bifix codes and sturmian words we prove new results concerning the relation between bifix codes episturmian words and subgroups of free groups we study bifix codes in factorial sets of words we generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set f of words f maximal bifix codes in the case of bifix codes contained in sturmian sets of words we obtain several new results let f be a sturmian set of words defined as the set of factors of a strict episturmian word our results express the fact that an f maximal bifix code of degree d behaves just as the set of words of f of length d an f maximal bifix code of degree d in a sturmian set of words on an alphabet with k letters has k 1 d 1 elements this generalizes the fact that a sturmian set contains k 1 d 1 words of length d moreover given an infinite word x if there is a finite maximal bifix code x of degree d such that x has at most d factors of length d in x then x is ultimately periodic our main result states that any f maximal bifix code of degree d on the alphabet a is the basis of a subgroup of index d of the free group on a

reversible christoffel factorizations we define a family of natural decompositions of sturmian words in christoffel words called reversible christoffel rc factorizations they arise from the observation that two sturmian words with the same language have almost always arbitrarily long abelian equivalent prefixes using the three gap theorem we prove that in each rc factorization only 2 or 3 distinct christoffel words may occur we begin the study of such factorizations considered as infinite words over 2 or 3 letters and show that in the general case they are either sturmian words or obtained by a three interval exchange transformation

tilings and quasiperiodicity quasiperiodic tilings are those tilings in which finite patterns appear regularly in the plane this property is a generalization of the periodicity it was introduced for representing quasicrystals and it is also motivated by the study of quasiperiodic words we prove that if a tile set can tile the plane then it can tile the plane quasiperiodically a surprising result that does not hold for periodicity in order to compare the regularity of quasiperiodic tilings we introduce and study a quasiperiodicity function and prove that it is bounded by x x c if and only if the considered tiling is periodic at last we prove that if a tile set can be used to form a quasiperiodic tiling which is not periodic then it can form an uncountable number of tilings

empirical likelihood ratio confidence regions than are those based on student s t statistic an effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable the method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain algorithms exist for finding the unique global minimum at a superlinear rate of convergence a byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points the multivariate empirical likelihood regions are justified for functions of several means such as variances correlations and regression parameters and for statistics with linear estimating equations an algorithm is given for computing profile empirical likelihoods for these statistics

asymptotic properties of weighted m estimators for variable probability samples i provide a systematic treatment of the asymptotic properties of weighted m estimators under variable probability stratified sampling the characterization of the sampling scheme and representation of the objective function allow for a straightforward analysis simple consistent asymptotic variance matrix estimators are proposed for a large class of problems when stratification is based on exogenous variables i show that the unweighted m estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality when population frequencies are known a more efficient weighting is possible i also show how the results carry over to multinomial sampling

adjustment by minimum discriminant information minimum discriminant information adjustment has primarily been used in the analysis of multinomial data however no such restriction is necessary let p be a distribution on ra and let be a convex set of distributions on ra let xi 1 c i c n be independent and identically distributed observations with common distribution p the minimum discriminant information adjustment mdia of p relative to is the element q of that is closest to p in the sense of kullback leibler discriminant information if pn is the empirical distribution of the xi 1 s i c n and qn is the mdia of pn relative to then qn is the maximum likelihood estimate in y let y consist of distributions a on r such that f t da t where t is a measurable transformation from r to rb and t e rb it is shown that under mild regularity conditions qn converges weakly to q the mdia of the true p with probability 1 and that en d f ddqn is an asymptotically normal and asymptotically unbiased estimate of e d f d dq

estimation of a fixed effect cobb douglas system using panel data abstract this paper considers the estimation of a model consisting of a cobb douglas production function and the associated first order conditions for cost minimization there is a fixed individual firm effect in each equation and the availability of panel data is assumed the conditional mle is derived and shown to be mle after a within transformation it can be written explicitly as a corrected ols estimator or as an iv estimator in the special case in which the errors in the production function are independent of the errors in the first order conditions the conditional mle does not require input price data

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

decompositions of unitary evolutions and entanglement dynamics of bipartite quantum systems we describe a decomposition of the lie group of unitary evolutions for a bipartite quantum system of arbitrary dimensions the decomposition is based on a recursive procedure that systematically uses the cartan classification of the symmetric spaces of the lie group so n the resulting factorization of unitary evolutions clearly displays the local and entangling character of each factor

combinatorial properties of smooth infinite words we describe some combinatorial properties of an intriguing class of infinite words called smooth connected with the kolakoski one k defined as the fixed point of the run length encoding it is based on a bijection on the free monoid over 1 2 that shows some surprising mixing properties all words contain the same finite number of square factors and consequently they are cube free this suggests that they have the same complexity as confirmed by extensive computations we further investigate the occurrences of palindromic subwords finally we show that there exist smooth words obtained as fixed points of substitutions realized by transducers as in the case of k

reversals and palindromes in continued fractions several results on continued fractions expansions are on indirect consequences of the mirror formula we survey occurrences of this formula for sturmian real numbers for simultaneous diophantine approximation and for formal power series

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

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

consistent estimation of linear panel data models with measurement error abstract measurement error causes a bias towards zero when estimating a panel data linear regression model the panel data context offers various opportunities to derive instrumental variables allowing for consistent estimation we consider three sources of moment conditions i restrictions on the covariance matrix of the errors in the equations ii nonzero third moments of the regressors and iii heteroskedasticity and nonlinearity in the relation between the error ridden regressor and another error free regressor in simulations these approaches appear to work well

standard sturmiam morphisms abstract we give a characterization of morphisms which preserve finite and infinite standard sturmian words the class of such morphisms coincides with the monoid d e of the endomorphisms of a where a a b generated by the two elementary morphisms e which interchanges the letter a with b and d which is the fibonacci morphism defined as d a ab d b a some new properties of these morphisms are shown in particular we prove the following result let x n n 0 be the sequence of sets inductively defined as x 0 e x 1 a x n 1 ax n for n 1 where e denotes the empty word and is the operator of left palindrome closure associating to each word w the shortest word having w as suffix for all n 0 one has x n 1 e g d x n a where denotes the identity map from this we derive a new characterization of the set per of all words w having two periods p and q which are coprimes and such that w p q 2 indeed one has that per aa xcup n 0 e l od x n a

gmm redundancy results for general missing data problems we consider questions of efficiency and redundancy in the gmm estimation problem in which we have two sets of moment conditions where two sets of parameters enter into one set of moment conditions while only one set of parameters enters into the other we then apply these results to a selectivity problem in which the first set of moment conditions is for the model of interest and the second set of moment conditions is for the selection process we use these results to explain the counterintuitive result in the literature that under an ignorability assumption that justifies gmm with weighted moment conditions weighting using estimated probabilities of selection is better than weighting using the true probabilities we also consider estimation under an exogeneity of selection assumption such that both the unweighted and the weighted moment conditions are valid and we show that when weighting is not needed for consistency it is also not useful for efficiency

almost any quantum logic gate is universal it is shown that if one can apply some hamiltonian repeatedly to a few variables at a time one can in general effect any desired unitary time evolution on an arbitrarily large number of variables as a result almost any quantum logic gate with two or more inputs is computationally universal in that copies of the gate can be wired together to effect any desired logic circuit and to perform any desired unitary transformation on a set of quantum variables

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

smooth words over arbitrary alphabets smooth infinite words over 1 2 are connected to the kolakoski word k 221121 defined as the fixpoint of the function that counts the length of the runs of 1 s and 2 s in this paper we extend the notion of smooth words to arbitrary alphabets and study some of their combinatorial properties using the run length encoding every word is represented by a word obtained from the iterations of in particular we provide a new representation of the infinite fibonacci word f as an eventually periodic word on the other hand the thue morse word is represented by a finite one

efficient estimation of average treatment effects using the estimated propensity score we are interested in estimating the average effect of a binary treatment on a scalar outcome if assignment to the treatment is exogenous or unconfounded that is independent of the potential outcomes given covariates biases associated with simple treatmentcontrol average comparisons can be removed by adjusting for differences in the covariates rosenbaum and rubin 1983 show that adjusting solely for differences between treated and control units in the propensity score removes all biases associated with differences in covariates although adjusting for differences in the propensity score removes all the bias this can come at the expense of efficiency as shown by hahn 1998 heckman ichimura and todd 1998 and robins mark and newey 1992 we show that weighting by the inverse of a nonparametric estimate of the propensity score rather than the true propensity score leads to an efficient estimate of the average treatment effect we provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score

on the least number of palindromes contained in an infinite word we investigate the least number of palindromic factors in an infinite word we first consider general alphabets and give answers to this problem for periodic and non periodic words closed or not under reversal of factors we then investigate the same problem when the alphabet has size two

canonical decompositions of n qubit quantum computations and concurrence the two qubit canonical decomposition su 4 su 2 su 2 su 2 su 2 writes any two qubit unitary operator as a composition of a local unitary a relative phasing of bell states and a second local unitary using lie theory we generalize this to an n qubit decomposition the concurrence canonical decomposition ccd su 2n kak the group k fixes a bilinear form related to the concurrence and in particular any unitary in k preserves the tangle i 1y i ny 2 for n even thus the ccd shows that any n qubit unitary is a composition of a unitary operator preserving this n tangle a unitary operator in a which applies relative phases to a set of ghz states and a second unitary operator which preserves the tangle as an application we study the extent to which a large random unitary may change concurrence the result states that for a randomly chosen a a su 22p the probability that a carries a state of tangle 0 to a state of maximum tangle approaches 1 as the even number of qubits approach

more efficient estimation under non normality when higher moments do not depend on the regressors using residual augmented least squares under normality least squares is efficient however if the errors are not normal we can gain efficiency from the assertion that higher moments do not depend on the regressors in this paper we show how the assumption that higher moments do not depend on the regressors can be exploited in a gmm framework and we provide simple estimators that are asymptotically equivalent to the gmm estimators these estimators can be calculated by linear regressions which have been augmented with functions of the least squares residuals

on k abelian palindromes abstract a word is called a palindrome if it is equal to its reversal in the paper we consider a k abelian modification of this notion two words are called k abelian equivalent if they contain the same number of occurrences of each factor of length at most k we say that a word is a k abelian palindrome if it is k abelian equivalent to its reversal a question we deal with is the following how many distinct palindromes can a word contain it is well known that a word of length n can contain at most n 1 distinct palindromes as its factors such words are called rich on the other hand there exist infinite words containing only finitely many distinct palindromes as their factors such words are called poor we show that in the k abelian case there exist infinite words containing finitely many distinct k abelian palindromic factors for rich words we show that there exist finite words of length n containing n 2 distinct k abelian palindromes as their factors

sturmian words lyndon words and trees abstract we prove some new combinatorial properties of the set per of all words w having two periods p and q which are coprimes and such that w p q 2 4 3 we show that aperb a b st lynd where st is the set of the finite factors of all infinite sturmian words and lynd is the set of the lyndon words on the alphabet a b it is also shown that aperb a b cp where cp is the set of christoffel primitive words such words can be defined in terms of the slope of the words and of their prefixes 1 from this result one can derive in a different way by using a theorem of borel and laubie that the elements of the set aperb are lyndon words we prove the following correspondence between the ratio p q of the periods p q p q of w per a a b and the slope w b 1 w a 1 of the corresponding christoffel primitive word awb if p q has the development in continued fractions 0 h 1 h n 1 h n 1 then has the development in continued fractions 0 h n h 2 h 1 1 this and other related results can be also derived by means of a theorem which relates the developments in continued fractions of the stern brocot and the raney numbers of a node in a complete binary tree however one needs some further results more precisely we label the binary tree with standard pairs standard tree christoffel pairs christoffel tree and the elements of per farey tree the main theorem is the following if the node w is labeled by the standard pair u v by the christoffel pair x y and by w per then uv wab xy awb the stern brocot number sb w is equal to the slope of the standard word uv and of the christoffel word xy while the raney number ra w is equal to the ratio of the minimal period of wa and the minimal period of wb some further auxiliary results are also derived

an information theoretic alternative to generalized method of moments estimation while optimally weighted gmm estimation has desirable large sample properties its small sample performance is poor in some applications we propose a computationally simple alternative for weakly dependent data generating mechanisms based on minimization of the kullback leibler information criterion conditions are derived under which the large sample properties of this estimator are similar to gmm i e the estimator will be consistent and asymptotically normal with the same asymptotic covariance matrix as gmm in addition we propose overidentifying and parametric restrictions tests as alternatives to analogous gmm procedures

on z factorization and c factorization of standard episturmian words abstract ziv lempel and crochemore factorization are two kinds of factorizations of words related to text processing in this paper we find these factorizations for standard epiesturmian words thus the previously known c factorization of characteristic sturmian words is provided as a special case moreover the two factorizations are compared

inequalities characterizing standard sturmian and episturmian words considering the smallest and the greatest factors with respect to the lexicographic order we associate to each infinite word r two other infinite words min r and max r in this paper we prove that the inequalities as min s max s bs characterize standard sturmian words proper ones and periodic ones and that the condition for any x a and lexicographic order satisfying x min a one has xs min s characterizes standard episturmian words

applications of standard sturmian words to elementary number theory this note is a short survey by no means intended to be complete of some of the descriptions which have been given of standard sturmian words and of some of the applications of these descriptions to elementary number theory elementary number theory is interpreted fairly broadly the descriptions and applications below have appeared before except for fact 4 application 1 and the proof of application 2

improved instrumental variables and generalized method of moments estimators this paper considers gmm estimation with moment conditions of the usual type plus extra moment conditions that do not depend on the parameters the extra moment conditions improve efficiency if they are correlated with the original set of moment conditions this is true in linear and nonlinear models but in linear models we provide simple explicit formulas for the improved estimators our results may be useful in rational expectations models in which an equation s error is a forecast error and the extra moment conditions would be forecast errors in related variables

constructive control of quantum systems using factorization of unitary operators we demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or gaussian wavepackets to illustrate the technique it is applied to find control schemes to achieve population transfers for pure state systems complete inversions of the ensemble populations for mixed state systems create arbitrary superposition states and optimize the ensemble average of dynamic observables

on christoffel and standard words and their derivatives we introduce and study natural derivatives for christoffel and standard words as well as for characteristic sturmian words these derivatives which are defined as inverse images under suitable morphisms preserve the aforementioned classes of words in the case of christoffel words the morphisms involved map a to a k 1 b resp a b k and b to a k b resp a b k 1 for a suitable k 0 as long as derivatives are not just a single letter higher order derivatives are naturally obtained we define the depth of a christoffel or of a standard word as the smallest order for which the derivative is a single letter we give several combinatorial and arithmetic descriptions of the depth and tight lower and upper bounds for it

dipole quadrupole decomposition of two coupled spin 1 systems using dipole and quadrupole operators as the orthogonal basis of su 3 algebra and successive cartan decompositions the decomposition of matrices for two coupled spin 1 systems is investigated so as to meet the requirements of some realistic quantum systems finally this kind of decomposition for the ternary swap gate is given specifically this method can be used to investigate the realization of two qutrit logic gates

palindromes in circular words there is a very short and beautiful proof that the number of distinct non empty palindromes in a word of length n is at most n in this paper we show with a very complicated proof that the number of distinct non empty palindromes with length at most n in a circular word of length n is less than 5n 3 for n divisible by 3 we present circular words of length n containing 5n 3 2 distinct palindromes so the bound is almost sharp the paper finishes with some open problems

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

model identification for spin networks abstract we consider the problem of determining the unknown parameters of the hamiltonian of a network of spin 1 2 particles in particular we study experiments in which the system is driven by an externally applied electro magnetic field and the expectation value of the total magnetization is measured under appropriate system theoretic assumptions we prove that if it is possible to prepare the system in a known initial state the above experiment allows to identify the parameters of the hamiltonian in the case where the initial state is itself an unknown parameter we characterize all the pairs hamiltonian initial state which give the same value of the magnetization for every form of the driving electro magnetic field the analysis is motivated by recent results on the isospectrality of hamiltonians describing magnetic molecules

sturmian words and the stern sequence central standard and christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart to characteristic sturmian words a natural arithmetization of the theory is obtained by representing central and christoffel words by irreducible fractions labeling respectively two binary trees the raney or calkin wilf tree and the stern brocot tree the sequence of denominators of the fractions in the raney tree is the famous stern diatomic numerical sequence an interpretation of the terms s n of stern s sequence as lengths of christoffel words when n is odd and as minimal periods of central words when n is even allows one to interpret several results on christoffel and central words in terms of stern s sequence and conversely to obtain new insight into the combinatorics of christoffel and central words one of our main results is a non commutative version of the alternating bit sets theorem by calkin and wilf we also study the length distribution of christoffel words corresponding to nodes of equal height in the tree obtaining some interesting bounds and inequalities

median estimation in the presence of auxiliary information summary this paper deals with the estimation under simple random sampling of a finite population median in the presence of an auxiliary variable an estimator which is a simple modification of the ratio estimator for the mean is studied along with two other proposed estimators derived from a different approach these estimators are applicable in situations where only the population median or a grouped frequency distribution of the auxiliary variable is known on the basis of the asymptotic properties derived and some simulation results the efficiencies of these estimators are compared it is shown that the two estimators proposed unlike that obtained from modifying the ratio estimator always dominate the sample median in mean squared error and that their efficiencies depend directly on the probability of concordance rather than on the validity of a linearity assumption between the survey variable and the auxiliary variable when the linearity assumption is approximately satisfied the three estimators are seen to have comparable efficiencies while the modified ratio estimator is found to be quite sensitive to departures from the linearity assumnption

palindromic language of thin discrete planes we work on the rveills hyperplane p v 0 with normal vectorvrd shift 0 and thicknessr such a hyperplane is connected as soon as is greater than some value v 0 called the connecting thickness of v with null shift in the case where v satisfies the so called kraaikamp and meester criterion at the connecting thickness the hyperplane has very specific properties first of all the adjacency graph of the voxels forms a tree this tree appeared in many works both in discrete geometry and in discrete dynamical systems in addition it is well known that for a finite coding of length n of discrete lines the number of palindromes in the language is exactly n 1 we extend this notion of language to labeled trees and we compute the number of distinct palindromes in fact for our voxel adjacency trees with n letters we show that the number of palindromes in the language is also n 1 this result establishes a first link between combinatorics on words palindromic languages voxel adjacency trees and connecting thickness of rveills hyperplanes it also provides a better understanding of the combinatorial structure of discrete planes