sampling and reconstruction of signals in a reproducing kernel subspace of lp rd in this paper we consider sampling and reconstruction of signals in a reproducing kernel subspace of lp rd 1 p associated with an idempotent integral operator whose kernel has certain off diagonal decay and regularity the space of p integrable non uniform splines and the shift invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of lp rd we show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively separated set with sufficiently small gap we also study the exponential convergence consistency and the asymptotic pointwise error estimate of the iterative approximation projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap

comparison of rigidity and connectivity percolation in two dimensions using a recently developed algorithm for generic rigidity of two dimensional graphs we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to l 4096 we compare three different universality classes the generic rigidity class the connectivity class and the generic open quotes braced square net close quotes gbsn we analyze the spanning cluster density p sub infinity the backbone density p sub b and the density of dangling ends p sub d in the generic rigidity gr and connectivity cases the load carrying component of the spanning cluster the backbone is ital fractal at p sub c so that the backbone density behaves as b approximately p minus p sub c sup beta sup prime for p gt p sub c we estimate beta sub gr sup prime 0 25 plus minus 0 02 for generic rigidity and beta sub c sup prime 0 467 plus minus 0 007 for the connectivity case we find the correlation length exponents nu sub gr 1 16 plus minus 0 03 for generic rigidity compared to the exact value for connectivity nu sub c 4 3 in contrast the gbsn undergoes a first order rigidity transition with the backbone density being extensive at p sub c and undergoing a jump discontinuity on reducing ital p across the transition we define a model which tunes continuously between the gbsn and gr classes and show that the gr class is typical copyright ital 1999 italmore the american physical society less

using post outcome measurement information in censoring by death problems summary many clinical studies on non mortality outcomes such as quality of life suffer from the problem that the non mortality outcome can be censored by death i e the non mortality outcome cannot be measured if the subject dies before the time of measurement to address the problem that this censoring by death is informative it is of interest to consider the average effect of the treatment on the non mortality outcome among subjects whose measurement would not be censored under either treatment or control which is called the survivor average causal effect sace the sace is not point identified under usual assumptions but bounds can be constructed the previous literature on bounding the sace uses only the survival information before the measurement of the non mortality outcome however survival information after the measurement of the non mortality outcome could also be informative for randomized trials we propose a set of ranked average score assumptions that make use of survival information before and after the measurement of the non mortality outcome which are plausibly satisfied in many studies and we develop a two step linear programming approach to obtain the closed form for bounds on the sace under our assumptions we also extend our method to randomized trials with non compliance or observational studies with a valid instrumental variable to obtain bounds on the complier sace which is presented in on line supplementary material we apply our method to a randomized trial of the effect of mechanical ventilation with lower tidal volume versus traditional tidal volume for acute lung injury patients our bounds on the sace are much shorter than the bounds that are obtained by using only the survival information before the measurement of the non mortality outcome

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

stability of localized operators let lp 1 p be the space of all p summable sequences and ca be the convolution operator associated with a summable sequence a it is known that the lp stability of the convolution operator ca for different 1 p are equivalent to each other i e if ca has lp stability for some 1 p then ca has lq stability for all 1 q in the study of spline approximation wavelet analysis time frequency analysis and sampling there are many localized operators of non convolution type whose stability is one of the basic assumptions in this paper we consider the stability of those localized operators including infinite matrices in the sjostrand class synthesis operators with generating functions enveloped by shifts of a function in the wiener amalgam space and integral operators with kernels having certain regularity and decay at infinity we show that the lp stability or lp stability of those three classes of localized operators are equivalent to each other and we also prove that the left inverse of those localized operators are well localized

algorithms for hyperbolic quadratic eigenvalue problems we consider the quadratic eigenvalue problem or the qep 2 a b c x 0 where a b and c are hermitian with a positive definite the qep is called hyperbolic if x bx 2 4 x ax x cx for all nonzero x c r we show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the qep a hyperbolic qep is overdamped if b is positive definite and c is positive semidefinite we show that a hyperbolic qep whose eigenvalues are necessarily real is overdamped if and only if its largest eigenvalue is nonpositive for overdamped qeps we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction we also present a new measure for the degree of hyperbolicity of a hyperbolic qep

maximum independent set on diluted triangular lattices core percolation and maximum independent set on random graphs have recently been characterized using the methods of statistical physics here we present a statistical physics study of these problems on bond diluted triangular lattices core percolation critical behavior is found to be consistent with the standard percolation values though there are strong finite size effects a transfer matrix method is developed and applied to find accurate values of the density and degeneracy of the maximum independent set on lattices of limited width but large length an extrapolation of these results to the infinite lattice limit yields high precision results which are tabulated these results are compared to results found using both vertex based and edge based local probability recursion algorithms which have proven useful in the analysis of hard computational problems such as the satisfiability problem

wiener s lemma for infinite matrices in this paper we introduce a class of infinite matrices related to the beurling algebra of periodic functions and we show that it is an inverse closed subalgebra of mathcal b ell q w the algebra of all bounded linear operators on the weighted sequence space ell q w for any 1 q and any discrete muckenhoupt aq weight w

varying the time frequency lattice of gabor frames a gabor or weyl heisenberg frame for l 2 r d is generated by time frequency shifts of a square integrable function the gabor atom along a time frequency lattice the dual frame is again a gabor frame generated by the dual atom in general gabor frames are not stable under a perturbation of the lattice constants that is even for arbitrarily small changes of the parameters the frame property can be lost in contrast as a main result we show that this kind of stability does hold for gabor frames generated by a gabor atom from the modulation space m 1 r d which is a dense subspace of l 2 r d moreover in this case the dual atom depends continuously on the lattice constants in fact we prove these results for more general weighted modulation spaces as a consequence we obtain for gabor atoms from the schwartz class that the continuous dependence of the dual atom holds even in the schwartz topology also we complement these main results by corresponding statements for gabor riesz sequences and their biorthogonal system

minimal vertex covers on finite connectivity random graphs a hard sphere lattice gas picture the minimal vertex cover or maximal independent set problem is studied on random graphs of finite connectivity analytical results are obtained by a mapping to a lattice gas of hard spheres of chemical radius 1 and they are found to be in excellent agreement with numerical simulations we give a detailed description of the replica symmetric phase including the size and entropy of the minimal vertex covers and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1 43 the replica symmetric solution breaks down at c e approximately 2 72 we give a simple one step replica symmetry broken solution and discuss the problems in the interpretation and generalization of this solution

sampling theorems for shift invariant spaces gabor frames and totally positive functions we study nonuniform sampling in shift invariant spaces and the construction of gabor frames with respect to the class of totally positive functions whose fourier transform factors as hat g xi prod j 1 n 1 2 pi i delta j xi 1 e c xi 2 for delta 1 ldots delta n in mathbb r c 0 in which case g is called totally positive of gaussian type in analogy to beurling s sampling theorem for the paley wiener space of entire functions we prove that every separated set with lower beurling density 1 is a sampling set for the shift invariant space generated by such a g in view of the known necessary density conditions this result is optimal and validates the heuristic reasonings in the engineering literature using a subtle connection between sampling in shift invariant spaces and the theory of gabor frames we show that the set of phase space shifts of g with respect to a rectangular lattice alpha mathbb z times beta mathbb z forms a frame if and only if alpha beta 1 this solves an open problem going back to daubechies in 1990 for the class of totally positive functions of gaussian type the proof strategy involves the connection between sampling in shift invariant spaces and gabor frames a new characterization of sampling sets without inequalities in the style of beurling new properties of totally positive functions and the interplay between zero sets of functions in a shift invariant space and functions in the bargmann fock space

number of guards needed by a museum a phase transition in vertex covering of random graphs in this letter we study the np complete vertex cover problem on finite connectivity random graphs when the allowed size of the cover set is decreased a discontinuous transition in solvability and typical case complexity occurs this transition is characterized by means of exact numerical simulations as well as by analytical replica calculations the replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity e where replica symmetry becomes locally unstable

eigenstructure of order one quasiseparable matrices three term and two term recurrence relations abstract this paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order one quasiseparable matrices various recursive relations for characteristic polynomials of their principal submatrices are derived the cost of evaluating the characteristic polynomial of an n n matrix and its derivative is only o n this leads immediately to several versions of a fast quasiseparable newton iteration algorithm in the hermitian case we extend the sturm property to the characteristic polynomials of order one quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm conditions guaranteeing that an eigenvalue of a order one quasiseparable matrix is simple are obtained and an explicit formula for the corresponding eigenvector is derived the method is further extended to the case when these conditions are not fulfilled several particular examples with tridiagonal almost unitary hessenberg and toeplitz matrices are considered the algorithms are based on new three term and two term recurrence relations for the characteristic polynomials of principal submatrices of order one quasiseparable matrices r it turns out that the latter new class of polynomials generalizes and includes two classical families i polynomials orthogonal on the real line that play a crucial role in a number of classical algorithms in numerical linear algebra and ii the szego polynomials that play a significant role in signal processing moreover new formulas can be seen as generalizations of the classical three term recurrence relations for the real orthogonal polynomials and of the two term recurrence relations for the szego polynomials

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

an almost periodic noncommutative wiener s lemma we develop a theory of almost periodic elements in banach algebras and present an abstract version of a noncommutative wiener s lemma the theory can be used for example to derive some of the recently obtained results in time frequency analysis such as the spectral properties of the finite linear combinations of time frequency shifts

estimation and inference for the causal effect of receiving treatment on a multinomial outcome this article considers the analysis of two arm randomized trials with noncompliance which have a multinomial outcome we first define the causal effect in these trials as some function of outcome distributions of compliers with and without treatment e g the complier average causal effect the measure of stochastic superiority of treatment over control for compliers then estimate the causal effect with the likelihood method next based on the likelihood ratio lr statistic we test those functions of or the equality of the outcome distributions of compliers with and without treatment although the corresponding lr statistic follows a chi squared chi 2 distribution asymptotically when the true values of parameters are in the interior of the parameter space under the null its asymptotic distribution is not chi 2 when the true values of parameters are on the boundary of the parameter space under the null therefore we propose a bootstrap double bootstrap version of a lr test for the causal effect in these trials the methods are illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices

two fast algorithms for solving diagonal plus semiseparable linear systems in this paper we discuss the structure of the factors of a qr and a urv factorization of a diagonal plus semiseparable matrix the q factor of a qr factorization has the diagonal plus semiseparable structure the ut and v factor of a urv factorization are semiseparable lower hessenberg orthogonal matrices the strictly upper triangular part of the r factor of a qr and of a urv factorization is the strictly upper triangular part of a rank 2 matrix this latter fact provides a tool to construct a fast qr solver and a fast urv solver for linear systems of the form d s x b

estimating the null distribution to adjust observed confidence levels for genome scale screening summary in a novel approach to the multiple testing problem efron 2004 journal of the american statistical association 99 96 104 2007a journal of the american statistical association 102 93 103 2007b annals of statistics 35 1351 1377 formulated estimators of the distribution of test statistics or nominal p values under a null distribution suitable for modeling the data of thousands of unaffected genes nonassociated single nucleotide polymorphisms or other biological features estimators of the null distribution can improve not only the empirical bayes procedure for which it was originally intended but also many other multiple comparison procedures such estimators in some cases improve the proposed multiple comparison procedure mcp based on a recent non bayesian framework of minimizing expected loss with respect to a confidence posterior a probability distribution of confidence levels the flexibility of that mcp is illustrated with a nonadditive loss function designed for genomic screening rather than for validation the merit of estimating the null distribution is examined from the vantage point of the confidence posterior mcp cpmcp in a generic simulation study of genome scale multiple testing conditioning the observed confidence level on the estimated null distribution as an approximate ancillary statistic markedly improved conditional inference specifically simulating gene expression data however indicates that estimation of the null distribution tends to exacerbate the conservative bias that results from modeling heavy tailed data distributions with the normal family to enable researchers to determine whether to rely on a particular estimated null distribution for inference or decision making an information theoretic score is provided as the sum of the degree of ancillarity and the degree of inferential relevance the score reflects the balance conditioning would strike between the two conflicting terms the cpmcp and other methods introduced are applied to gene expression microarray data

calibrating confidence coefficients abstract two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed the first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level this is achieved through a combination of computer simulation and density estimation the asymptotic validity of the procedure is proved for a number of common situations a classical example is the nonparametric estimation of the variance of a population using the normal theory interval here it is shown that the estimated coverage probability consistently estimates the true coverage probability if the population distribution possesses a finite sixth moment the second approach is more traditional it is a procedure for modifying an interval to yield improved coverage properties given a confidence interval its estimated coverage probability obtained in the first approach is used to alter the nominal level of the interval

polynomial iterative algorithms for coloring and analyzing random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find with a one step replica symmetry breaking approximation the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c in c d c q in which ground states spontaneously divide into an exponential number of clusters furthermore we extended our considerations to the case of single instances showing consistent results this leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region i e when c in c d c q

bootstrap confidence intervals for a class of parametric problems summary we consider the following class of problems having observed a multivariate normal data vector y with unknown mean vector q covariance matrix the identity find an approximate confidence interval for 0 t q a real valued function of q a simple geometric construction is given which leads to highly accurate solutions this construction shows that the standard approximation based on maximum likelihood theory 04 cz00 can be quite misleading when 0 is nonlinear in q we discuss bootstrap based confidence intervals which remove most of the error in the standard approximation at the expense of considerably more calculation the bootstrap intervals are invariant under transformation of both y and i and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality without requiring knowledge of the normalizing transformation

characterization of coorbit spaces with phase space covers abstract we show that coorbit spaces can be characterized in terms of arbitrary phase space covers which are families of phase space multipliers associated with partitions of unity this generalizes previously known results for time frequency analysis to include time scale decompositions as a by product we extend the existing results for time frequency analysis to an irregular setting

slanted matrices banach frames and sampling in this paper we present a rare combination of ab stract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sam pling problems we show that for a large class of slanted matrices boundedness below of the corresponding operator in p for some p implies boundedness below in p for all p we use the established result to enrich our understanding of banach frames and obtain new results for irregular sampling problems we also present a version of a non commutative wiener s lemma for slanted matri ces

pseudodifferential operators on locally compact abelian groups and sjoestrand s symbol class we investigate pseudodifferential operators on arbitrary locally compact abelian groups as symbol classes for the kohn nirenberg calculus we introduce a version of sjoestrand s class pseudodifferential operators with such symbols form a banach algebra that is closed under inversion since hard analysis techniques are not available on locally compact abelian groups a new time frequency approach is used with the emphasis on modulation spaces gabor frames and banach algebras of matrices sjoestrand s original results are thus understood as a phenomenon of abstract harmonic analysis rather than hard analysis and are proved in their natural context and generality

polynomial control on stability inversion and powers of matrices on simple graphs abstract spatially distributed networks of large size arise in a variety of science and engineering problems such as wireless sensor networks and smart power grids most of their features can be described by properties of their state space matrices whose entries have indices in the vertex set of a graph in this paper we introduce novel algebras of beurling type that contain matrices on a connected simple graph having polynomial off diagonal decay and we show that they are banach subalgebras of b l p 1 p the space of all bounded operators on the space l p of all p summable sequences the l p stability of state space matrices is an essential hypothesis for the robustness of spatially distributed networks in this paper we establish the equivalence among l p stabilities of matrices in beurling algebras for different exponents 1 p with quantitative analysis for the lower stability bounds admission of norm control inversion plays a crucial role in some engineering practice in this paper we prove that matrices in beurling subalgebras of b l 2 have norm controlled inversion and we find a norm controlled polynomial with close to optimal degree polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks in this paper we apply our results on norm controlled inversion to obtain a polynomial estimate to powers of matrices in beurling algebras the polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary markov chain on a spatially distributed network

stability of qr based fast system solvers for a subclass of quasiseparable rank one matrices the development of fast algorithms to perform computations with quasiseparable matrices has received a lot of attention in the last decade many different algorithms have been presented by several research groups all over the world despite this intense activity to the best of our knowledge there is no rounding error analysis published for these fast algorithms in this paper we present error analyses for two fast solvers of quasiseparable linear systems when they are applied on order one quasiseparable matrices that include the diagonal in the lower triangular rank structure both solvers are based on computing first the qr factorization of the coefficient matrix and their error analyses require novel structured techniques for proving rigorously that only one of the considered algorithms is backward stable while the other one is not two fundamental consequences of this work are i users should employ with caution fast algorithms for quasiseparable matrices since they may be unstable and ii a lot of work has to be done to identify which fast algorithms for quasiseparable matrices are backward stable among the large family available in the literature

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

memory estimation of inverse operators we use methods of harmonic analysis and group rep resentation theory to estimate memory decay of the inverse oper ators in banach spaces the memory of the operators is defined using the notion of the beurling spectrum we obtain a general continuous non commutative version of the celebrated wiener s tauberian lemma with estimates of the fourier coefficients of inverse operators in particular we generalize various estimates of the elements of the inverse matrices the results are illus trated with a variety of examples including integral and integro differential operators

perturbed sampling formulas and local reconstruction in shift invariant spaces abstract let v be a closed subspace of l 2 r generated from the integer shifts of a continuous function with a certain decay at infinity and a non vanishing property for the function n z n e i n on in this paper we determine a positive number so that the set n n n z is a set of stable sampling for the space v for any selection of the elements n within the ranges we demonstrate the resulting sampling formula called perturbation formula for functions f v and also we establish a finite reconstruction formula approximating f on bounded intervals we compute the corresponding error and we provide estimates for the jitter error as well

distributed multi parametric quadratic programming one of the fundamental problems in the area of large scale optimization is to study locality features of spatially distributed optimization problems in which the variables are coupled in the cost function as well as constraints such problems can motivate the development of fast and well conditioned distributed algorithms in this paper we study spatial locality features of large scale multi parametric quadratic programming mpqp problems with linear inequality constraints our main application focus is receding horizon control of spatially distributed linear systems with input and state constraints we propose a new approach for analysis of large scale mpqp problems by blending tools from duality theory with operator theory the class of spatially decaying matrices is introduced to capture couplings between optimization variables in the cost function and the constraints we show that the optimal solution of a convex mpqp is piecewise affine represented as convolution sums more importantly we prove that the kernel of each convolution sum decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the optimization problem

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

deformation of gabor systems abstract we introduce a new notion for the deformation of gabor systems such deformations are in general nonlinear and in particular include the standard jitter error and linear deformations of phase space with this new notion we prove a strong deformation result for gabor frames and gabor riesz sequences that covers the known perturbation and deformation results our proof of the deformation theorem requires a new characterization of gabor frames and gabor riesz sequences it is in the style of beurling s characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of gabor frames without inequalities from lattices to non uniform sets

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

representation of the exact solution for infinite system of linear equations in this paper representation of the exact solution for the infinite system of linear equations is given and we give a sufficient and necessary condition for the infinite system of linear equations which has solutions

the problem of regions in the problem of regions we wish to know which one of a discrete set of possibilities applies to a continuous parameter vector this problem arises in the following way we compute a descriptive statistic from a set of data notice an interesting feature and wish to assign a confidence level to that feature for example we compute a density estimate and notice that the estimate is bimodal what confidence can we assign to bimodality a natural way to measure confidence is via the bootstrap we compute our descriptive statistic on a large number of bootstrap data sets and record the proportion of times that the feature appears this seems like a plausible measure of confidence for the feature the paper studies the construction of such confidence values and examines to what extent they approximate frequentist p values and bayesian a posteriori probabilities we derive more accurate confidence levels using both frequentist and objective bayesian approaches the methods are illustrated with a number of examples including polynomial model selection and estimating the number of modes of a density

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

approximately unbiased tests of regions using multistep multiscale bootstrap resampling approximately unbiased tests based on bootstrap probabilities are considered for the exponential family of distributions with unknown expectation parameter vector where the null hypothesis is represented as an arbitraryshaped region with smooth boundaries this problem has been discussed previously in efron and tibshirani ann statist 26 1998 1687 1718 and a corrected p value with second order asymptotic accuracy is calculated by the two level bootstrap of efron halloran and holmes proc natl acad sci u s a 93 1996 13429 13434 based on the abc bias correction of efron j amer statist assoc 82 1987 171 185 our argument is an extension of their asymptotic theory where the geometry such as the signed distance and the curvature of the boundary plays an important role we give another calculation of the corrected p value without finding the nearest point on the boundary to the observation which is required in the two level bootstrap and is an implementational burden in complicated problems the key idea is to alter the sample size of the replicated dataset from that of the observed dataset the frequency of the replicates falling in the region is counted for several sample sizes and then the p value is calculated by looking at the change in the frequencies along the changing sample sizes this is the multiscale bootstrap of shimodaira systematic biology 51 2002 492 508 which is third order accurate for the multivariate normal model here we introduce a newly devised multistep multiscale bootstrap calculating a third order accurate p value for the exponential family of distributions in fact our p value is asymptotically equivalent to those obtained by the double bootstrap of hall the bootstrap and edgeworth expansion 1992 springer new york and the modified signed likelihood ratio of barndorff nielsen biometrika 73 1986 307 322 ignoring o n 3 2 terms yet the computation is less demanding and free from model specification the algorithm is remarkably simple despite complexity of the theory behind it the differences of the pvalues are illustrated in simple examples and the accuracies of the bootstrap methods are shown in a systematic way

time sampling and reconstruction in weighted reproducing kernel subspaces abstract this paper mainly studies the time sampling and reconstruction of signals in weighted reproducing kernel subspaces of l p firstly a class of examples for weighted reproducing kernel subspaces are given then using samples produced by a crossing time encoding machine or an integrate and fire time encoding machine satisfying some density constraints iterative approximation algorithms are established for perfect recovery of signals in weighted reproducing kernel subspaces finally an integrate and fire sampling scheme is discussed and an approximate reconstruction procedure is proposed for signals in weighted reproducing kernel subspaces moreover we give detailed analysis for the approximate reconstruction method in the setting of shift invariant subspaces

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

symmetry and inverse closedness of matrix algebras and functional calculus for infinite matrices we investigate the symbolic calculus for a large class of matrix algebras that are defined by the off diagonal decay of infinite matrices applications are given to the symmetry of some highly non commutative banach algebras to the analysis of twisted convolution and to the theory of localized frames

the noncommutative wiener lemma linear independence and spectral properties of the algebra of time frequency shift operators in this paper we analyze the banach algebra of time frequency shifts with absolutely summable coecients we prove a noncommutative version of the wiener lemma we also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators as a corollary we obtain a special case of the heil ramanathan topiwala conjecture regarding linear independence of nitely many time frequency shifts of one l 2 function we also estimate the coecient decay of the inverse of nite linear combinations of time frequency shifts

bounds on causal effects in three arm trials with non compliance summary the paper considers the analysis of three arm randomized trials with non compliance in these trials the average causal effects of treatments within principal strata of compliance behaviour are of interest for better understanding the effect of the treatment unfortunately even with the usual assumptions the average causal effects of treatments within principal strata are not point identified however the observable data do provide useful information on the bounds of the identification regions of the parameters of interest under two sets of assumptions we derive sharp bounds for the causal effects within principal strata for binary outcomes and construct confidence intervals to cover the identification regions the methods are illustrated by an analysis of data from a randomized study of treatments for alcohol dependence

wiener s lemma for twisted convolution and gabor frames as a consequence ca is invertible and bounded on all p zd for 1 p oo simultaneously in this article we study several non commutative generalizations of wiener s lemma and their application to gabor theory the paper is divided into two parts the first part sections 2 and 3 is devoted to abstract harmonic analysis and extends wiener s lemma to twisted convolution the second part section 4 is devoted to the theory of gabor frames specifically to the design of dual windows with good time frequency localization in particular we solve a conjecture of janssen feichtinger and one of us 17 18 9 these two topics appear to be completely disjoint but they are not the solution of the conjectures about gabor frames is an unexpected application of methods from non commutative harmonic analysis to application oriented mathematics it turns out that the connection between twisted convolution and the heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis to be more concrete we formulate some of our main results first and will deal with the details and the technical background later

structured eigenvalue condition numbers for parameterized quasiseparable matrices the development of fast algorithms for performing computations with n times n n n low rank structured matrices has been a very active area of research during the last two decades as a consequence of the numerous applications where these matrices arise the key ideas behind these fast algorithms are that low rank structured matrices can be described in terms of o n parameters and that these algorithms operate on the parameters instead on the matrix entries therefore the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer since this determines the maximum accuracy of a given fast computation in other words it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low rank structured matrices but as far as we know this has not yet been accomplished in any case in this paper we derive structured relative eigenvalue condition numbers for the important class of low rank structured matrices known as 1 1 1 1 quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the givens vector representations of these matrices and we provide fast algorithms for computing them comparisons among the new structured condition numbers and the unstructured one are also presented as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge in addition the approach presented in this paper is general and may be extended to other problems and classes of low rank structured matrices

left inverses of matrices with polynomial decay abstract it is known that the algebra of schur operators on l 2 namely operators bounded on both l 1 and l is not inverse closed when l 2 l 2 x where x is a metric space one can consider elements of the schur algebra with certain decay at infinity for instance if x has the doubling property then q sun has proved that the weighted schur algebra a x for a strictly polynomial weight is inverse closed in this paper we prove a sharp result on left invertibility of the these operators namely if an operator a a x satisfies a f p f p for some 1 p then it admits a left inverse in a x the main difficulty here is to obtain the above inequality in l 2 the author was both motivated and inspired by a previous work of aldroubi baskarov and krishtal 2008 1 where similar results were obtained through different methods for x z d under additional conditions on the decay