the decomposition of metric n lie algebras and the uniqueness in this paper we show that the orthogonal decomposition of any metric n lie algebra into indecomposable nondegenerate ideals is unique up to an isometry moreover the decomposition is unique up to the order of the ideals if the center of the n lie algebra is zero

a trace minimization algorithm for the generalized eigenvalue problem an algorithm for computing a few of the smallest or largest eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem ax lambda bx is presented the matrices a and b are assumed to be symmetric and haphazardly sparse with b being positive definite the problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace the rate of convergence of the method is established and a technique for improving it is discussed numerical experiments and comparisons with other methods are presented

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

group sequential enrichment design incorporating subgroup selection an important component of clinical trials in drug development is the analysis of treatment efficacy in patient subgroups subpopulations because of concerns of multiplicity and of the small sample sizes often involved such analyses can present substantial statistical challenges and may lead to misleading conclusions as a confirmatory seamless phase ii iii design we propose an adaptive enrichment group sequential procedure whereby resources can be concentrated on subgroups most likely to respond to treatment stopping boundaries are defined through upper and lower spending functions the procedure is presented in terms of the efficient score enabling the analysis of normal binary or time to event data it addresses the dilution effect by eliminating populations at the first stage that appear likely to derive no therapeutic benefit it subsequently proceeds with the definitive assessment of treatment efficacy among the remaining pooled populations using a group sequential design the procedure provides strong protection of familywise type i error rate and we employ a bootstrap algorithm to obtain point and interval estimates that are adjusted for the selection bias we give examples to demonstrate how the design is used we make comparisons with adaptive two stage combination test procedures and with a group sequential test that does not account for the presence of subgroups numerical results show that the procedure has high power to detect subgroup specific effects and the use of multiple interim analysis points can lead to substantial sample size savings

a consistency adjusted alpha adaptive strategy for sequential testing in a clinical trial with two clinically important endpoints each of which can fully characterize a treatment benefit to support an efficacy claim by itself a minimum degree of consistency in the findings is expected otherwise interpretation of study findings can be problematic clinical trial literature contains examples where lack of consistency in the findings of clinically relevant endpoints led to difficulties in interpreting study results the aim of this paper is to introduce this consistency concept at the study design stage and investigate the consequences of its implementation in the statistical analysis plan the proposed methodology allows testing of hierarchically ordered endpoints to proceed as long as a pre specified consistency criterion is met in addition while an initial allocation of the alpha level is specified for the ordered endpoints at the design stage the methodology allows the alpha level allocated to the second endpoint to be adaptive to the findings of the first endpoint in addition the methodology takes into account the correlation between the endpoints in calculating the significance level and the power of the test for the next endpoint the proposed consistency adjusted alpha adaptive strategy caaas is very general several of the well known multiplicity adjustment approaches arise as special cases of this strategy by appropriate selection of the consistency level and the form of alpha adaptation function we discuss control of the type i error rate as well as power of the proposed methodology and consider its application to clinical trial data

a flexible strategy for testing subgroups and overall population subgroup analyses in addition to the total study population analysis are common in clinical trials however it is well recognized that findings from subgroup analyses do not provide confirmatory evidence for subgroup treatment effects without placing a priori criteria for ensuring that their findings are scientifically sound in this paper we address some of the common pitfalls of subgroup analyses subgroups analyses inherently have low power for detecting treatment effects we investigate the power interplay for a subgroup analysis and that for the total study population and list factors that impact the power of a subgroup analysis then we introduce a flexible statistical strategy for testing a pre specified sequence of hypotheses for both the overall and a subgroup the proposed method strongly controls the familywise type i error rate and enjoys higher power than other traditional methods this testing strategy allows testing for a subgroup once a pre specified degree of consistency in the efficacy findings between the subgroup and the overall study population is met in addition it accounts for the dependency between test statistics for the subgroup and the overall study population we discuss the power performance of this new method and provide significance levels for subgroup analysis finally we illustrate its application through retrospective analysis of data from three published clinical trials

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

a projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a hermitian matrix we present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured hermitian matrix a we are interested in the case in which the dimension of the invariant subspace is large e g over several hundreds or thousands even though it may still be small relative to the dimension of a these problems arise from for example density functional theory dft based electronic structure calculations for complex materials the key feature of our algorithm is that it performs fewer rayleigh ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the davidson algorithm it is a block algorithm and hence can take advantage of efficient blas3 operations and be implemented with multiple levels of concurrency we discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer

an efficient iterative method for the generalized stokes problem the generalized stokes problem which arises frequently in the simulation of time dependent navier stokes equations for incompressible fluid flow gives rise to symmetric linear systems of equations these systems are indefinite due to a set of linear constraints on the velocity causing difficulty for most preconditioners and iterative methods this paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method this new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite numerical experiments indicating superior convergence compared to existing methods are presented a natural extension of this method to elliptic problems is also proposed along with theoretical bounds on the rate of convergence and results of experiments demonstrating robust and effective preconditioning

restricted rank modification of the symmetric eigenvalue problem theoretical considerations the problem is considered how to obtain the eigenvalues and vectors of a matrix a vvt where a is a symmetric matrix with known spectral decomposition and vvt is a positive semidefinite matrix of low rank it is shown that the eigenvalues of a vvt can easily be located to any desired accuracy by means of the inertia of the matrix i vt a 1v the problem of determining the eigenvalues of a restricted to r v can be treated likewise

two stage selection and testing designs for comparative clinical trials summary a two stage design which selects the best of several experimental treatments and compares it to a standard control is proposed the design allows early termination with acceptance of the global null hypothesis optimal sample size and cut off parameters are obtained by minimizing expected total sample size for fixed significance level and power with many other factors moreover the usual error rate computations associated with comparative testing do not account for the preliminary selection process consequently the overall procedure may be neither effective nor efficient for identifying an experimental treatment which is an improvement over c in this paper we propose a new approach to the problem of identifying the best of k experimental treatments and determining whether it is superior to a control we deal with the binomial setting where patient response may be characterized as either success or failure with ok the success probability for ek k 0 corresponds to c for ease of notation assume 01 ok we propose a two stage procedure which allows early termination with acceptance of h0 o 01 ok with design parameters chosen to minimize expected total sample size the design and accompanying generalized definitions of size a and power 1 3 are given in 2 the algorithm used for optimization is described in 3 numerical results are presented in 4 followed by a discussion of the relative merits of the proposed design

a bayesian approach for envelope models the envelope model is a new paradigm to address estimation and prediction in multivariate analysis using sufficient dimension reduction techniques it has the potential to achieve substantial efficiency gains compared to standard models this model was first introduced by statist sinica 20 2010 927 960 for multivariate linear regression and has since been adapted to many other contexts however a bayesian approach for analyzing envelope models has not yet been investigated in the literature in this paper we develop a comprehensive bayesian framework for estimation and model selection in envelope models in the context of multivariate linear regression our framework has the following attractive features first we use the matrix bingham distribution to construct a prior on the orthogonal basis matrix of the envelope subspace this prior respects the manifold structure of the envelope model and can directly incorporate prior information about the envelope subspace through the specification of hyperparamaters this feature has potential applications in the broader bayesian sufficient dimension reduction area second sampling from the resulting posterior distribution can be achieved by using a block gibbs sampler with standard associated conditionals this in turn facilitates computationally efficient estimation and model selection third unlike the current frequentist approach our approach can accommodate situations where the sample size is smaller than the number of responses lastly the bayesian approach inherently offers comprehensive uncertainty characterization through the posterior distribution we illustrate the utility of our approach on simulated and real datasets

block minimum perturbation algorithm based on block arnoldi process for nonsymmetric linear systems with multiple right hand sides abstract in many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right hand sides customarily one uses the residual error as a stopping condition but small residue does not imply accurate approximate solutions therefore minimizing quantities other than the residual norm may be more adequate in this context based on the above consideration a block minimum perturbation krylov subspace method bminpet for the solution xm satisfying a a x m b b at each step of nonsymmetric linear systems a x b is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm a b f of the matrix a b and a stopping criterion is given for consistent system with multiple right hand sides to get a more robust way to stop the iterations this process is a generalization of minimum perturbation algorithm minpert by kasenally and simoncini 1997 analysis of a minimum perturbation algorithm for nonsymmetric linear systems siam j numer anal 34 48 66 to multiple right hand sides for the benefit of the amount of calculation we give lower and upper bounds about min a b f that are easier to compute as a by product we come up with a conception of function f g which is a generalization of function that is a symmetric gauge sg function the algorithm is used to analyze the performance of bgmres method by evaluating the proximity of their solutions to block joint backward perturbation optimality the relationships between bminpet method and relative methods are investigated numerical examples show us that bminpert shows its superiority compared with bfgmres s m pf gsgmres bl bicg rq bgmres and barnoldi in solving the large sparse ill conditioned problems

matrices vector spaces and information retrieval the evolution of digital libraries and the internet has dramatically transformed the processing storage and retrieval of information efforts to digitize text images video and audio now consume a substantial portion of both academic and industrial activity even when there is no shortage of textual materials on a particular topic procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking recently developed information retrieval technologies are based on the concept of a vector space data are modeled as a matrix and a user s query of the database is represented as a vector relevant documents in the database are then identified via simple vector operations orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself the purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections

unconstrained energy functionals for electronic structure calculations the performance of conjugate gradient schemes for minimizing unconstrained energy functionals in the context of condensed matter electronic structure density functional calculations is studied the unconstrained functionals allow a straightforward application of conjugate gradients by removing the explicit orthonormality constraints on the quantum mechanical wave functions however the removal of the constraints can lead to slow convergence in particular when preconditioning is used the convergence properties of two previously suggested energy functionals are analyzed and a new functional is proposed which unifies some of the advantages of the other functionals a numerical example derived from a diamond crystal confirms the analysis

numerical methods for electronic structure calculations of materials the goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems the paper is intended for a diverse scientific computing audience for this reason we assume the reader does not have an extensive background in the related physics our overview focuses on the nature of the numerical problems to be solved their origin and the methods used to solve the resulting linear algebra or nonlinear optimization problems it is common knowledge that the behavior of matter at the nanoscale is in principle entirely determined by the schrodinger equation in practice this equation in its original form is not tractable successful but approximate versions of this equation which allow one to study nontrivial systems took about five or six decades to develop in particular the last two decades saw a flurry of activity in developing effective software one of the main practical variants of the schrodinger equation is based on what is referred to as density functional theory dft the combination of dft with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials this article will emphasize pseudopotential density functional theory but other techniques will be discussed as well

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

evaluation of experiments with adaptive interim analyses summary a general method for statistical testing in experiments with an adaptive interim analysis is proposed the method is based on the observed error probabilities from the disjoint subsamples before and after the interim analysis formally an intersection of individual null hypotheses is tested by combining the twop values into a global test statistic stopping rules for fisher s product criterion in terms of critical limits for the p value in the first subsample are introduced including early stopping in the case of missing effects the control of qualitative treatment stage interactions is considered a generalization to three stages is outlined the loss of power when using the product criterion instead of the optimal classical test on the whole sample is calculated for the test of the mean of a normal distribution depending on increasing proportions of the first subsample in relation to the total sample size an upper bound on the loss of power due to early stopping is derived a general example is presented and rules for assessing the sample size in the second stage of the trial are given the problems of interpretation and precautions to be taken for applications are discussed finally the sources of bias for estimation in such designs are described

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

a neuro fuzzy system modeling with self constructing rule generationand hybrid svd based learning we propose an approach for neuro fuzzy system modeling a neuro fuzzy system for a given set of input output data is obtained in two steps first the data set is partitioned automatically into a set of clusters based on input similarity and output similarity tests membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster then a fuzzy if then rule is extracted from each cluster to form a fuzzy rule base second a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule base to decrease the size of the search space and to speed up the convergence we develop a hybrid learning algorithm which combines a recursive singular value decomposition based least squares estimator and the gradient descent method the proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points besides it learns faster consumes less memory and produces lower approximation errors than other methods

lorentzian lie n algebras we classify lie n algebras possessing an invariant lorentzian inner product we classify lie n algebras possessing an invariant lorentzian inner product

adaptive group sequential design with population enrichment in phase 3 randomized controlled trials with two binary co primary endpoints the use of co primary endpoints in drug development allows investigators to capture an experimental intervention s multidimensional effect more comprehensively than a single primary endpoint we propose the theoretical basis and development of an adaptive population enrichment design with co primary endpoints provide stage wise boundary values for futility and efficacy and discuss power under different efficacy configurations subgroup prevalence and analysis times using a pre specified decision criterion we considered a two arm two stage parallel group design where population enrichment occurs at the interim analysis by dropping any non responsive subgroups a test for efficacy is conducted only in the enriched population two binary endpoints are evaluated as co primary endpoints our trial objective is to determine whether the experimental intervention is superior to the control intervention with superiority required in both endpoints we define the stopping boundary using alpha spending functions using a 0 025 significance level for each endpoint we obtain the stage i threshold boundary values for futility and efficacy as 0 1040 and 2 2761 respectively and the stage ii boundary value for futility and efficacy is 2 2419 we show that in the presence of substantial heterogeneity of treatment effect we gain more power to observe an effect in the subgroup where the benefits are greater by allowing the dropping of non responsive subgroups at an early stage our design reduces the likelihood of obtaining false negative results due to inclusion of the heterogeneous treatment effects of both subgroups which would dilute the responsive subgroup s results

estimation in flexible two stage designs adaptive test designs for clinical trials allow for a wide range of data driven design adaptations using all information gathered until an interim analysis the basic principle is to use a test statistics which is invariant with respect to the design adaptations under the null hypothesis this allows for a control of the type i error rate for the primary hypothesis even for adaptations not specified a priori in the study protocol estimation is usually another important part of a clinical trial however is more difficult in adaptive designs in this research paper we give an overview of point and interval estimates for flexible designs and compare methods for typical sample size rules we also make some proposals for confidence intervals which have nominal coverage probability also after an unforeseen design adaptation and which contain the maximum likelihood estimate and the usual unadjusted confidence interval

modification of sample size in group sequential clinical trials in group sequential clinical trials sample size reestimation can be a complicated issue when it allows for change of sample size to be influenced by an observed sample path our simulation studies show that increasing sample size based on an interim estimate of the treatment difference can substantially inflate the probability of type i error in most practical situations a new group sequential test procedure is developed by modifying the weights used in the traditional repeated significance two sample mean test the new test has the type i error probability preserved at the target level and can provide a substantial gain in power with the increase of sample size generalization of the new procedure is discussed

a robust ilu with pivoting based on monitoring the growth of the inverse factors abstract an incomplete lu decomposition with pivoting is presented that progressively monitors the growth of the inverse factors of l u the information on the growth of the inverse factors is used as feedback for dropping entries in l and u this method often yields a robust and effective preconditioner especially when the system is highly indefinite numerical examples demonstrate the effectiveness of this approach

adaptive sample size calculations in group sequential trials a method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed without exaggerating the type i error rate this method enables data driven sample size reassessments during the course of the study it uses the stopping boundaries of the classical group sequential tests furthermore exact test procedures may be derived for a wide range of applications the procedure is compared with the classical designs in terms of power and expected sample size

simple facts concerning nambu algebras a class of substitution equations arising in the extension of jacobi identity for n gebras is studied and solved graded bracket and cohomology adapted to the study of formal deformations are presented new identities in the case of nambu lie algebras are proved the triviality in the gerstenhaber sense of certain deformed n skew symmetric brackets satisfying the leibniz rule with respect to a star product is shown for n 3

on the modification of an eigenvalue problem that preserves an eigenspace eigenvalue problems arise in many application areas ranging from computational fluid dynamics to information retrieval in these fields we are often interested in only a few eigenvalues and corresponding eigenvectors of a sparse matrix in this paper we comment on the modifications of the eigenvalue problem that can simplify the computation of those eigenpairs these transformations allow us to avoid difficulties associated with non hermitian eigenvalue problems such as the lack of reliable non hermitian eigenvalue solvers by mapping them into generalized hermitian eigenvalue problems also they allow us to expose and explore parallelism they require knowledge of a selected eigenvalue and preserve its eigenspace the positive definiteness of the hermitian part is inherited by the matrices in the generalized hermitian eigenvalue problem the position of the selected eigenspace in the ordering of the eigenvalues is also preserved under certain conditions the effect of using approximate eigenvalues in the transformation is analyzed and numerical experiments are presented

on the bias of maximum likelihood estimation following a sequential test summary tlhe bias of maximum likelihood estimates calculated at the end of a sequential procedure is investigated for the two sequential designs considered in detail the sequential probability ratio test and the triangular test this bias is appreciable a method of calculating an adjusted estimate with reduced bias is described and an approximation to the standard error of the new estimate is provided examples of the implementation of the method are given and its advantages and disadvantages relative to alternative approaches are discussed