniques not based on orderings are considered and modified to obtain improved accuracy a bias adjusted maximum likelihood estimate together with a new and general method for setting confidence limits are discussed the realistic scenario of group sequential monitoring is assumed and methods for exact estimation are given accuracy of the methodology after a triangular test and an o brien fleming test are demonstrated through simulation

estimation of a secondary parameter in a group sequential clinical trial summary in this article we investigate estimation of a secondary parameter in group sequential tests we study the model in which the secondary parameter is the mean of the normal distribution in a subgroup of the subjects the bias of the naive secondary parameter estimator is studied it is shown that the sampling proportions of the subgroup have a crucial effect on the bias as the sampling proportion of the subgroup at or just before the stopping time increases the bias of the naive subgroup parameter estimator increases as well an unbiased estimator for the subgroup parameter and an unbiased estimator for its variance are derived using simulations we compare the mean squared error of the unbiased estimator to that of the naive estimator and we show that the differences are negligible as an example the methods of estimation are applied to an actual group sequential clinical trial the beta blocker heart attack trial

tests and confidence intervals for secondary endpoints in sequential clinical trials in a sequential clinical trial whose stopping rule depends on the primary endpoint inference on secondary endpoints is an important long standing problem ignoring the possibility of early stopping based on the primary endpoint may result in substantial bias to address this problem a commonly used approach is to develop bias correction by estimating the bias in the case of bivariate normal outcomes and appealing to joint asymptotic normality of the statistics associated with the primary and secondary endpoints we propose herein a new approach that uses resampling and a novel ordering scheme in the sample space of sequential statistics observed up to a stopping time this approach is shown to provide accurate inference in complex clinical trials including time sequential trials with survival endpoints and covariates

confidence interval calculation for a sequential clinical trial of binary responses consider a sequential test applied to two streams of binary responses in a comparative clinical trial after the completion of such a trial woodroofe s 1992 technique for accurate confidence interval estimation for a treatment difference can be applied in this paper the technique is extended to provide confidence intervals for the individual absolute success probabilities of the two treatments in the trial for which no current methodology exists accuracy is explored by simulation of coverage probabilities and individual confidence limit probabilities the simulations concern a triangular test and an o brien fleming test

estimation after sequential testing a simple approach for a truncated sequential probability ratio test an approximate pivot is constructed for the problem of estimating a normal mean following a truncated sequential probability ratio test and shown to provide a useful method for constructing confidence bounds and intervals letting t denote the sample size and s t the sum of observations the approximate pivot is constructed by standardizing s t t 1 2 s t t the mean and variance of which are no longer 0 and 1 due to the optional stopping

effects of the reference set on frequentist inferences summary we employ second order likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function referring to this as the effect of the reference set there are two aspects of higher order corrections to first order likelihood methods namely i that involving effects of fitting nuisance parameters and leading to the modified profile likelihood and ii another part pertaining to limitation in adjusted information generally each of these involves a first order adjustment depending on the reference set however we show that for some important settings likelihood irrelevant model specifications have a second order effect on both of these adjustments this result includes specification of the censoring model for survival data on the other hand for sequential experiments the likelihood irrelevant specification of the stopping rule has a second order effect on adjustment i but a first order effect on adjustment ii these matters raise the issue of what are ideal frequentist inferences since consideration of exact frequentist inferences will not suffice we indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic without commonly considered adjustments thereto

estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials pour des essais de confirmation destines a prendre une decision correcte il est important que des experiences adaptees fournissent une inference avec un niveau nominal correct des estimateurs sans biais et des intervalles de confiance pour comparer les traitements dans les essais en cours mais une estimation ponctuelle simple et son intervalle de confiance sont souvent biaises dans des experiences sequentielles adaptatives nous developpons une nouvelle procedure d estimation suivant un test a partir d une taille d echantillon d une experience de re estimation pour obtenir un intervalle de confiance exact et une estimation ponctuelle la methode et basee sur la propriete generale de la distribution d une fonction pivot d un essai clinique sequentiel de groupe de shen et fisher 1999 on propose un estimateur modifie tenant explicitement compte de la limite d arret avec biais reduits quand les tailles de blocs sont petites nous montrons que les estimateurs sont convergents le calcul des estimations est elementaire nous fournissons aussi une fonction de ponderation modifiee pour ameliorer la puissance du test de nombreuses etudes de simulation montrent que les intervalles de confiance exacts ont une probabilite nominale de recouvrement precise et les estimateurs ponctuels sont pratiquement sans biais pour les tailles d echantillon habituels

bilinear forms on fermionic novikov algebras novikov algebras were introduced in connection with the poisson brackets of hydrodynamic type and hamiltonian operators in formal variational calculus fermionic novikov algebras correspond to a certain hamiltonian super operator in a super variable in this paper we show that there is a remarkable geometry on fermionic novikov algebras with non degenerate invariant symmetric bilinear forms which we call pseudo riemannian fermionic novikov algebras they are related to pseudo riemannian lie algebras furthermore we obtain a procedure to classify pseudo riemannian fermionic novikov algebras as an application we give the classification in dimension 4 motivated by the one in dimension 4 we construct some examples in high dimensions

the classification of novikov algebras in low dimensions invariant bilinear forms in this note we give a complete classification of the non degenerate symmetric invariant bilinear forms on novikov algebras in dimension 2 and 3 which can be regarded as an addendum of the classification of novikov algebras in low dimensions given in our previous work bai c m and meng d j 2001 j phys a math gen 34 1581 94

confidence intervals for secondary parameters following a sequential test in sequential studies formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study however the completed data set may warrant a more detailed analysis involving the estimation of parameters corresponding to effects that were not used to determine when to stop and yet correlated with those that were this paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used the method assumes that information accumulates on the primary and secondary parameters at proportional rates this requirement will be valid in many potential applications but only in limited situations in survival analysis

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

on simple novikov algebras and their irreducible modules abstract we give a complete classification of finite dimensional simple novikov algebras and their irreducible modules over an algebraically closed field with prime characteristic moreover we introduce novikov poisson algebras and their tensor theory our tensor theory enables us to understand better certain finite dimensional simple novikov algebras and their irreducible modules

alpha continuity properties of the symmetric alpha stable process let d be a domain of finite lebesgue measure in d and let x t d be the symmetric stable process killed upon exiting d each element of the set i i 1 of eigenvalues associated to x t d regarded as a function of a 0 2 is right continuous in addition if d is lipschitz and bounded then each i is continuous in and the set of associated eigenfunctions is precompact

the first exit time of planar brownian motion from the interior of a parabola let d be the interior of a parabola in mathbb r 2 and tau d the first exit time of brownian motion from d we show log p tau d t behaves like t 1 3 as t to infty

a comparison of adaptive allocation rules for group sequential binary response clinical trials in clinical trials to compare two or more treatments with dichotomous responses group sequential designs may reduce the total number of patients involved in the trial and response adaptive designs may result in fewer patients being assigned to the inferior treatments in this paper we combine group sequential and response adaptive designs extending recent work on sample size re estimation in trials to compare two treatments with normally distributed responses to analogous binary response trials we consider the use of two parameters of interest in the group sequential design the log odds ratio and the simple difference between the probabilities of success in terms of the adaptive sampling rules we study two urn models the drop the loser rule and the randomized polya urn rule and compare their properties with those of two sequential maximum likelihood estimation rules which minimize the expected number of treatment failures we investigate two ways in which adaptive urn designs can be used in conjunction with group sequential designs the first method updates the urn at each interim analysis and the second method continually updates the urn after each patient response assuming immediate patient responses our simulation results show that the group sequential design which uses the drop the loser rule applied fully sequentially is the most effective method for reducing the expected number of treatment failures and the average sample number whilst still maintaining the nominal error rates over a range of success probabilities

variational calculus of supervariables and related algebraic structures abstract we establish a formal variational calculus of supervariables which is a combination of the bosonic theory of gel fand dikii and the fermionic theory in our earlier work certain interesting new algebraic structures are found in connection with hamiltonian superoperators in terms of our theory in particular we find connections between hamiltonian superoperators and novikov poisson algebras that we introduced in our earlier work in order to establish a tensor theory of novikov algebras furthermore we prove that an odd linear hamiltonian superoperator in our variational calculus induces a lie superalgebra which is a natural generalization of the super virasoro algebra under certain conditions

hom novikov algebras we study a twisted generalization of novikov algebras called hom novikov algebras in which the two defining identities are twisted by a linear map it is shown that hom novikov algebras can be obtained from novikov algebras by twisting along any algebra endomorphism all algebra endomorphisms on complex novikov algebras of dimensions 2 or 3 are computed and their associated hom novikov algebras are described explicitly another class of hom novikov algebras is constructed from hom commutative algebras together with a derivation generalizing a construction due to dorfman and gel fand two other classes of hom novikov algebras are constructed from hom lie algebras together with a suitable linear endomorphism generalizing a construction due to bai and meng

on the realization of transitive novikov algebras novikov algebras were introduced in connection with the poisson brackets of hydrodynamic type and hamiltonian operators in the formal variational calculus it is well known that the radical of a finite dimensional novikov algebra is transitive in this paper we prove that a kind realization of novikov algebras given by s gel fand is transitive and we give a deformation theory of novikov algebras in two and three dimensions we find that all transitive novikov algebras can be realized as the novikov algebras given by s gel fand and their compatible infinitesimal deformations

the automorphisms of novikov algebras in low dimensions novikov algebras were introduced in connection with poisson brackets of hydrodynamic type and hamiltonian operators in the formal variational calculus they also correspond to a class of vertex algebras an automorphism of a novikov algebra is a linear isomorphism satisfying xy x y which keeps the algebraic structure the set of automorphisms of a novikov algebra is a lie group whose lie algebra is just the novikov algebra s derivation algebra the theory of automorphisms plays an important role in the study of novikov algebras in this paper we study the automorphisms of novikov algebras we get some results on their properties and classification in low dimensions these results are fundamental in a certain sense and they will serve as a guide for further development moreover we apply these results to classify gel fand dorfman bialgebras and novikov poisson algebras these results also can be used to study certain phase spaces and geometric classical r matrices

brownian motion in twisted domains the tail behavior of a brownian motion s exit time from an unbounded domain depends upon the growth of the inner radius of the domain in this article we quantify this idea by introducing the notion of a twisted domain in the plane roughly speaking such a domain is generated by a planar curve as follows as a traveler proceeds out along the curve the boundary curves of the domain are obtained by moving out g r units along the unit normal to the curve when the traveler is r units away from the origin the function g is called the growth radius such domains can be highly nonconvex and asymmetric we give a detailed account of the case g r r p 0 p 1 when p 1 a twisted domain can reasonably be interpreted as a twisted cone