positive decreasing solutions of quasi linear difference equations the second order nonlinear difference equation is considered a full characterization of limit behavior of all positive decreasing solutions is established the obtained results answer some open problems formulated for sturm liouville discrete operator a comparison with the continuous case jointly with similarities and discrepancies is given as well

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

positive solutions of a class of nonlinear ordinary differential equations on considere l equation differentielle non lineaire x n f t x o t t 0 ou n 2 t 1 ou t 1 et f t 0 o r est une fonction continue telle que f t x o pour tout t x t 0 o

stepwise signal extraction via marginal likelihood this article studies the estimation of a stepwise signal to determine the number and locations of change points of the stepwise signal we formulate a maximum marginal likelihood estimator which can be computed with a quadratic cost using dynamic programming we carry out an extensive investigation on the choice of the prior distribution and study the asymptotic properties of the maximum marginal likelihood estimator we propose to treat each possible set of change points equally and adopt an empirical bayes approach to specify the prior distribution of segment parameters a detailed simulation study is performed to compare the effectiveness of this method with other existing methods we demonstrate our method on single molecule enzyme reaction data and on dna array comparative genomic hybridization cgh data our study shows that this method is applicable to a wide range of models and offers appealing results in practice supplementary materials for this article are available online

integral criteria for a classification of solutions of linear differential equations are positive continuous function on the interval 0 co by solution we always mean a continuously differentiable function such that r t x t has a continuous derivative satisfying e when the func tion r is continuous but does not have a continuous derivative e can be interpreted as the first order system for the vector x y x rx 381

periodic solutions of a class of nonlinear difference equations via critical point method in this paper we obtain some new sufficient conditions for the existence of nontrivial m periodic solutions of the following nonlinear difference equation d p n dx n 1 d f n x n n z by using the critical point method where f zxr r is continuous in the second variable m 2 is a given positive integer p n m p n for any n z and f t m z f t z for any t z zxr 1 d 1 and d 0

asymptotic integration of nonlinear differential equations abstract for a wide class of higher order nonlinear differential equations global existence of solutions which behave at infinity like polynomials is established by combining two fixed point results

local extremes runs strings and multiresolution the paper considers the problem of nonparametric regression with emphasis on controlling the number of local extremes two methods the run method and the taut string multiresolution method are introduced and analyzed on standard test beds it is shown that the number and locations of local extreme values are consistently estimated rates of convergence are proved for both methods the run method converges slowly but can withstand blocks as well as a high proportion of isolated outliers the rate of convergence of the taut string multiresolution method is almost optimal the method is extremely sensitive and can detect very low power peaks section 1 contains an introduction with special reference to the number of local extreme values the run method is described in section 2 and the taut string multiresolution method in section 3 low power peaks are considered in section 4 section 5 contains a comparison with other methods and section 6 a short conclusion the proofs are given in section 7 and the taut string algorithm is described in the appendix

monotone solutions of a class of second order nonlinear differential equations on etudie le comportement asymptotique des solutions de l equation p t f x t x t g t g x t on donne des conditions necessaires et suffisantes pour que ces solutions soient bornees

scale space view of curve estimation scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation the family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface the smoothing parameter here plays the same role as that played by the scale of resolution in a visual system in this paper we study in detail various features of that surface from a statistical viewpoint weak convergence of the empirical scale space surface to its theoretical counterpart and some related asymptotic results have been established under appropriate regularity conditions our theoretical analysis provides new insights into nonparametric smoothing procedures and yields useful techniques for statistical exploration of features in the data in particular we have used the scale space approach for the development of an effective exploratory data analytic tool called sizer

change point estimation using nonparametric regression we consider a regression model in which the mean function may have a discontinuity at an unknown point we propose an estimate of the location of the discontinuity based on one side nonparametric regression estimates of the mean function the change point estimate is shown to converge in probability at rate 0 n 1 and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models confidence regions for the location and size of the change are also discussed

edge preserving and peak preserving smoothing an alternative procedure is developed to the smoothed linear fitting method of mcdonald and owen the procedure is based on the detection of discontinuities by comparing at any given position three smooth fits diagnostics are used to detect discontinuities in the regression function itself edge detection or in its first derivative peak detection an application in electron microscopy is discussed

improved normal form for grammars with one sided contexts formal grammars equipped with operators for specifying the form of the context of a substring were recently studied by barash and okhotin an extension of context free grammars with one sided context specifications inform and comput 2014 further extending the author s earlier work on propositional connectives in grammars a okhotin conjunctive grammars j autom lang comb 2001 these grammars allow two types of context specifications for a substring v of a string uvx a proper left context operator d states that u is of the form described by d and an extended left context operator e states that uv is described by e this paper establishes a normal form for these grammars in which extended left contexts are never used whereas left contexts may be applied only to individual symbols so that all rules are of the form a b 1 c 1 b m c m or a a d this eliminates circular dependencies in the grammar and leads to a new clearer version of the known cubic time parsing algorithm with some further improvements the algorithm is accelerated from time o n 3 to time o n 3 log n

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

positive solutions of higher order nonlinear delay difference equations abstract classification schemes for positive solutions of a class of higher order nonlinear delay difference equations are given in terms of their asymptotic behavior and necessary as well as sufficient conditions for the existence of these solutions are also provided

wild binary segmentation for multiple change point detection we propose a new technique called wild binary segmentation wbs for consistent estimation of the number and locations of multiple change points in data we assume that the number of change points can increase to infinity with the sample size due to a certain random localisation mechanism wbs works even for very short spacings between the change points and or very small jump magnitudes unlike standard binary segmentation on the other hand despite its use of localisation wbs does not require the choice of a window or span parameter and does not lead to a significant increase in computational complexity wbs is also easy to code we propose two stopping criteria for wbs one based on thresholding and the other based on what we term the strengthened schwarz information criterion we provide default recommended values of the parameters of the procedure and show that it offers very good practical performance in comparison with the state of the art the wbs methodology is implemented in the r package wbs available on cran in addition we provide a new proof of consistency of binary segmentation with improved rates of convergence as well as a corresponding result for wbs

minors of the wronskian of the differential equation lny p x y 0 ii dominance of solutions on etudie les equations l n y p x y 0 ou l n est un operateur differentiel disconjugue et p x est de signe fixe on definit une base de l espace solution et on ordonne ses elements selon leurs grandeurs relatives a l infini

discontinuous versus smooth regression given measurements x i y i i 1 n we discuss methods to assess whether an underlying regression function is smooth continuous or differentiable or whether it has discontinuities the variance of the measurements is assumed to be unknown and is estimated simultaneously by regressing squared differences of the data formed with various span sizes on the span size itself we obtain an asymptotic linear model with dependent errors the parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements both parameters can be consistently estimated with mean squared error rates of convergence of n 2 3 for the sum of squared jump sizes and n 1 for the error variance we derive the asymptotic constants of the mean squared error mse and discuss the dependence of mse on the maximum span size l the test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0 the asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for u statistics we motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps

tail greedy bottom up data decompositions and fast multiple change point detection this article proposes a tail greedy bottom up transform for one dimensional data which results in a nonlinear but conditionally orthonormal multiscale decomposition of the data with respect to an adaptively chosen unbalanced haar wavelet basis the tail greediness of the decomposition algorithm whereby multiple greedy steps are taken in a single pass through the data both enables fast computation and makes the algorithm applicable in the problem of consistent estimation of the number and locations of multiple changepoints in data the resulting agglomerative change point detection method avoids the disadvantages of the classical divisive binary segmentation and offers very good practical performance it is implemented in the r package breakfast available from cran

half linear equations and characteristic properties of the principal solution abstract the characteristic properties of the principal solution for half linear differential equation a t x b t x 0 where the functions a b are positive and continuous for t 0 and u u p 2u p 1 are investigated in the linear case it is well known that the principal solution is the smallest one in a neighbourhood of infinity we show that this property continues to hold in the half linear case in addition it is proved that the principal solutions can be fully characterized by means of two different integral criteria which reduce to that one well known in the linear case

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

on continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals soit x t f t x t t j x s ou jcr est un intervalle donne f j r n r n est une fonction continue et s un sous ensemble de c j r n l espace des fonctions continues de j dans r n on considere l equation operateur associee x t x ou t est un operateur non lineaire discontinu dans son domaine

sizer for exploration of structures in curves abstract in the use of smoothing methods in data analysis an important question is which observed features are really there as opposed to being spurious sampling artifacts an approach is described based on scale space ideas originally developed in the computer vision literature assessment of significant zero crossings of derivatives results in the sizer map a graphical device for display of significance of features with respect to both location and scale here scale means level of resolution that is bandwidth

existence theorems for second order discrete boundary value problems abstract this article is concerned with the existence of solutions of boundary value problems for nonlinear second order difference equations of the type p n x n 1 q n x n f n x n where 0 is the ratio of odd positive integers p n and q n are real sequences the authors apply the linking theorem and the mountain pass lemma in the critical point theory and give some new results for the existence of solutions

wedgelets nearly minimax estimation of edges we study a simple horizon model for the problem of recovering an image from noisy data in this model the image has an edge with fi holder regularity adopting the viewpoint of computational harmonic analysis we develop an overcomplete collection of atoms called wedgelets dyadically organized indicator functions with a variety of locations scales and orientations the wedgelet representation provides nearly optimal representations of objects in the horizon model as measured by minimax description length we show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet decorated recursive partition which minimizes a complexity penalized sum of squares this estimate using su cient sub pixel resolution achieves nearly the minimax mean squared error in the horizon model in fact the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon 1 fi 2 wedgelet analysis and de noising may be used successfully outside the horizon model we study images modelled as indicators of star shaped sets with smooth bound aries and show that complexity penalized wedgelet partitioning achieves nearly the minimax risk in that setting also

asymptotic properties of kneser solutions to nonlinear second order odes with regularly varying coefficients in this work we investigate properties of a class of solutions to the second order ode p t u t q t f u t 0 on the interval a a 0 where p and q are functions regularly varying at infinity and f satisfies f l 0 f 0 f l 0 with l0 0

multiple changepoint fitting via quasilikelihood with application to dna sequence segmentation summary we consider situations where a step function with a variable number of steps provides an adequate model for a regression relationship while the variance of the observations depends on their mean this model provides for discontinuous jumps at changepoints and for constant means and error variances in between changepoints the basic statistical problem consists of identification of the number of changepoints their locations and the levels the function assumes in between we embed this problem into a quasilikelihood formulation and utilise the minimum deviance criterion to fit the model for the choice of the number of changepoints we discuss a modified schwarz criterion a dynamic programming algorithm makes the segmentation feasible for sequences of moderate length the performance of the segmentation method is demonstrated in an application to the segmentation of the bacteriophage l sequence

consistencies and rates of convergence of jump penalized least squares estimators we study the asymptotics for jump penalized least squares regression aiming at approximating a regression function by piecewise constant functions besides conventional consistency and convergence rates of the estimates in l 2 0 1 our results cover other metrics like skorokhod metric on the space of cadlag functions and uniform metrics on c 0 1 we will show that these estimators are in an adaptive sense rate optimal over certain classes of approximation spaces special cases are the class of functions of bounded variation piecewise holder continuous functions of order 0 1 and the class of step functions with a finite but arbitrary number of jumps in the latter setting we will also deduce the rates known from change point analysis for detecting the jumps finally the issue of fully automatic selection of the smoothing parameter is addressed

asymptotic behavior of nonoscillatory solutions to n th order nonlinear neutral differential equations abstract for a class of n th order nonlinear neutral differential equations sufficient conditions for all nonoscillatory solutions to satisfy lim t t 1 n x t a are established for another class of equations necessary and sufficient conditions for nonoscillatory solutions to satisfy the above condition are obtained our theorems complement and extend recent results reported in the literature

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

cart and best ortho basis a connection we study what we call dyadic cart a method of nonparametric regression which constructs a recursive partition by optimizing a complexity penalized sum of squares where the optimization is over all recursive partitions arising from midpoint splits we show that the method is adaptive to unknown degrees of anisotropic smoothness specifically consider the anisotropic smoothness classes of nikol skii consisting of bivariate functions f xl x2 whose finite difference of distance h in direction i is bounded in lp norm by ch8i i 1 2 we show that dyadic cart with an appropriate complexity penalty parameter a u2 const log n is within logarithmic terms of minimax over every anisotropic smoothness class 0 c x 0 81 82 1 the proof shows that dyadic cart is identical to a certain adaptive best ortho basis algorithm based on the library of all anisotropic haar bases then it applies empirical basis selection ideas of donoho and johnstone the basis empirically selected by dyadic cart is shown to be nearly as good as a basis ideally adapted to the underlying f the risk of estimation in an ideally adapted anisotropic haar basis is shown to be comparable to the minimax risk over anisotropic smoothness classes underlying the success of this argument is harmonic analysis of anisotropic smoothness classes we show that for each anisotropic smoothness class there is an anisotropic haar basis which is a best orthogonal basis for representing that smoothness class the basis is optimal not just within the library of anisotropic haar bases but among all orthogonal bases of l2 0 1 2

minimal penalties for gaussian model selection this paper is mainly devoted to a precise analysis of what kind of penalties should be used in order to perform model selection via the minimiza tion of a penalized least squares type criterion within some general gaussian framework including the classical ones as compared to our previous paper on this topic birge and massart in j eur math soc 3 203 268 2001 more elaborate forms of the penalties are given which are shown to be in some sense optimal we indeed provide more precise upper bounds for the risk of the penalized estimators and lower bounds for the penalty terms showing that the use of smaller penalties may lead to disastrous results these lower bounds may also be used to design a practical strategy that allows to estimate the penalty from the data when the amount of noise is unknown we provide an illustra tion of the method for the problem of estimating a piecewise constant signal in gaussian noise when neither the number nor the location of the change points are known

nonparametric maximum likelihood approach to multiple change point problems in multiple change point problems different data segments often follow different distributions for which the changes may occur in the mean scale or the entire distribution from one segment to another without the need to know the number of change points in advance we propose a nonparametric maximum likelihood approach to detecting multiple change points our method does not impose any parametric assumption on the underlying distributions of the data sequence which is thus suitable for detection of any changes in the distributions the number of change points is determined by the bayesian information criterion and the locations of the change points can be estimated via the dynamic programming algorithm and the use of the intrinsic order structure of the likelihood function under some mild conditions we show that the new method provides consistent estimation with an optimal rate we also suggest a prescreening procedure to exclude most of the irrelevant points prior to the implementation of the nonparametric likelihood method simulation studies show that the proposed method has satisfactory performance of identifying multiple change points in terms of estimation accuracy and computation time

on monotonic solutions of systems of nonlinear second order differential equations abstract bounded monotonic and asymptotic properties of solutions for a class of second order nonlinear differential equations are discussed some necessary and sufficient conditions for boundedness of all solutions are established moreover all solutions are classified into four disjoint subsets which are characterized in terms of the convergence and divergence of several integrals the obtained results extend and improve many known results of various authors

sparse estimators and the oracle property or the return of hodges estimator abstract we point out some pitfalls related to the concept of an oracle property as used in fan and li 2001 variable selection via nonconcave penalized likelihood and its oracle properties journal of the american statistical association 96 1348 1360 2002 variable selection for cox s proportional hazards model and frailty model annals of statistics 30 74 99 2004 new estimation and model selection procedures for semiparametric modeling in longitudinal data analysis journal of the american statistical association 99 710 723 which are reminiscent of the well known pitfalls related to hodges estimator the oracle property is often a consequence of sparsity of an estimator we show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function in particular the maximal risk diverges to infinity whenever the loss function is unbounded for ease of presentation the result is set in the framework of a linear regression model but generalizes far beyond that setting in a monte carlo study we also assess the extent of the problem in finite samples for the smoothly clipped absolute deviation scad estimator introduced in fan and li 2001 variable selection via nonconcave penalized likelihood and its oracle properties journal of the american statistical association 96 1348 1360 we find that this estimator can perform rather poorly in finite samples and that its worst case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity

properties of the third order trinomial differential equations with delay argument abstract in this paper we study properties of the third order trinomial delay differential equation y t p t y t g t y t 0 by transforming this equation to binomial second third order differential equation employing suitable comparison theorems we establish new results on asymptotic behavior of solutions of the studied equation

on an extension of the fubini theorem and its applications in odes the paper deals with the extension of the fubini theorem there are given applications for the second order differential nonlinear equations the first application brings comparison results for oscillation of two generalized emden fowler equations the second one shows the similarities and discrepancies for nonoscillatory half linear and linear differential equations

heterogeneous change point inference summary we propose a heterogeneous simultaneous multiscale change point estimator called h smuce for the detection of multiple change points of the signal in a heterogeneous gaussian regression model a piecewise constant function is estimated by minimizing the number of change points over the acceptance region of a multiscale test which locally adapts to changes in the variance the multiscale test is a combination of local likelihood ratio tests which are properly calibrated by scale dependent critical values to keep a global nominal level even for finite samples we show that h smuce controls the error of overestimation and underestimation of the number of change points for this new deviation bounds for f type statistics are derived moreover we obtain confidence sets for the whole signal all results are non asymptotic and uniform over a large class of heterogeneous change point models h smuce is fast to compute achieves the optimal detection rate and estimates the number of change points at almost optimal accuracy for vanishing signals while still being robust we compare h smuce with several state of the art methods in simulations and analyse current recordings of a transmembrane protein in the bacterial outer membrane with pronounced heterogeneity for its states an r package is available on line

multiple change point analysis fast implementation and strong consistency one of the main challenges in identifying structural changes in stochastic processes is to carry out analysis for time series with dependency structure in a computationally tractable way another challenge is that the number of true change points is usually unknown requiring a suitable model selection criterion to arrive at informative conclusions to address the first challenge we model the data generating process as a segment wise autoregression which is composed of several segments time epochs each of which modeled by an autoregressive model we propose a multiwindow method that is both effective and efficient for discovering the structural changes the proposed approach was motivated by transforming a segment wise autoregression into a multivariate time series that is asymptotically segment wise independent and identically distributed to address the second challenge we derive theoretical guarantees for almost surely selecting the true number of change points of segment wise independent multivariate time series specifically under mild assumptions we show that a bayesian information criterion like criterion gives a strongly consistent selection of the optimal number of change points while an akaike information criterion like criterion cannot finally we demonstrate the theory and strength of the proposed algorithms by experiments on both synthetic and real world data including the eastern u s temperature data and the el nino data the experiment leads to some interesting discoveries about temporal variability of the summer time temperature over the eastern u s and about the most dominant factor of ocean influence on climate which were also discovered by environmental scientists

parsing by matrix multiplication generalized to boolean grammars the well known parsing algorithm for context free grammars due to valiant 1975 25 is analyzed and extended to handle the more general boolean grammars which are context free grammars augmented with conjunction and negation operators in the rules the algorithm reduces construction of a parsing table to computing multiple products of boolean matrices of various sizes its time complexity on an input string of length n is o bmm n logn where bmm n is the number of operations needed to multiply two boolean matrices of size nxn which is o n w with w 2 373 as per the current knowledge a parse tree can be constructed in time mm n log o 1 n where mm n is the complexity of multiplying two integer matrices by applying a known efficient procedure for determining witnesses for boolean matrix multiplication the algorithm has a succinct proof of correctness and is ready to be implemented

a classification of the solutions of a differential equation according to their behaviour at infinity ii the solutions of the differential equation l n y p y 0 where l n is a disconjugate operator are classified according to their behaviour as the solution space is decomposed into disjoint sets we study the dominance properties of the solutions which belong to different sets

estimation of a function with discontinuities via local polynomial fit with an adaptive window choice we propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error at the same time the estimator is shown to be very sensitive to discontinuities or change points of the underlying function f or its derivatives for instance in the case of a jump of a regression function beyond the intervals of length in order n 1 log n around change points the quality of estimation is essentially the same as if locations of jumps were known the method is fully adaptive and no assumptions are imposed on the design number and size of jumps the results are formulated in a nonasymptotic way and can therefore be applied for an arbitrary sample size

asymptotic decay of solutions of a nonlinear second order differential equation with deviating argument on etudie l existence et le comportement asymptotique des solutions decroissantes des equations differentielles fonctionnelles du type p t h x x t f t x g t

oscillation constants for euler type differential equations involving the p t laplacian abstract this paper deals with the oscillation problem for the nonlinear differential equation x p t 2 x t p t x p t 2 x 0 where is a positive parameter and p t 1 is a nondecreasing and smooth function some non oscillation criteria are given for this equation the obtained results are the best possible in a certain sense we use the riccati technique and the function sequence technique moreover we propose a conjecture concerning the oscillation problem for this equation in the case when p t tends to infinity as t

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

on intermediate solutions and the wronskian for half linear differential equations the asymptotic behavior of nonoscillatory solutions of the half linear differential equation is studied in particular two wronskian type functions which have some interesting properties similar to the one of the wronskian in the linear case are given using these properties and suitable integral inequalities the existence of the so called intermediate solutions is examined and an open problem is solved

epi convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator using the variational properties of epi convergence together with suitable results on the measurability of multifunctions and integrands we prove a strong law of large numbers for sequences of integrands from which we deduce a general theorem of almost sure convergence strong consistency for the maximum likelihood estimator

properties and refinements of the fused lasso we consider estimating an unknown signal both blocky and sparse which is corrupted by additive noise we study three interrelated least squares procedures and their asymptotic properties the first procedure is the fused lasso put forward by friedman et al ann appl statist 1 2007 302 332 which we modify into a different estimator called the fused adaptive lasso with better properties the other two estimators we discuss solve least squares problems on sieves one constrains the maximal l 1 norm and the maximal total variation seminorm and the other restricts the number of blocks and the number of nonzero coordinates of the signal we derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso and we derive convergence rates for the sieve estimators explicitly in terms of the constraining parameters

multiscale change point inference summary we introduce a new estimator the simultaneous multiscale change point estimator smuce for the change point problem in exponential family regression an unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level the probability of overestimating the true number of change points k is controlled by the asymptotic null distribution of the multiscale test statistic further we derive exponential bounds for the probability of underestimating k by balancing these quantities will be chosen such that the probability of correctly estimating k is maximized all results are even non asymptotic for the normal case on the basis of these bounds we construct asymptotically honest confidence sets for the unknown step function and its change points at the same time we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log term finally the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n even for an unbounded number of change points we illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions the performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy

note on bounded lp solutions to x a t xc 0 abstract a theorem is presented giving sufficent conditions for the existence of a solution x t to x a t xc 0 to be in lc 1 b and approach 0 as t also x t is shown to be in l2 b and approach 0 as t finally an explicit solution to x xc 0 is given and a discussion of boundedness of solutions under less restrictive conditions is presented

edge preserving smoothers for image processing abstract classical smoothers have limited usefulness in image processing because sharp edges tend to be blurred there is a literature on edge preserving smoothers but these all require moderately large smooth stretches here we discuss an approach to this problem called sigma filtering and propose an improvement based on running m estimation both computational and theoretical aspects are developed for image processing the methods have a niche between standard filtering approaches and bayes markov random field analysis

image denoising pointwise adaptive approach a new method of pointwise adaptation has been proposed and studied in spokoiny 1998 ann statist 26 1356 1378 in the context of estimation of piecewise smooth univariate functions the present paper extends that method to estimation of bivariate grey scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design the proposed estimator hat f x at a point x is simply the average of observations over a window widehat u x selected in a data driven way the theoretical properties of the procedure are studied for the case of piecewise constant images we present a nonasymptotic bound for the accuracy of estimation at a specific grid point x as a function of the number of pixels n of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary it is also shown that the proposed method provides a near optimal rate of estimation near edges and inside homogeneous regions we briefly discuss algorithmic aspects and the complexity of the procedure the numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues an example from satellite sar imaging illustrates the applicability of the method

growth properties of solutions to a linear differential equation on compare les proprietes de dominance des solutions de lny p x y 0 ou ln est un operateur non conjugue aux proprietes de dominance des solutions de son adjointe

limit point type results for nonlinear fourth order differential equations sufficient conditons are given under which the nonlinear fourth order differential equation is of the nonlinear limit point type

iterative potts and blake zisserman minimization for the recovery of functions with discontinuities from indirect measurements signals with discontinuities appear in many problems in the applied sciences ranging from mechanics electrical engineering to biology and medicine the concrete data acquired are typically discrete indirect and noisy measurements of some quantities describing the signal under consideration the task is to restore the signal and in particular the discontinuities in this respect classical methods perform rather poor whereas non convex non smooth variational methods seem to be the correct choice examples are methods based on mumford shah and piecewise constant mumford shah functionals and discretized versions which are known as blake zisserman and potts functionals owing to their non convexity minimization of such functionals is challenging in this paper we propose a new iterative minimization strategy for blake zisserman as well as potts functionals and a related jump sparsity problem dealing with indirect noisy measurements we provide a convergence analysis and underpin our findings with numerical experiments

on multiple context free grammars abstract multiple context free grammars mcfg s is a subclass of generalized context free grammars introduced by pollard 1984 in order to describe the syntax of natural languages the class of languages generated by mcfg s called multiple context free languages or shortly mcfl s properly includes the class of context free languages and is properly included in the class of context sensitive languages first the paper presents results on the generative capacity of mcfg s and also on the properties of mcfl s such as formal language theoretic closure properties next it is shown that the time complexity of the membership problem for multiple context free languages is o ne where n is the length of an input string and e is a constant called the degree of a given mcfg head grammars hg s introduced by pollard and tree adjoining grammars tag s introduced by joshi et al 1975 are also grammatical formalisms to describe the syntax of natural languages the paper also presents the following results on the generative capacities of hg s tag s and 2 mcfg s which are a subclass of mcfg s 1 the class hl of languages generated by hg s is the same as the one generated by tag s 2 hl is the same as the one generated by left wrapping hg s or right wrapping hg s which is a proper subclass of hg s 3 hl is properly included in the one generated by 2 mcfg s as a corollary of 1 it is also shown that hl is a substitution closed full afl

on super linear emden fowler type differential equations we study the second order emden fowler type differential equation in the super linear case using a holder type inequality we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions subdominant intermediate and dominant solutions jointly with this sufficient conditions for the existence of globally positive intermediate solutions are established some of our results are new also for the emden fowler equation

the l 1 potts functional for robust jump sparse reconstruction we investigate the nonsmooth and nonconvex l 1 potts functional in discrete and continuous time we show gamma convergence of discrete l 1 potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer for the discrete l 1 potts problem we introduce an o n 2 time and o n space algorithm to compute an exact minimizer we apply l 1 potts minimization to the problem of recovering piecewise constant signals from noisy measurements f it turns out that the l 1 potts functional has a quite interesting blind deconvolution property in fact we show that mildly blurred jump sparse signals are reconstructed by minimizing the l 1 potts functional furthermore for strongly blurred signals and a known blurring operator we derive an iterative reconstruction algorithm

oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations the equation to be considered iswhere pi t 0 i n and q t are continuous and positive on some half line a it is known that always has strictly monotone nonoscillatory solutions defined on a so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of in this paper sufficient conditions are given for this situation to hold for the structure of the solution space of is also studied

positive solutions of functional differential equations with singular nonlinear terms soit p i a 0 0 i n des fonctions continues on definit les operateurs differentiels l i 0 i n par l 0 y t y t p 0 t l i y t 1 p i t dl i 1 y t dt 1 i n et on considere les equations l n y t f t y g t 0 l n y t f t y g t 0 ou f a 0 0 et g a 0 sont des fonctions continues et lim t g t on etudie la structure des solutions de ces equations definies et positives au voisinage de l infini

oscillation results for emden fowler type differential equations abstract the third order nonlinear differential equation x a t x b t f x 0 is considered we present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature in particular some results recently stated by m gregus and m gregus jr j math anal appl 181 1994 575 585 in addition contributions to the classification of solutions are given the techniques used are based on a transformation which reduces to a suitable disconjugate form to this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to are stated they are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives

change points in nonparametric regression analysis an otherwise smooth regression model are proposed the assumptions needed are much weaker than those made in parametric models the proposed estimators apply as well to the detection of discontinuities in derivatives and therefore to the detection of change points of slope and of higher order curvature the proposed estimators are based on a comparison of left and right one sided kernel smoothers weak convergence of a stochastic process in local differences to a gaussian process is established for properly scaled versions of estimators of the location of a change point the continuous mapping theorem can then be invoked to obtain asymptotic distributions and corresponding rates of convergence for change point estimators these rates are typically faster than n 1 2 rates of global lp convergence of curve estimates with appropriate kernel modifications adapting to estimated change points are derived as a consequence it is shown that these rates of convergence are the same as if the location of the change point was known the methods are illustrated by means of the well known data on the annual flow volume of the nile river between 1871 and 1970 1 introduction nonparametric regression methods are usually applied in order to obtain a smooth fit of a regression curve without having to specify a parametric class of regression functions sometimes a generally smooth curve might contain an isolated discontinuity or change point in the curve or in a possibly higher order derivative and in many cases interest focuses on the occurrence of such change points in parametric approaches to the regression change point problem simple linear regressions before and after a possible change point are assumed and then the possibility of a discontinuity in the form of a jump or of a jump in the first derivative or equivalently a slope change is incorporated into the model see for instance hinkley 1969 and brown durbin and evans 1975 the analysis of change points which describe sudden localized changes typically occurring in economics medicine and the physical sciences has recently found increasing interest general smoothness assumptions allowing for a large class of regression functions to be considered seem to be more appropriate in a variety of applied problems than parametric modelling an

bounded and zero convergent solutions of second order difference equations abstract in this paper we study boundedness and monotonicity properties of a homogeneous second order linear difference equation under certain conditions it is shown that every solution must eventually be monotonic the question of when such monotonic solutions diverge to infinity or converge to zero and the corresponding impact this behavior has on recessive and dominant solutions is investigated a necessary and sufficient condition for all solutions to be bounded is presented comparison theorems are also discussed