on gamma function inequalities on montre que certaines fonctions comportant les quotients de fonctions gamma sont completement monotones on en deduit des inegalites on etablit aussi la divisibilite infinie de plusieurs distributions de probabilite dont les transformees de laplace comportent des quotients de fonctions gamma

estimation of a covariance matrix using the reference prior estimation of a covariance matrix sum is a notoriously difficult problem the standard unbiased estimator can be substantially suboptimal we approach the problem from a noninformative prior bayesian perspective developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting bayes estimators these expressions involve the computation of high dimensional posterior expectations which is done using a recent markov chain simulation tool the hit and run sampler frequentist risk comparisons with previously suggested estimators are also given and determination of the accuracy of the estimators is addressed

levy processes and fourier multipliers abstract we study fourier multipliers which result from modulating jumps of levy processes using the theory of martingale transforms we prove that these operators are bounded in l p r d for 1 p and we obtain the same explicit bound for their norm as the one known for the second order riesz transforms

parsimonious covariance matrix estimation for longitudinal data this article proposes a data driven method to identify parsimony in the covariance matrix of longitudinal data and to exploit any such parsimony to produce a statistically efficient estimator of the covariance matrix the approach parameterizes the covariance matrix through the cholesky decomposition of its inverse for longitudinal data this is a one step ahead predictive representation and the cholesky factor is likely to have off diagonal elements that are zero or close to zero a hierarchical bayesian model is used to identify any such zeros in the cholesky factor similar to approaches that have been successful in bayesian variable selection the model is estimated using a markov chain monte carlo sampling scheme that is computationally efficient and can be applied to covariance matrices of high dimension it is demonstrated through simulations that the proposed method compares favorably in terms of statistical efficiency with a highly regarded competing approach the estimator is applied to three

on the number of factors of sturmian words abstract we prove that for m 1 card a m 1 m i 1 m i 1 i where a m is the set of factors of length m of all the sturmian words and is the euler function this result was conjectured by dulucq and gouyou beauchamps 1987 who proved that this result implies that the language m 0 a m c is inherently ambiguous we also give a combinatorial version of the riemann hypothesis

balance properties of multi dimensional words a word u is called 1 balanced if for any two factors v and w of u of equal length we have 1 v i w i1 for each letter i where v i denotes the number of occurrences of i in the factor v the aim of this paper is to extend the notion of balance to multi dimensional words we first characterize all 1 balanced words on zn in particular we prove they are fully periodic for n 1 we then give a quantitative measure of non balancedness for some words on z2 with irrational density including two dimensional sturmian words

studies on finite sturmian words several properties of finite sturmian words are proved the inverse of the richomme seebold bijection between factor sets of given length of sturmian sequences and left special sturmian words is constructed using circular factors of christoffel words and studied the labbe bijection between christoffel words and left special sturmian words is introduced and studied factor sets are classified by two types the two classes are characterized by conjugation and periodicity properties the two classes are related to the rauzy graph a normal form for finite sturmian words is given following a result of de luca and de luca several classes of sturmian words are characterized through this normal form left or right special words bispecial words palindromes central words the normal form is used to derive a proof of the lipatov mignosi formula for the number of sturmian words of given length to produce a linear algorithm which checks if a word is sturmian and which computes its normal form to construct algorithmically the contraction and completion in various classes of sturmian words to compute the de luca palindromic closure of any sturmian word and prove that it is sturmian the completion is related to the de luca palindromization function and it is shown that the sets of directive words of central words of even and odd length are rational group languages christoffel words are characterized in several ways an extension of a result of chuan an extension of a result of droubay pirillo by counting circular factors by counting lyndon factors a special christoffel word is constructed having length a given markoff number a bivariate count of special sturmian words is given using ideas of bedaride domenjoud jamet and remy open problems are given

combinatorics of patterns of a bidimensional sturmian sequence resume nous etudions une generalisation des suites sturmiennes en construisant une surface plissee donnee par l approximation d un plan par trois sortes de faces carrees orientees suivant les trois plans de coordonnees a cette surface on associe par projection un pavage du plan par trois sortes de losanges on definit sur ce pavage une fonction de complexite en comptant le nombre de motifs distincts d une fenetre de taille donnee nous donnons en etudiant les prolongements des motifs la forme explicite de cette fonction dans le cas d une fenetre triangulaire et d une fenetre en forme de parallelogramme

needles and straw in haystacks empirical bayes estimates of possibly sparse sequences an empirical bayes approach to the estimation of possibly sparse sequences observed in gaussian white noise is set out and investigated the prior considered is a mixture of an atom of probability at zero and a heavy tailed density y with the mixing weight chosen by marginal maximum likelihood in the hope of adapting between sparse and dense sequences if estimation is then carried out using the posterior median this is a random thresholding procedure other thresholding rules employing the same threshold can also be used probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n allowing for sparsity of various kinds and degrees the signal classes considered are nearly black sequences where only a proportion is allowed to be nonzero and sequences with normalized p norm bounded by for 0 and 0 1 simulations show excellent performance for appropriately chosen functions y the method is computationally tractable and software is available the extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered

on covering a digital disc with concentric circles in z2 digital circles and digital discs satisfy many bizarre anisotropic properties understanding of which is essential for solving various problems in image analysis and computer graphics in this paper we study the underlying properties of absentee pixels that appear while covering a digital disc with concentric digital circles we present for the first time a mathematical characterization of these pixels based on number theory and digital geometry interestingly the absentees occur in multitude and we show that their count varies quadratically with the radius the notion of infimum parabola and supremum parabola has been used to derive the count of these absentees using this parabolic characterization we derive a necessary and sufficient condition for a pixel to be a disc absentee and obtain the geometric properties of the absentees an algorithm to locate the absentees is presented we show that the ratio of the absentee pixels to the total number of disc pixels approaches a constant with increasing radius test results have been furnished to substantiate our theoretical findings

repetitions in sturmian strings in this paper we apply a simple representation of sturmian strings which we call a reduction sequence to three algorithms the first algorithm accepts as input a given finite string x and determines in time whether or not x is sturmian the second algorithm is a modification of the first that in the case that x is sturmian outputs a reduction sequence for a superstring u of x that is a prefix of an infinite sturmian string the third algorithm uses the reduction sequence of u to compute all the repetitions in u in time thus extending a recent result for fibonacci strings the third algorithm is also based on a characterization of the repetitions in a sturmian string that describes them compactly in terms of runs finally for every integer r 4 we show how to construct an infinite sturmian string that contains maximal repetitions of exponents 2 3 r 1 but none of exponent r

infinite words with linear subword complexity abstract in section 1 we study the relations among some combinatorial properties of infinite words especially in the case of infinite words with linear subword complexity the main result of section 1 concerns a permutation property of infinite words with linear complexity in section 2 we investigate the special case of the sturmian infinite words the main result is that a sturmian infinite word associated to a real number contains no k th powers iff has a continued fraction expansion with bounded partial quotients in section 3 it is shown that some of the results of sections 1 and 2 are optimal

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

some combinatorial properties of sturmian words abstract in this paper we give a characterization of finite sturmian words by palindrome words which generalizes a property of the fibonacci words we prove that the set st of finite sturmian words coincides with the set of the factors of all the words w such that w ab cxy with a b c palindromes x y a b and x y moreover using this result we prove that st is equal to the set of the factors of all words w having two periods p and q which are coprimes and such that w p q 2 several other combinatorial properties concerning special and bispecial elements of st are shown as a consequence we give a new and purely combinatorial proof of the enumeration formula of st

combinatorial properties of sequences defined by the billiard in paved triangles resume nous etudions certaines proprietes combinatoires de la dynamique symbolique du billard dans le triangle rectangle isocele et le demi triangle equilateral dans le cas du triangle rectangle isocele nous developpons un algorithme de construction des suites obtenues comme codage et nous calculons la complexite de ces suites lorsqu elles ne sont pas periodiques nous donnons des regles de construction des mots infinis qui codent les trajectoires de billard dans le demi triangle equilateral ceci nous a conduit a etablir un algorithme de construction du codage d un point sous une rotation munie d une partition en intervalles

efficient estimation of covariance selection models summary a bayesian method is proposed for estimating an inverse covariance matrix from gaussian data the method is based on a prior that allows the off diagonal elements of the inverse covariance matrix to be zero and in many applications results in a parsimonious parameterisation of the covariance matrix no assumption is made about the structure of the corresponding graphical model so the method applies to both nondecomposable and decomposable graphs all the parameters are estimated by model averaging using an efficient metropolis hastings sampling scheme a simulation study demonstrates that the method produces statistically efficient estimators of the covariance matrix when the inverse covariance matrix is sparse the methodology is illustrated by applying it to three examples that are high dimensional relative to the sample size

on a combinatorial property of sturmian words recall that a semigroup has the property pn if for any sequence of n of its elements two differently permuted products of these n elements are equal let s be an infinite sturmian word on a 2 letter alphabet a we prove that the rees quotient of a by the set of the non factors of s has p4 and that this result is the best possible we prove also that if st is the set of all finite sturmian words then the rees quotient a a st has p8

factors of characteristic words of irrational numbers let be an irrational number between 0 and 1 the characteristic word f of is defined to be the infinite word over 0 1 whose nth letter is n 1 n n 1 it is well known that for each m 1 f has exactly m 1 distinct factors of length m in this paper we shall develop a method to construct these factors under our construction the 1 sets of these m 1 factors x0 m x1 m xm m are determined these factors are increasing in the lexicographic order and their moments m x0 m m x1 m m xm m form an increasing sequence of m 1 consecutive integers some known results about generating factors of f using the unbordered words and their conjugates turn out to be consequences of our main theorem

the complexity of functions on lattices abstract let f z 0 1 be a given function in 1938 morse and hedlund observed that if the number of distinct vectors f x 1 f x n x z called complexity is at most n for some positive integer n then f is periodic with period at most n this result is best possible functions with low complexity have been studied to a large extent and relations with or applications to many branches of mathematics computer science and physics are known in the present paper we discuss the above phenomenon in greater generality to begin with we observe that f is periodic if the number of distinct vectors f x a 1 f x a n x z is at most n for some n and given integers a 1 n but that the period cannot be bounded as a function of n only our main topic are multi dimensional functions f z k 0 1 with the property that for some n and distinct vectors a 1 a n z k the number of distinct vectors f x a 1 f x a n x z k is bounded by n we show that such a function with arbitrary k is periodic if n 3 for n 4 there are non periodic examples which we determine completely finally limitations to the general periodicity principle are discussed a conjecture for convex bodies a 1 a n in z 2 is made and we prove it for n 4

a new geometric approach to sturmian words we introduce a new geometric approach to sturmian words by means of a mapping that associates certain lines in the nxn grid and sets of finite sturmian words of length n using this mapping we give new proofs of the formulas enumerating the finite sturmian words and the palindromic finite sturmian words of a given length we also give a new proof for the well known result that a factor of a sturmian word has precisely two return words

martingale transforms and related singular integrals byrodrigo banuelosabstract the operators obtained by taking conditional expectation of continuoustime martingale transforms are studied both on the circle t and on r using aburkholder gundy inequality for vector valued martingales it is shown that thevector formed by any number of these operators is bounded on lp r 1 p oo with constants that depend only on p and the norms of the matrices involved as acorollary we obtain a recent result of stein on the boundedness of the riesztransforms on lp k 1 p oo with constants independent of n

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

weak repetitions in sturmian strings in this paper we analyze the weak repetitions in sturmian strings and show that an optimally efficient algorithm to compute the weak repetitions in sturmian strings according to the output encodings defined in the literature is quadratic in the string length finally we present an encoding that leads to a linear time algorithm

fast polygonal approximation of digital curves using relaxed straightness properties several existing digital straight line segment dss recognition algorithms can be used to determine the digital straightness of a given one pixel thick digital curve because of the inherent geometric constraints of digital straightness these algorithms often produce a large number of segments to cover a given digital curve representing a real life object image thus a curve segment which is not exactly digitally straight but appears to be visually straight is fragmented into multiple dss when these algorithms are run in this paper a new concept of approximate straightness is introduced by relaxing certain conditions of dss and an algorithm is described to extract those segments from a digital curve the number of such segments required to cover the curve is found to be significantly fewer than that of the exact dss cover as a result the data set required for representing a curve also reduces to a large extent the extracted set of segments can further be combined to determine a compact polygonal approximation of a digital curve based on certain approximation criteria and a specified error tolerance the proposed algorithm involves only primitive integer operations and thus runs very fast compared to those based on exact dss the overall time complexity becomes linear in the number of points present in the representative set experimental results on several digital curves demonstrate the speed elegance and efficacy of the proposed method

balancedness and coboundaries in symbolic systems abstract this paper studies balancedness for infinite words and subshifts both for letters and factors balancedness is a measure of disorder that amounts to strong convergence properties for frequencies it measures the difference between the numbers of occurrences of a given word in factors of the same length we focus on two families of words namely dendric words and words generated by substitutions the family of dendric words includes sturmian and arnoux rauzy words as well as codings of regular interval exchanges we prove that dendric words are balanced on letters if and only if they are balanced on words in the substitutive case we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness

on the fragmentary complexity of symbolic sequences a measure of the ability of a symbolic sequence to be covered by initial fragments of another symbolic sequence is introduced and its basic properties are investigated applications to the characterization of symbolic sequences associated with shift mappings on a torus corresponding to a special partitioning of the torus and to multirate systems of coprocessors are considered

estimation of a covariance matrix under stein s loss on decrit la technique generale de stein pour ameliorer l estimateur sans biais invariant le meilleur et l estimateur minimax de la matrice de covariance normale

hardy stein identity for non symmetric levy processes and fourier multipliers abstract using ito s formula for processes with jumps we extend the hardy stein identity proved in 7 to non symmetric pure jump levy processes and derive its martingale version by a symmetrization argument for littlewood paley functions the l p boundedness of fourier multipliers arising from non symmetric pure jump levy processes is proved

iterated floor function algebraic numbers discrete chaos beatty subsequences semigroups for a real number a the floor function laj is the integer part of a the sequence lmaj m 1 2 3 is the beatty sequence of ac identities are proved which express the sum of the iterated floor functional ai for 1 i n operating on a nonzero algebraic number a of degree n in terms of only ai lmcaj m and a bounded term applications include discrete chaos discrete dynamical systems explicit construction of infinite nonchaotic subsequences of chaotic sequences discrete order identities explicit construction of nontrivial beatty subsequences and certain arithmetical semigroups beatty sequences have a large literature in combinatorics they have also been used in nonperiodic tilings quasicrystallography periodic scheduling computer vision digital lines and formal language theory

repetitive perhaps but certainly not boring in this paper some of the work done on repetitions in strings is surveyed especially that of an algorithmic nature several open problems are described and conjectures formulated about some of them

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

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

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

on the factors of the sturmian sequences resume cet article a pour objet l etude d une construction associant a toute droite de pente p q p et q premiers entre eux et q n un mot de longueur n sur l alphabet 0 1 nous montrons que nous obtenons par cette construction le langage constitue de tous les facteurs des suites de sturm nous formulons apres avoir obtenu une equation fonctionelle dont la solution est la serie generatrice de ce langage une conjecture reliant cette serie generatrice a la fonction d euler

enumeration and structure of trapezoidal words trapezoidal words are words having at most n 1 distinct factors of length n for every n 0 they therefore encompass finite sturmian words we give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration we then separate trapezoidal words into two disjoint classes open and closed a trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix otherwise it is open we investigate open and closed trapezoidal words in relation with their special factors we prove that sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a sturmian palindrome if and only if its longest repeated prefix is a palindrome we also define a new class of words semicentral words and show that they are characterized by the property that they can be written as uxyu for a central word u and two different letters x y finally we investigate the prefixes of the fibonacci word with respect to the property of being open or closed trapezoidal words and show that the sequence of open and closed prefixes of the fibonacci word follows the fibonacci sequence

some properties of the factors of sturmian sequences in this paper we introduce the singular words of sturmian sequences which play an important role in studying the properties of the factors of sturmian sequence we also completely determine the powers of the factors the overlaps of the factors and the structure of the palindromes of the factors

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

on the sparsity of signals in a random sample this article proposes a method of moments technique for estimating the sparsity of signals in a random sample this involves estimating the largest eigenvalue of a large hermitian trigonometric matrix under mild conditions as illustration the method is applied to two well known problems the first focuses on the sparsity of a large covariance matrix and the second investigates the sparsity of a sequence of signals observed with stationary weakly dependent noise simulation shows that the proposed estimators can have significantly smaller mean absolute errors than their main competitors copyright 2012 oxford university press

sturmian morphisms and a words abstract in this paper we prove that the set w of words over 0 1 coincides with the set of images of 0 under sturmian morphisms this gives an alternative approach of generating words for each w w a sturmian solution h of the equation w h 0 which is a composition of sturmian morphisms of special forms can be constructed explicitly in order to demonstrate how results on sturmian morphic images are reformulated as results on words and vice versa we give a sturmian proof for the conjugation and reversion theorem of words obtained by the present author in an earlier paper

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

hierarchical structures in sturmian dynamical systems the paper is concerned with hierarchical stuctures in subshifis over a finite alphabet in particular we present a hierarchy based approach to sturmian systems this approach is then used to characterize the linearly repetitive sturmian systems among the sturmian systems by uniform positivity of certain weights more generally we discuss various bounds on weights and their relationship finally we combine hierarchies and bounds on a weights to study a thermodynamic formalism for lattice gas models associated to arbitrary linearly repetitive subshifts