saddle point least squares iterative solvers for the time harmonic maxwell equations abstract we introduce a least squares method for discretizing the time harmonic maxwell equations written as a first order system the method belongs to the recently introduced least squares approach called the saddle point least squares spls method see bacuta and qirko 2015 2017 it is related to two known methods the bramble and pasciak least squares method for solving electromagnetic problems bramble and pasciak 2004 bramble et al 2005 and the minimum residual dpg method of demkowicz and gopalakrishnan 2010 2013 the proposed discretization is based on reduction to a symmetric saddle point formulation on spaces of piecewise polynomial functions the edge elements are avoided a discrete test space is chosen first among standard conforming finite element spaces and the discrete trial space is chosen second as the image of the first order maxwell differential operator on the discrete test space for the proposed iterative processes a nodal basis for the trial space is not required and the main inversion at each step is done only on the test space the method is efficient for both convex and non convex domains and for a various range of frequencies as shown by the numerical results

a saddle point least squares approach to mixed methods we investigate new pde discretization approaches for solving variational formulations with different types of trial and test spaces the general mixed formulation we consider assumes a stability lbb condition and a data compatibility condition at the continuous level we expand on the bramble pasciak s least square formulation for solving such problems by providing new ways to choose approximation spaces and new iterative processes to solve the discrete formulations our proposed method has the advantage that a discrete inf sup condition is automatically satisfied by natural choices of test spaces first and corresponding trial spaces second in addition for the proposed iterative solver a nodal basis for the trial space is not required applications of the new approach include discretization of first order systems of pdes such as div curl systems and time harmonic maxwell equations

false discovery control in large scale spatial multiple testing summary the paper develops a unified theoretical and computational framework for false discovery control in multiple testing of spatial signals we consider both pointwise and clusterwise spatial analyses and derive oracle procedures which optimally control the false discovery rate false discovery exceedance and false cluster rate a data driven finite approximation strategy is developed to mimic the oracle procedures on a continuous spatial domain our multiple testing procedures are asymptotically valid and can be effectively implemented using bayesian computational algorithms for analysis of large spatial data sets numerical results show that the procedures proposed lead to more accurate error control and better power performance than conventional methods we demonstrate our methods for analysing the time trends in tropospheric ozone in eastern usa

multiple testing with heterogeneous multinomial distributions summary standard false discovery rate fdr procedures can provide misleading inference when testing multiple null hypotheses with heterogeneous multinomial data for example in the motivating study the goal is to identify species of bacteria near the roots of wheat plants rhizobacteria that are moderately or strongly associated with productivity however standard procedures discover the most abundant species even when their association is weak and fail to discover many moderate and strong associations when the species are not abundant this article provides a new fdr controlling method based on a finite mixture of multinomial distributions and shows that it tends to discover more moderate and strong associations and fewer weak associations when the data are heterogeneous across tests the new method is applied to the rhizobacteria data and performs favorably over competing methods

online rules for control of false discovery rate and false discovery exceedance multiple hypothesis testing is a core problem in statistical inference and arises in almost every scientific field given a set of null hypotheses mathcal h n h 1 ldots h n benjamini and hochberg j r stat soc ser b stat methodol 57 1995 289 300 introduced the false discovery rate mathrm fdr which is the expected proportion of false positives among rejected null hypotheses and proposed a testing procedure that controls mathrm fdr below a pre assigned significance level nowadays mathrm fdr is the criterion of choice for large scale multiple hypothesis testing in this paper we consider the problem of controlling mathrm fdr in an online manner concretely we consider an ordered possibly infinite sequence of null hypotheses mathcal h h 1 h 2 h 3 ldots where at each step i the statistician must decide whether to reject hypothesis h i having access only to the previous decisions this model was introduced by foster and stine j r stat soc ser b stat methodol 70 2008 429 444 we study a class of generalized alpha investing procedures first introduced by aharoni and rosset j r stat soc ser b stat methodol 76 2014 771 794 we prove that any rule in this class controls online mathrm fdr provided p values corresponding to true nulls are independent from the other p values earlier work only established mathrm mfdr control next we obtain conditions under which generalized alpha investing controls mathrm fdr in the presence of general p values dependencies we also develop a modified set of procedures that allow to control the false discovery exceedance the tail of the proportion of false discoveries finally we evaluate the performance of online procedures on both synthetic and real data comparing them with offline approaches such as adaptive benjamini hochberg

a saddle point least squares approach for primal mixed formulations of second order pdes abstract we present a saddle point least squares spls method for discretizing second order elliptic problems written as primal mixed variational formulations a stability lbb condition and a data compatibility condition at the continuous level are automatically satisfied the proposed discretization method follows a general spls approach and has the advantage that a discrete inf sup condition is automatically satisfied for standard choices of the test and trial spaces for the proposed iterative processes a nodal basis for the trial space is not required efficient preconditioning techniques that involve inversion only on the test space can be considered stability and approximation properties for two choices of discrete spaces are investigated applications of the new approach include discretization of second order problems with highly oscillatory coefficient interface problems and higher order approximation of the flux for elliptic problems with smooth coefficients

nonvanishing of siegel poincare series we prove that under suitable conditions certain siegel poincare series of exponential type of even integer weight and degree 2 do not vanish identically we also find estimates for twisted kloosterman sums and salie sums in all generality

exceedance control of the false discovery proportion multiple testing methods to control the false discovery rate the expected proportion of falsely rejected null hypotheses among all rejections have received much attention it can be valuable to control not the mean of this false discovery proportion fdp but rather the probability that the fdp exceeds a specified bound in this article we construct a general class of methods for exceedance control of fdp based on inverting tests of uniformity the method also produces a confidence envelope for the fdp as a function of rejection threshold we discuss how to select a procedure with good power

false discovery control for random fields this article extends false discovery rates to random fields for which there are uncountably many hypothesis tests we develop a method for finding regions in the field s domain where there is a significant signal while controlling either the proportion of area or the proportion of clusters in which false rejections occur the method produces confidence envelopes for the proportion of false discoveries as a function of the rejection threshold from the confidence envelopes we derive threshold procedures to control either the mean or the specified tail probabilities of the false discovery proportion an essential ingredient of this construnction is a new algorithm to compute a confidence superset for the set of all true null locations we demonstrate our method with applications to scan statistics and functional neuroimaging

comparing connectomes across subjects and populations at different scales brain connectivity can be represented by a network that enables the comparison of the different patterns of structural and functional connectivity among individuals in the literature two levels of statistical analysis have been considered in comparing brain connectivity across groups and subjects 1 the global comparison where a single measure that summarizes the information of each brain is used in a statistical test 2 the local analysis where a single test is performed either for each node connection which implies a multiplicity correction or for each group of nodes connections where each subset is summarized by one single test in order to reduce the number of tests to avoid a penalizing multiplicity correction we comment on the different levels of analysis and present some methods that have been proposed at each scale we highlight as well the possible factors that could influence the statistical results and the questions that have to be addressed in such an analysis

permutation based simultaneous confidence bounds for the false discovery proportion when multiple hypotheses are tested interest is often in ensuring that the proportion of false discoveries fdp is small with high confidence in this paper confidence upper bounds for the fdp are constructed which are simultaneous over all rejection cut offs in particular this allows the user to select a set of hypotheses post hoc such that the fdp lies below some constant with high confidence our method uses permutations to account for the dependence structure in the data so far only meinshausen provided an exact permutation based and computationally feasible method for simultaneous fdp bounds we provide an exact method which uniformly improves this procedure further we provide a generalization of this method it lets the user select the shape of the simultaneous confidence bounds this gives the user more freedom in determining the power properties of the method interestingly several existing permutation methods such as significance analysis of microarrays sam and westfall and young s maxt method are obtained as special cases

adaptive wavelet methods for saddle point problems optimal convergence rates in this paper an adaptive wavelet scheme for saddle point problems is developed and analyzed under the assumption that the underlying continuous problem satisfies the inf sup condition it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity this means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the error of the best wavelet n term approximation of the solution with respect to the relevant norms moreover the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom it is remarkable that compatibility constraints on the trial spaces such as the ladyzhenskaya babuska brezzi lbb condition do not arise in the second part the general results are applied to the stokes problem aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon the regularity of the solutions in certain besov scales is analyzed these results reveal under which circumstances the work accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements this in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the stokes problem

schur complements on hilbert spaces and saddle point systems for any continuous bilinear form defined on a pair of hilbert spaces satisfying the compatibility ladyshenskaya babusca brezzi condition symmetric schur complement operators can be defined on each of the two hilbert spaces in this paper we find bounds for the spectrum of the schur operators only in terms of the compatibility and continuity constants in light of the new spectral results for the schur complements we review the classical babusca brezzi theory find sharp stability estimates and improve a convergence result for the inexact uzawa algorithm we prove that for any symmetric saddle point problem the inexact uzawa algorithm converges provided that the inexact process for inverting the residual at each step has the relative error smaller than 1 3 as a consequence we provide a new type of algorithm for discretizing saddle point problems which combines the inexact uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions

an adaptive uzawa fem for the stokes problem convergence without the inf sup condition we introduce and study an adaptive finite element method fem for the stokes system based on an uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity we show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree k for velocity whereas for pressure the elements can be either discontinuous of degree k 1 or continuous of degree k 1 and k the popular taylor hood family is the sole example of stable elements included in the theory which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf sup condition we discuss the realization and complexity of the elliptic adaptive inner solver and provide consistent computational evidence that the resulting meshes are quasi optimal

strong control conservative point estimation and simultaneous conservative consistency of false discovery rates a unified approach summary the false discovery rate fdr is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses benjamini and hochberg introduced this quantity and proved that a particular step up p value method controls the fdr storey introduced a point estimate of the fdr for fixed significance regions the former approach conservatively controls the fdr at a fixed predetermined level and the latter provides a conservatively biased estimate of the fdr for a fixed predetermined significance region in this work we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent in particular the fdr point estimates can be used to define valid fdr controlling procedures in the asymptotic setting we also show that the point estimates can be used to estimate the fdr conservatively over all significance regions simultaneously which is equivalent to controlling the fdr at all levels simultaneously the main tool that we use is to translate existing fdr methods into procedures involving empirical processes this simplifies finite sample proofs provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence

complex multiplication cycles and a conjecture of beilinson and bloch a generalization of the conjecture of birch and swinnerton dyer is investigated using complex multiplication cycles on a particular kuga fiber variety a weak finiteness result consistent with the conjecture is proved the image of complex multiplication cycles under the abel jacobi map is computed explicitly the results provide numerical evidence supporting the conjecture they also give evidence for a relationship between complex multiplication cycles and a modular form of weight 5 2 and raise questions for further investigation

harmonic maass forms of weight 1 the object of this paper is to initiate a study of the fourier coefficients of a weight one harmonic maass form and relate them to the complex galois representation associated to a weight one newform which is the form s image under a certain differential operator in this paper our focus will be on weight one dihedral newforms of prime level p 3 mod 4 in this case we give properties of the fourier coefficients that are similar to and sometimes reduce to cases of stark s conjectures on derivatives of l functions we also give a new modular interpretation of certain products of differences of singular moduli studied by gross and zagier finally we provide some numerical evidence that the fourier coefficients of a mock modular form whose shadow is exotic are similarly related to the associated complex galois representation

operating characteristics and extensions of the false discovery rate procedure summary we investigate the operating characteristics of the benjamini hochberg false discovery rate procedure for multiple testing this is a distribution free method that controls the expected fraction of falsely rejected null hypotheses among those rejected the paper provides a framework for understanding more about this procedure we first study the asymptotic properties of the deciding point d that determines the critical p value from this we obtain explicit asymptotic expressions for a particular risk function we introduce the dual notion of false non rejections and we consider a risk function that combines the false discovery rate and false non rejections we also consider the optimal procedure with respect to a measure of conditional risk

cycle integrals of the j function and mock modular forms in this paper we construct certain mock modular forms of weight 1 2 whose fourier coecients are given in terms of cycle integrals of the modular j function their shadows are weakly holomorphic forms of weight 3 2 these new mock modular forms occur as holomorphic parts of weakly harmonic maass forms we also construct a generalized mock modular form of weight 1 2 having a real quadratic class number times a regulator as a fourier coecient as an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers the fourier coecients of these modular integrals are given in terms of cycle integrals of modular functions such a modular integral can be interpreted in terms of a shimura type lift of a mock modular form of weight 1 2 and yields a real quadratic analogue of a borcherds product

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

cfl condition and boundary conditions for dgm approximation of convection diffusion we propose a general method for the design of discontinuous galerkin methods dgms for nonstationary linear equations the method is based on a particular splitting of the bilinear forms that appear in the weak dgm we prove that an appropriate time splitting gives a stable linear explicit scheme whatever the order of the polynomial approximation numerical results are presented

a unified approach for uzawa algorithms we present a unified approach in analyzing uzawa iterative algorithms for saddle point problems we study the classical uzawa method the augmented lagrangian method and two versions of inexact uzawa algorithms the target application is the stokes system but other saddle point systems e g arising from mortar methods or lagrange multipliers methods can benefit from our study we prove convergence of uzawa algorithms and find optimal rates of convergence in an abstract setting on finite or infinite dimensional hilbert spaces the results can be used to design multilevel or adaptive algorithms for solving saddle point problems the discrete spaces do not have to satisfy the lbb stability condition

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

saddle point least squares preconditioning of mixed methods abstract we present a simple way to discretize and precondition mixed variational formulations our theory connects with and takes advantage of the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric positive definite operators efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete spaces that are always compatible are provided for the proposed discrete spaces and solvers a basis is needed only for the test spaces and assembly of a global saddle point system is avoided we prove sharp approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete inf sup and sup sup constants of the pair of discrete spaces

simultaneous control of all false discovery proportions in large scale multiple hypothesis testing closed testing procedures are classically used for familywise error rate fwer control but they can also be used to obtain simultaneous confidence bounds for the false discovery proportion fdp in all subsets of the hypotheses in this paper we investigate the special case of closed testing with simes local tests we construct a novel fast and exact shortcut which we use to investigate the power of this method when the number of hypotheses goes to infinity we show that if a minimal amount of signal is present the average power to detect false hypotheses at any desired fdp level does not vanish additionally we show that the confidence bounds for fdp are consistent estimators for the true fdp for every non vanishing subset for the case of a finite number of hypotheses we show connections between simes based closed testing and the procedure of benjamini and hochberg

on quasi shift splitting iteration method for a class of saddle point problems abstract motivated by a technique of scale splitting for matrix we put forward a quasi shift splitting named after qss iteration method to address a class of large scale sparse saddle point problems this novel method also naturally leads to the corresponding qss preconditioner in addition an extrapolated variant of qss named after eqss iteration method is constructed further some useful convergence properties of the qss and eqss iteration methods and spectral properties of qss preconditioner are studied finally a saddle point equations stem from model navier stokes problem is supplied to illustrate the effectiveness of our new two methods and qss preconditioner

multiple hypotheses testing and expected number of type i errors the performance of multiple test procedures with respect to error control is an old issue assuming that all hypotheses are true we investigate the behavior of the expected number of type i errors ene as a characteristic of certain multiple tests controlling the familywise error rate fwer or the false discovery rate fdr at a prespecified level we derive explicit formulas for the distribution of the number of false rejections as well as for the ene for single step step down and step up procedures based on independent p values moreover we determine the corresponding asymptotic distributions of the number of false rejections as well as explicit formulae for the ene if the number of hypotheses tends to infinity in case of fwer control we mostly obtain poisson distributions and in one case a geometric distribution as limiting distributions in case of fdr control we obtain limiting distributions which are apparently not named in the literature surprisingly the ene is bounded by a small number regardless of the number of hypotheses under consideration finally it turns out that in case of dependent test statistics the ene behaves completely differently compared to the case of independent test statistics

selective inference on multiple families of hypotheses summary in many complex multiple testing problems the hypotheses are divided into families given the data families with evidence for true discoveries are selected and hypotheses within them are tested neither controlling the error rate in each family separately nor controlling the error rate over all hypotheses together can assure some level of confidence about the filtration of errors within the selected families we formulate this concern about selective inference in its generality for a very wide class of error rates and for any selection criterion and present an adjustment of the testing level inside the selected families that retains control of the expected average error over the selected families

prym varieties and teichmuller curves this paper gives a uniform construction of infinitely many primitive teichmuller curves v mg for g 2 3 and 4

petersson norms of not necessarily cuspidal jacobi modular forms and applications abstract we extend the usual notion of petersson inner product on the space of cuspidal jacobi forms to include non cuspidal forms as well this is done by examining carefully the relation between certain growth killing invariant differential operators on h 2 and those on h 1 c here h n denotes the siegel upper half space of degree n as applications we can understand better the growth of petersson norms of fourier jacobi coefficients of klingen eisenstein series which in turn has applications to finer issues about representation numbers of quadratic forms and as a by product we also show that any siegel modular form of degree 2 is determined by its fundamental fourier coefficients

residual reduction algorithms for nonsymmetric saddle point problems in this paper we introduce and analyze uzawa algorithms for non symmetric saddle point systems convergence for the algorithms is established based on new spectral results about schur complements a new uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state navier stokes equations

notes on a saddle point reformulation of mixed variational problems abstract we summarize some general ideas regarding approximation of mixed variational problems using saddle point reformulation we consider the concepts of optimal and almost optimal or test norm and provide estimates for the continuity and stability constants a preconditioning strategy for solving the discrete mixed formulations is used in combination with the special test norms we further provide a choice for a discrete trial space that depends on the choice of a standard test space and leads to discrete stability when using the appropriate test norm examples to illustrate how the stability of the saddle point discretization can be improved using special test norms are included

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

some results on false discovery rate in stepwise multiple testing procedures the concept of false discovery rate fdr has been receiving increasing attention by researchers in multiple hypotheses testing this paper produces some theoretical results on the fdr in the context of stepwise multiple testing procedures with dependent test statistics it was recently shown by benjamini and yekutieli that the benjamini hochberg step up procedure controls the fdr when the test statistics are positively dependent in a certain sense this paper strengthens their work by showing that the critical values of that procedure can be used in a much more general stepwise procedure under similar positive dependency it is also shown that the fdr controlling benjamini liu step down procedure originally developed for independent test statistics works even when the test statistics are positively dependent in some sense an explicit expression for the fdr of a generalized stepwise procedure and an upper bound to the fdr of a step down procedure are obtained in terms of probability distributions of ordered components of dependent random variables before establishing the main results 1 introduction the main purpose of this paper is to present some formulas and inequalities providing theoretical insight into when and what stepwise procedures control the false discovery rate fdr at a prespecified level in multiple hypotheses testing the fdr first introduced in multiple testing by benjamini and hochberg 1995 is the expected proportion of erroneously rejected null hypotheses with v and r representing respectively the number of true null hypotheses rejected and the total number of null hypotheses rejected in a multiple testing procedure let q v r if r 0a nd 0i fr 0 then e q is the fdr of that procedure as they have argued it is an appropriate error rate to control in many practical situations particularly where a large number of null hypotheses are involved they also put forward an fdr controlling stepup procedure that is more powerful than comparable procedures controlling the traditional familywise error fwe rate when the underlying test statistics are independent benjamini and liu 1999 recently introduced a step down procedure with the same property the concept of fdr is now being used in a wide variety of applications abramovich and benjamini 1996 basford and tukey 1997 drigalenko

on the nonlinear inexact uzawa algorithm for saddle point problems this paper discusses convergence behavior of the nonlinear inexact uzawa algorithm for solving saddle point problems presented in a recent paper of bramble pasciak and vassilev siam j numer anal 34 1997 pp 1072 1092 it is shown that this algorithm converges under a condition weaker than that stated in their paper

cascadic multilevel algorithms for symmetric saddle point systems abstract in this paper we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems the algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems on each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a schur complement is implemented the level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error we prove new estimates that relate the iteration error and the residual for the constraint equation the new estimates are the key ingredients in imposing an efficient level change criterion the first iteration on each new level uses information about the best approximation of the discrete solution from the previous level the theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non increasing number of iterations on each level even though numerical results supporting the efficiency of the algorithms are presented for the stokes system the algorithms can be applied to a large class of boundary value problems including first order systems that can be reformulated at the continuous level as symmetric saddle point problems such as the maxwell equations

a stochastic process approach to false discovery control this paper extends the theory of false discovery rates fdr pioneered by benjamini and hochberg j roy statist soc ser b 57 1995 289 300 we develop a framework in which the false discovery proportion fdp the number of false rejections divided by the number of rejections is treated as a stochastic process after obtaining the limiting distribution of the process we demonstrate the validity of a class of procedures for controlling the false discovery rate the expected fdp we construct a confidence envelope for the whole fdp process from these envelopes we derive confidence thresholds for controlling the quantiles of the distribution of the fdp as well as controlling the number of false discoveries we also investigate methods for estimating the p value distribution

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

computational scales of sobolev norms with application to preconditioning this paper provides a framework for developing computationally efficient multilevel preconditioners and representations for sobolev norms specifically given a hilbert space v and a nested sequence of subspaces v 1 c v 2 c c v we construct operators which are spectrally equivalent to those of the form a k k q k q k 1 here k k 1 2 are positive numbers and q k is the orthogonal projector onto v k with q 0 0 we first present abstract results which show when a is spectrally equivalent to a similarly constructed operator a defined in terms of an approximation q k of q k for k 1 2 we show that these results lead to efficient preconditioners for discretizations of differential and pscudo differential operators of positive and negative order these results extend to sums of operators for example singularly perturbed problems such as i e can be preconditioned uniformly independently of the parameter e we also show how to precondition an operator which results from tikhonov regularization of a problem with noisy data finally we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition

a direct approach to false discovery rates summary multiple hypothesis testing involves guarding against much more complicated errors than single hypothesis testing whereas we typically control the type i error rate for a single hypothesis test a compound error rate is controlled for multiple hypothesis tests for example controlling the false discovery rate fdr traditionally involves intricate sequential p value rejection methods based on the observed data whereas a sequential p value method fixes the error rate and estimates its corresponding rejection region we propose the opposite approach we fix the rejection region and then estimate its corresponding error rate this new approach offers increased applicability accuracy and power we apply the methodology to both the positive false discovery rate pfdr and fdr and provide evidence for its benefits it is shown that pfdr is probably the quantity of interest over fdr also discussed is the calculation of the q value the pfdr analogue of the p value which eliminates the need to set the error rate beforehand as is traditionally done some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the benjamini hochberg fdr method

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

global voxel and cluster tests by theory and permutation for a difference between two groups of structural mr images of the brain the authors describe almost entirely automated procedures for estimation of global voxel and cluster level statistics to test the null hypothesis of zero neuroanatomical difference between two groups of structural magnetic resonance imaging mri data theoretical distributions under the null hypothesis are available for 1 global tissue class volumes 2 standardized linear model analysis of variance anova and ancova coefficients estimated at each voxel and 3 an area of spatially connected clusters generated by applying an arbitrary threshold to a two dimensional 2 d map of normal statistics at voxel level the authors describe novel methods for economically ascertaining probability distributions under the null hypothesis with fewer assumptions by permutation of the observed data nominal type i error control by permutation testing is generally excellent whereas theoretical distributions may be over conservative permutation has the additional advantage that it can be used to test any statistic of interest such as the sum of suprathreshold voxel statistics in a cluster or cluster mass regardless of its theoretical tractability under the null hypothesis these issues are illustrated by application to mri data acquired from 18 adolescents with hyperkinetic disorder and 16 control subjects matched for age and gender

estimates for fourier coefficients of siegel cusp forms the proof of 1 was based on appropriate estimates both for the fourier coefficients of cusp forms on the jacobi group f1 2 more precisely let

false discovery proportion estimation by permutations confidence for significance analysis of microarrays summary significance analysis of microarrays is a highly popular permutation based multiple testing method that estimates the false discovery proportion fdp the fraction of false positive results among all rejected hypotheses perhaps surprisingly until now this method had no known properties this paper extends significance analysis of microarrays by providing 1 upper confidence bounds for the fdp so that exact confidence statements can be made as a special case an estimate of the fdp is obtained that underestimates the fdp with probability at most 0 5 moreover using a closed testing procedure this paper decreases the upper bounds and estimates in such a way that the confidence level is maintained we base our methods on a general result on exact testing with random permutations

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

arithmetic of elliptic curves upon quadratic extension this paper is a study of variations in the rank of the mordell weil group of an elliptic curve e defined over a number field f as one passes to quadratic extensions k of f let s k be the selmer group for multiplication by 2 on e k in analogy with genus theory we describe s k in terms of various objects defined over f and the local norm indices dimf2 ft norm ah for each completion fv of f in particular we show that dim s k dim e k 2 has the same parity as zi we compute i when e has good or multiplicative reduction modulo v assuming that the 2 primary component of the tate shafare vitch group u1 k is finite as conjectured we obtain the parity of rank e k for semistable elliptic curves defined over q and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of birch and swinnerton dyer 1 introduction let e be an elliptic curve defined over a number field f our motivating question is this what can be said about variations in the rank of the mordell weil group e k over quadratic extensions k f dl 2 1 let e d denote the twist of e which becomes isomorphic to e over k but not over f concretely if we choose for e a model over f of the form y2 f x then a model for d is given by dy2 f x if a denotes the generator of gal k f then e f can be identified with the l eigenspace and e d f with the l eigenspace of a acting on e k it follows that rank e k rank e f rank e d f an equiv alent question therefore is to describe changes in the rank of e d f as d varies for certain specific curves defined over q this question has been discussed for

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

two new variants of nonlinear inexact uzawa algorithms for saddle point problems in this paper we consider some nonlinear inexact uzawa methods for iteratively solving linear saddle point problems by means of a new technique we first give an essential improvement on the convergence results of bramble paschiak vassilev for a known nonlinear inexact uzawa algorithm then we propose two new algorithms which can be viewed as a combination of the known nonlinear inexact uzawa method with the classical steepest descent method and conjugate gradient method respectively the two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved including the preconditioned schur complement numerical results of the algorithms applied for the stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms

on a theta lift related to the shintani lift abstract we study the millson theta lift which maps weight 2 k to weight 1 2 k harmonic weak maass forms for k z k 0 and which is closely related to the classical shintani lift from weight 2 k 2 to weight k 3 2 cusp forms we compute the fourier expansion of the theta lift and show that it involves twisted traces of cm values and geodesic cycle integrals of the input function as an application we recover zagier s generating series of twisted traces of singular moduli of weight 1 2 as millson lifts

on the p adic height of heegner cycles 0 1 in gr za gross and zagier proved a remarkable formula which relates the first derivative of the l function of a modular form f of weight 2 on 0 n over a suitable imaginary quadratic field and the neron tate height of a heegner point over the same quadratic field on the f part of the jacobian of the modular curve x0 n later perrin riou pr 2 proved a p adic version of this formula kolyvagin s method of euler systems ko combined with the formula of gross zagier proves the conjecture of birch and swinnerton dyer up to a controlled rational factor for all modular elliptic curves over q with analytic rank 1

lifting maps from a vector space of jacobi cusp forms to a subspace of elliptic modular forms in this paper we construct a lifting map from a vector space of generalized jacobi cusp forms to a certain subspace of elliptic cusp forms and vice versa such that both mappings are adjoint with respect to the petersson scalar products

a new approximation technique for div curl systems in this paper we describe an approximation technique for div curl systems based in l where is a domain in 3 we formulate this problem as a general variational problem with different test and trial spaces the analysis requires the verification of an appropriate inf sup condition this results is a very weak formulation where the solution space is l and the data reside in various negative norm spaces subsequently we consider finite element approximations based on this weak formulation the main approach of this paper involves the development of stable pairs of discrete test and trial spaces with this approach we enlarge the test space so that the discrete inf sup condition holds and use a negative norm least squares formulation to reduce to a uniquely solvable linear system this leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low sobolev regularity e g h with 0 s 1 2 the resulting algebraic equations are symmetric positive definite and well conditioned a second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned some numerical results are also presented