ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

continuous subgroups of the generalized schrodinger groups some general results on the subalgebras of the lie algebra asch n of the generalized schrodinger group sch n and on the subalgebras of the lie algebra asch n of the generalized extended schrodinger group sch n have been obtained the subalgebra structure of the algebras asch n and asch n are studied with respect to inner automorphisms of the groups sch n and sch n respectively the maximal abelian subalgebras and the one dimensional subalgebras of the algebras asch n and asch n have been explicitly found the full classification of the subalgebras of the algebras asch 3 asch 3 which are nonconjugate to the subalgebras of asch 2 asch 2 respectively has been carried out

estimating the null and the proportion of nonnull effects in large scale multiple comparisons an important issue raised by efron in the context of large scale multiple comparisons is that in many applications the usual assumption that the null distribution is known is incorrect and seemingly negligible differences in the null may result in large differences in subsequent studies this suggests that a careful study of estimation of the null is indispensable in this article we consider the problem of estimating a null normal distribution and a closely related problem estimation of the proportion of nonnull effects we develop an approach based on the empirical characteristic function and fourier analysis the estimators are shown to be uniformly consistent over a wide class of parameters we investigate the numerical performance of the estimators using both simulated and real data in particular we apply our procedure to the analysis of breast cancer and human immunodeficiency virus microarray datasets the estimators perform favorably compared with existing methods

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

the exotic conformal galilei algebra and nonlinear partial differential equations abstract the conformal galilei algebra cga and the exotic conformal galilei algebra ecga are applied to construct partial differential equations pdes and systems of pdes which admit these algebras we show that there are no single second order pdes invariant under the cga but systems of pdes can admit this algebra moreover a wide class of nonlinear pdes exists which are conditionally invariant under cga it is further shown that there are systems of nonlinear pdes admitting ecga with the realisation obtained very recently in d martelli y tachikawa comments on galilei conformal field theories and their geometric realisation preprint arxiv 0903 5184v2 hep th 2009 moreover wide classes of nonlinear systems invariant under two different 10 dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented

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

nonclassical symmetry reductions of the three dimensional incompressible navier stokes equations the nonclassical reduction method as pioneered by bluman and cole j meth mech 18 1025 42 is used to examine symmetries of the full three dimensional unsteady incompressible navier stokes equations of fluid mechanics the procedure when applied to a system of partial differential equations yields reduced sets of equations with one fewer independent variables we find eight possibilities for reducing the navier stokes equations in the three spatial and one temporal dimensions to sets of partial differential equations in three independent variables some of these reductions are derivable using the lie group method of classical symmetries but the remainder are genuinely nonclassical further investigations of one of our eight forms shows how it is possible to derive novel exact solutions of the navier stokes equations by the nonclassical method

on the benjamini hochberg method we investigate the properties of the benjamini hochberg method for multiple testing and of a variant of storey s generalization of it extending and complementing the asymptotic and exact results available in the literature results are obtained under two different sets of assumptions and include asymptotic and exact expressions and bounds for the proportion of rejections the proportion of incorrect rejections out of all rejections and two other proportions used to quantify the efficacy of the method

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

optimal rates of convergence for estimating the null density and proportion of nonnull effects in large scale multiple testing an important estimation problem that is closely related to large scale multiple testing is that of estimating the null density and the proportion of nonnull effects a few estimators have been introduced in the literature however several important problems including the evaluation of the minimax rate of convergence and the construction of rate optimal estimators remain open in this paper we consider optimal estimation of the null density and the proportion of nonnull effects both minimax lower and upper bounds are derived the lower bound is established by a two point testing argument where at the core is the novel construction of two least favorable marginal densities f 1 and f 2 the density f 1 is heavy tailed both in the spatial and frequency domains and f 2 is a perturbation of f 1 such that the characteristic functions associated with f 1 and f 2 match each other in low frequencies the minimax upper bound is obtained by constructing estimators which rely on the empirical characteristic function and fourier analysis the estimator is shown to be minimax rate optimal compared to existing methods in the literature the proposed procedure not only provides more precise estimates of the null density and the proportion of the nonnull effects but also yields more accurate results when used inside some multiple testing procedures which aim at controlling the false discovery rate fdr the procedure is easy to implement and numerical results are given

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

reduction and exact solutions of the navier stokes equations the authors construct a complete set of g 1 3 inequivalent ansatze of codimension 1 for the navier stokes ns field which reduce the ns equations to systems of ordinary differential equations ode having solved these odes they thereby obtain solutions of the ns equations formulae of group multiplication of solutions are given several non lie ansatze are discussed

on the classification of subalgebras of the conformal algebra with respect to inner automorphisms we give a complete justification of the classification of inequivalent subalgebras of the conformal algebra with respect to the inner automorphisms of the conformal group and we perform the classification of the subalgebras of the conformal algebra ac 1 3

multiple testing of composite null hypotheses in heteroscedastic models in large scale studies the true effect sizes often range continuously from zero to small to large and are observed with heteroscedastic errors in practical situations where the failure to reject small deviations from the null is inconsequential specifying an indifference region or forming composite null hypotheses can greatly reduce the number of unimportant discoveries in multiple testing the heteroscedasticity issue poses new challenges for multiple testing with composite nulls in particular the conventional framework in multiple testing which involves rescaling or standardization is likely to distort the scientific question we propose the concept of a composite null distribution for heteroscedastic models and develop an optimal testing procedure that minimizes the false nondiscovery rate subject to a constraint on the false discovery rate the proposed approach is different from conventional methods in that the effect size statistical significance and multiplicity issues are addressed in

controlling the familywise error rate with plug in estimator for the proportion of true null hypotheses summary estimation of the number or proportion of true null hypotheses in multiple testing problems has become an interesting area of research the first important work in this field was performed by schweder and spjotvoll among others they proposed to use plug in estimates for the proportion of true null hypotheses in multiple test procedures to improve the power we investigate the problem of controlling the familywise error rate fwer when such estimators are used as plug in estimators in single step or step down multiple test procedures first we investigate the case of independent p values under the null hypotheses and show that a suitable choice of plug in estimates leads to control of fwer in single step procedures we also investigate the power and study the asymptotic behaviour of the number of false rejections although step down procedures are more difficult to handle we briefly consider a possible solution to this problem anyhow plug in step down procedures are not recommended here for dependent p values we derive a condition for asymptotic control of fwer and provide some simulations with respect to fwer and power for various models and hypotheses

bistable kinetic model driven by correlated noises steady state analysis a simple rule to obtain the fokker planck equation for a general one dimensional system driven by correlated gaussian white noises is proved by the functional method the fokker planck equation obtained in this paper is applied to the bistable kinetic model we find the following for the steady state 1 in the ensuremath alpha d parameter plane ensuremath alpha is the strength of the additive noise and d is the multiplicative noise strength the critical curve separating the unimodal and bimodal regions of the stationary probability distribution spd of the model is shown to be affected by ensuremath lambda the degree of correlation of the noises as ensuremath lambda is increased the area of the bimodal region in the ensuremath alpha d plane is contracted 2 when we take a point fixed in the ensuremath alpha d plane and increase ensuremath lambda the form of the spd changes from a bimodal to a unimodal structure 3 the positions of the extreme value of the spd of the model sensitively depend on the strength of the multiplicative noise and weakly depend on the additive noise strength 4 for ensuremath lambda 1 the case of perfectly correlated noises when the parameters ensuremath alpha and d take values in the neighborhood of the line ensuremath alpha d in the ensuremath alpha d plane the spd s corresponding to the points ensuremath alpha dg1 and ensuremath alpha d1 exhibit a very different shape of divergence therefore the ratio ensuremath alpha d 1 plays the role of a critical ratio

plots of p values to evaluate many tests simultaneously summary when a large number of tests are made possibly on the same data it is proposed to base a simultaneous evaluation of all the tests on a plot of cumulative p values using the observed significance probabilities the points corresponding to true null hypotheses should form a straight line while those for false null hypotheses should deviate from this line the line may be used to estimate the number of true hypotheses the properties of the method are studied in some detail for the problems of comparing all pairs of means in a one way layout testing all correlation coefficients in a large correlation matrix and the evaluation of all 2 x 2 subtables in a contingency table the plot is illustrated on real data

vortices and invariant surfaces generated by symmetries for the 3d navier stokes equations abstract we show that certain infinitesimal operators of the lie point symmetries of the incompressible 3d navier stokes equations give rise to vortex solutions with different characteristics this approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity and the vortex stretching vector s where s is the strain matrix the symmetry algebra associated with the navier stokes equations turns out to be infinite dimensional new vortical structures generalizing in some cases well known configurations such as for example the burgers and lundgren solutions are obtained and discussed in relation to the value of the dynamic angle arctan s s a systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed and the corresponding invariant surfaces are recognized

optimal screening and discovery of sparse signals with applications to multistage high throughput studies summary a common feature in large scale scientific studies is that signals are sparse and it is desirable to narrow down significantly the focus to a much smaller subset in a sequential manner we consider two related data screening problems one is to find the smallest subset such that it virtually contains all signals and another is to find the largest subset such that it essentially contains only signals these screening problems are closely connected to but distinct from the more conventional signal detection or multiple testing problems we develop phase transition diagrams to characterize the fundamental limits in simultaneous inference and derive data driven screening procedures which control the error rates with near optimality properties applications in the context of multistage high throughput studies are discussed

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

lower bounds for the number of false null hypotheses for multiple testing of associations under general dependence structures we propose probabilistic lower bounds for the number of false null hypotheses when testing multiple hypotheses of association simultaneously the bounds are valid under general and unknown dependence structures between the test statistics the power of the proposed estimator to detect the full proportion of false null hypotheses is discussed and compared to other estimators the proposed estimator is shown to deliver a tight probabilistic lower bound for the number of false null hypotheses in a multiple testing situation even under strong dependence between test statistics

on subalgebras of the lie algebra of the extended poincare group p 1 n some general results on the subalgebras of the lie algebra ap 1 n of the extended poincare group p 1 n n 2 with respect to p 1 n conjugation have been obtained all subalgebras of ap 1 4 that are nonconjugate to the subalgebras of ap 1 4 are classified with respect to p 1 4 conjugation the list of representatives of each conjugacy class is presented

colored noise enhanced stability in a tumor cell growth system under immune response in this paper we investigate a mathematical model for describing the growth of tumor cell under immune response which is driven by cross correlation between multiplicative and additive colored noises as well as the nonzero cross correlation in between the expression of the mean first passage time mfpt is obtained by virtue of the steepest descent approximation it is found i when the noises are negatively cross correlated 0 then the escape is slower than in the case with no correlation moreover in the case of positive cross correlation the escape time has a maximum for a certain intensity of one of the noises i e the maximum for mfpt identifies the noise enhanced stability of the cancer state ii the effect of the cross correlation time 3 on the mfpt is completely opposite for 0 and 0 iii the self correlation times 1 and 2 of colored noises can enhance stability of the cancer state while the immune rate can reduce it

proportion of non zero normal means universal oracle equivalences and uniformly consistent estimators d0 we propose a general approach to construct the universal oracle equivalence of the proportion the construction is based on the underlying characteristic function the oracle equivalence reduces the problem of estimating the proportion to the problem of estimating the oracle which is relatively easier to handle in fact the oracle equivalence naturally yields a family of estimators for the proportion which are consistent under mild conditions uniformly across a wide class of parameters the approach compares favourably with recent works by meinshausen and rice and genovese and wasserman in particular the consistency is proved for an unprecedentedly broad class of situations the class is almost the largest that can be hoped for without further constraints on the model we also discuss various extensions of the approach report results on simulation experiments and make connections between the approach and several recent procedures in large scale multiple testing including the false discovery rate approach and the local false discovery rate approach

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

the limiting behavior of a modified maximal symmetric 2s spacing with applications this paper presents a solution to an open problem in astrophysics namely that of estimating nonparametrically the strength of the pulsed signal in a series of high energy photon arrival times the newly proposed estimator based on a modified maximal symmetric 2s spacing is shown to be strongly consistent and asymptotically normally distributed and a monte carlo study shows that its small and moderate sample behavior is very satisfactory additionally new results regarding the weak and strong limiting behavior of modified maximal 2 s spacings are derived

correlated noise in a logistic growth model the logistic differential equation is used to analyze cancer cell population in the presence of a correlated gaussian white noise we study the steady state properties of tumor cell growth and discuss the effects of the correlated noise it is found that the degree of correlation of the noise can cause tumor cell extinction

stochastic resonance in a bistable system subject to multiplicative and additive noise the stochastic resonance sr phenomenon in a bistable system under the simultaneous action of multiplicative and additive noise and periodic signal is studied by using the theory of signal to noise ratio snr in the adiabatic limit two cases have been considered the case of no correlations between multiplicative and additive noise and the case of correlations between two noises the expressions of the snr for both cases are obtained the effects of intensity of multiplicative and additive noise and the intensity of the correlations between noises on the snr are discussed for both cases respectively it is found that the existence of a maximum in the snr is the identifying characteristic of the sr phenomenon in the case of no correlations between multiplicative and additive noise the snr is independent of the initial condition of the system however the snr is not only dependent on the intensity of correlations between noises but also on the initial condition of the system in the presence of correlations between two noises

estimating the proportion of false null hypotheses among a large number of independently tested hypotheses we consider the problem of estimating the number of false null hypotheses among a very large number of independently tested hypotheses focusing on the situation in which the proportion of false null hypotheses is very small we propose a family of methods for establishing lower 100 1 a confidence bounds for this proportion based on the empirical distribution of the p values of the tests methods in this family are then compared in terms of ability to consistently estimate the proportion by letting 0 as the number of hypothesis tests increases and the proportion decreases this work is motivated by a signal detection problem that occurs in astronomy

continuous subgroups of the poincare group p 1 4 an exhaustive description of the nonsplitting subalgebras of the lp 1 4 algebra with respect to p 1 4 conjugation is presented

the optimal discovery procedure a new approach to simultaneous significance testing summary the neyman pearson lemma provides a simple procedure for optimally testing a single hypothesis when the null and alternative distributions are known this result has played a major role in the development of significance testing strategies that are used in practice most of the work extending single testing strategies to multiple tests has focused on formulating and estimating new types of significance measures such as the false discovery rate these methods tend to be based on p values that are calculated from each test individually ignoring information from the other tests i show here that one can improve the overall performance of multiple significance tests by borrowing information across all the tests when assessing the relative significance of each one rather than calculating p values for each test individually the optimal discovery procedure is introduced which shows how to maximize the number of expected true positive results for each fixed number of expected false positive results the optimality that is achieved by this procedure is shown to be closely related to optimality in terms of the false discovery rate the optimal discovery procedure motivates a new approach to testing multiple hypotheses especially when the tests are related as a simple example a new simultaneous procedure for testing several normal means is defined this is surprisingly demonstrated to outperform the optimal single test procedure showing that a method which is optimal for single tests may no longer be optimal for multiple tests connections to other concepts in statistics are discussed including stein s paradox shrinkage estimation and the bayesian approach to hypothesis testing

properties of higher criticism under strong dependence the problem of signal detection using sparse faint information is closely related to a variety of contemporary statistical problems including the control of false discovery rate and classification using very high dimensional data each problem can be solved by conducting a large number of simultaneous hypothesis tests the properties of which are readily accessed under the assumption of independence in this paper we address the case of dependent data in the context of higher criticism methods for signal detection short range dependence has no first order impact on performance but the situation changes dramatically under strong dependence there although higher criticism can continue to perform well it can be bettered using methods based on differences of signal values or on the maximum of the data the relatively inferior performance of higher criticism in such cases can be explained in terms of the fact that under strong dependence the higher criticism statistic behaves as though the data were partitioned into very large blocks with all but a single representative of each block being eliminated from the dataset

nonlinear galilei invariant pdes with infinite dimensional lie symmetry a class of nonlinear galilei invariant generalizations of the heat equation admitting infinite dimensional lie symmetry is presented lie symmetries and examples of exact solutions are found for some nonlinear equations generalizing the heat equation full sets of the absolute differential invariants for two non equivalent representations of the galilei algebra are constructed

class of self limiting growth models in the presence of nonlinear diffusion the source term in a reaction diffusion system in general does not involve explicit time dependence a class of self limiting growth models dealing with animal and tumor growth and bacterial population in a culture on the other hand are described by kinetics with explicit functions of time we analyze a reaction diffusion system to study the propagation of spatial front for these models

adaptive linear step up procedures that control the false discovery rate we provide a new two stage procedure in which the linear step up procedure is used in stage one to estimate mo providing a new level q which is used in the linear step up procedure in the second stage we prove that a general form of the two stage procedure controls the false discovery rate at the desired level q this framework enables us to study analytically the properties of other procedures that exist in the literature a simulation study is presented that shows that two stage adaptive procedures improve in power over the original procedure mainly because they provide tighter control of the false discovery rate we further study the performance of the current suggestions some variations of the procedures and previous suggestions in the case where the test statistics are positively dependent a case for which the original procedure controls the false discovery rate in the setting studied here the newly proposed two stage procedure is the only one that controls the false discovery rate the procedures are illustrated with two examples of biological importance

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

robustness and accuracy of methods for high dimensional data analysis based on student s t statistic summary student s t statistic is finding applications today that were never envisaged when it was introduced more than a century ago many of these applications rely on properties e g robustness against heavy tailed sampling distributions that were not explicitly considered until relatively recently we explore these features of the t statistic in the context of its application to very high dimensional problems including feature selection and ranking the simultaneous testing of many different hypotheses and sparse high dimensional signal detection robustness properties of the t ratio are highlighted and it is established that those properties are preserved under applications of the bootstrap in particular bootstrap methods correct for skewness and therefore lead to second order accuracy even in the extreme tails indeed it is shown that the bootstrap and also the more popular but less accurate t distribution and normal approximations are more effective in the tails than towards the middle of the distribution these properties motivate new methods e g bootstrap based techniques for signal detection that confine attention to the significant tail of a statistic

on integrable doebner goldin equations we integrate sub families of a family of nonlinear schrodinger equations proposed by doebner and goldin in the 1 1 dimensional case which have exceptional lie symmetries since the method of integration involves non local transformations of dependent and independent variables general solutions obtained include implicitly determined functions by properly specifying one of the arbitrary functions contained in these solutions we obtain broad classes of explicit square integrable solutions the physical significance and some analytical properties of the solutions obtained are briefly discussed

group invariant solutions without transversality we present a generalization of lie s method for finding the group invariant solutions to a system of partial differential equations our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables the theoretical basis for our method is provided by a general existence theorem for the invariant sections both local and global of a bundle on which a finite dimensional lie group acts a simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations the characterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics harmonic maps and general relativity this work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions

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

estimation and confidence sets for sparse normal mixtures for high dimensional statistical models researchers have begun to focus on situations which can be described as having relatively few moderately large coefficients such situations lead to some very subtle statistical problems in particular ingster and donoho and jin have considered a sparse normal means testing problem in which they described the precise demarcation or detection boundary meinshausen and rice have shown that it is even possible to estimate consistently the fraction of nonzero coordinates on a subset of the detectable region but leave unanswered the question of exactly in which parts of the detectable region consistent estimation is possible in the present paper we develop a new approach for estimating the fraction of nonzero means for problems where the nonzero means are moderately large we show that the detection region described by ingster and donoho and jin turns out to be the region where it is possible to consistently estimate the expected fraction of nonzero coordinates this theory is developed further and minimax rates of convergence are derived a procedure is constructed which attains the optimal rate of convergence in this setting furthermore the procedure also provides an honest lower bound for confidence intervals while minimizing the expected length of such an interval simulations are used to enable comparison with the work of meinshausen and rice where a procedure is given but where rates of convergence have not been discussed extensions to more general gaussian mixture models are also given

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

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

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

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

cross correlation enhanced stability in a tumor cell growth model with immune surveillance driven by cross correlated noises the transient properties of a tumor cell growth model with immune surveillance driven by cross correlated multiplicative and additive noises are investigated the explicit expression of extinction rate from the state of a stable tumor to the state of extinction is obtained based on the numerical computations we find the following i the intensity of multiplicative noise d and the intensity of additive noise enhance the extinction rate for the case of 0 i e denotes cross correlation intensity between two noises but for the case of 0 a critical noise intensity d or exists at which the extinction rate is the smallest d and at first weaken the extinction rate and then enhance it ii the immune rate and the cross correlation intensity play opposite roles on the extinction rate i e enhances the extinction rate of the tumor cell while weakens the extinction rate of the tumor cell namely the immune rate can enhance the extinction of the tumor cell and the cross correlation between two noises can enhance stability of the cancer state

adaptive false discovery rate control under independence and dependence in the context of multiple hypothesis testing the proportion 0 of true null hypotheses in the pool of hypotheses to test often plays a crucial role although it is generally unknown a priori a testing procedure using an implicit or explicit estimate of this quantity in order to improve its efficency is called adaptive in this paper we focus on the issue of false discovery rate fdr control and we present new adaptive multiple testing procedures with control of the fdr in a first part assuming independence of the p values we present two new procedures and give a unified review of other existing adaptive procedures that have provably controlled fdr we report extensive simulation results comparing these procedures and testing their robustness when the independence assumption is violated the new proposed procedures appear competitive with existing ones the overall best though is reported to be storey s estimator albeit for a specific parameter setting that does not appear to have been considered before in a second part we propose adaptive versions of step up procedures that have provably controlled fdr under positive dependence and unspecified dependence of the p values respectively in the latter case while simulations only show an improvement over non adaptive procedures in limited situations these are to our knowledge among the first theoretically founded adaptive multiple testing procedures that control the fdr when the p values are not independent

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

intertwining operators for l conformal galilei algebras and hierarchy of invariant equations the l conformal galilei algebra denoted by is a non semisimple lie algebra specified by a pair of parameters d l the algebra is regarded as a nonrelativistic analogue of the conformal algebra we derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry this is done by developing a representation theory such as verma modules singular vectors of and vector field representations for d 1 2

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

asymptotic results on adaptive false discovery rate controlling procedures based on kernel estimators the false discovery rate fdr is a commonly used type i error rate in multiple testing problems it is defined as the expected false discovery proportion fdp that is the expected fraction of false positives among rejected hypotheses when the hypotheses are independent the benjamini hochberg procedure achieves fdr control at any pre specified level by construction fdr control offers no guarantee in terms of power or type ii error a number of alternative procedures have been developed including plug in procedures that aim at gaining power by incorporating an estimate of the proportion of true null hypotheses in this paper we study the asymptotic behavior of a class of plug in procedures based on kernel estimators of the density of the p values as the number m of tested hypotheses grows to infinity in a setting where the hypotheses tested are independent we prove that these procedures are asymptotically more powerful in two respects i a tighter asymptotic fdr control for any target fdr level and ii a broader range of target levels yielding positive asymptotic power we also show that this increased asymptotic power comes at the price of slower non parametric convergence rates for the fdp these rates are of the form m k 2k 1 where k is determined by the regularity of the density of the p value distribution or equivalently of the test statistics distribution these results are applied to one and two sided tests statistics for gaussian and laplace location models and for the student model

a group analysis of the 2d navier stokes fourier equations abstract we study a 2 1 dimensional model of an incompressible thermoconducting fluid named navier stokes fourier system we apply a group theoretical analysis in correspondence of the generators of the symmetry group allowed by this model exact solutions are found some of them show possible interesting physical interpretations in our first exploration this feature is illustrated by dealing with simple special cases

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

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

stochastic resonance driven by two different kinds of colored noise in a bistable system the phenomenon of stochastic resonance in a bistable nonlinear system is investigated when both the multiplicative noise and the coupling between additive and multiplicative noise are colored with different values of noise correlation time ensuremath tau 1 and ensuremath tau 2 combining the functional analysis and unified colored noise approximation the two different kinds of colored noise in the nonlinear system can be simplified the signal to noise ratio is calculated when a weakly periodic signal is added to the system it is found that there appears a transition between one peak and two peaks in the curve of the signal to noise ratio when either the noise correlation time ensuremath tau 1 and ensuremath tau 2 or the coupling strength ensuremath lambda between additive and multiplicative noise is increased the transition between one and two peaks depending on ensuremath tau 1 and ensuremath lambda is more complex than that depending on ensuremath tau 2

estimation of a two component mixture model with applications to multiple testing we consider a two component mixture model with one known component we develop methods for estimating the mixing proportion and the other unknown distribution nonparametrically given i i d data from the mixture model we use ideas from shape restricted function estimation and develop tuning parameter free estimators that are easily implementable and have good finite sample performance we establish the consistency of our procedures distribution free finite sample lower confidence bounds are developed for the mixing proportion the identifiability of the model and the estimation of the density of the unknown mixing distribution are also addressed we discuss the connection with the problem of multiple testing and compare our procedure with some of the existing methods in that area through simulation studies we also analyse two data sets one arising from an application in astronomy and the other from a microarray experiment