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

global attractor for degenerate damped hyperbolic equations abstract the aim of this paper is to consider a class of degenerate damped hyperbolic equations with the critical nonlinearity involving an operator l that is x elliptic with respect to a family of vector fields x we prove the global existence of solutions and characterize their long time behavior in particular we show the semigroup generated by the equation has a compact connected and invariant attractor

burrows wheeler transform and palindromic richness the investigation of the extremal case of the burrows wheeler transform leads to study the words w over an ordered alphabet a a 1 a 2 a k with a 1

quantifying long range correlations with a multiscale ordinal pattern approach abstract in this paper we use the ordinal patterns probabilities associated with fractional brownian motions for estimating the hurst exponent of artificially generated and experimentally measured data numerical analysis show a reliable estimation of this scaling parameter even when data with low resolution are analysed robustness to observational noise is also obtained several experimental applications allow us to confirm the practical utility of the proposed approach we contrast results obtained by implementing this multiscale symbolic tool with those obtained from the classical detrended fluctuation analysis

wiener criterion for x elliptic operators abstract in this note we prove a wiener criterion of regularity of boundary points for the dirichlet problem related to x elliptic operators in divergence form enjoying the doubling condition and the poincare inequality as a step towards this result we exhibit some other characterizations of regularity in terms of the capacitary potentials finally we also show that a cone type criterion holds true in our setting

factors of characteristic words location and decompositions let a be an irrational number with 0 a 1 and let f a be the characteristic word of a the location of a factor w of f a is defined to be the set of all positions in f a at which w occurs in this paper an explicit formula of the location of each factor w of f a is obtained the decompositions red a w and sod a w of f a associated with w are established the former one involves all return words of w while the latter one involves all occurrences of w and the corresponding separate factors and overlap factors these results generalize the work by the present authors 2005 z x wen and z y wen 1994 g melancon 1999 w t cao and z y wen 2003 i m araujo and v bruyere 2005 and a glen 2006

episturmian words and episturmian morphisms infinite episturmian words are a generalization of sturmian words which includes the arnoux rauzy sequences we continue their study and that of episturmian morphisms begun previously in relation with the action of the shift operator palindromic and periodic factors of these words are described we consider in particular the case where these words are generated by morphisms and introduce then a notion of intercept generalizing that of sturmian words finally we prove that the frequencies of the factors in a strong sense do exist for all episturmian words

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

an invariant harnack inequality for a class of subelliptic operators under global doubling and poincare assumptions and applications abstract the aim of this paper is to prove an invariant non homogeneous harnack inequality for a class of subelliptic operators l in divergence form with low regular coefficients the main assumption whose geometric meaning is well known in the literature on harnack inequalities is the requirement that l be naturally associated with a carnot caratheodory doubling metric space where a poincare inequality also holds both doubling and poincare conditions are assumed to hold globally for every cc ball accordingly the harnack inequality will hold true on every cc ball applications to inner and boundary holder estimates are provided together with pertinent results on the green function for l an explicit example of a class of operators for which our results are fulfilled is also given via the green function for l the global nature of the harnack inequality can be applied to the study of the existence of a fundamental solution for l globally defined out of the diagonal of r n r n

long range dependence in tree ring width time series of austrocedrus chilensis revealed by means of the detrended fluctuation analysis by using the detrended fluctuation analysis and detrended moving average method 823 time series of tree ring widths in austrocedrus chilensis in patagonia were analyzed the tree ring widths of a chilensis have been widely used for climatological studies the results point out to the presence of significant scaling in the temporal fluctuations of tree ring which is not due to singular probability density function of the widths but due to the presence of long range correlations such results are in good agreement with those concerning the evidence of long range dependencies in weather time series

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

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

a new characteristic property of rich words originally introduced and studied by the third and fourth authors together with j justin and s widmer 2008 rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes several characterizations of rich words have already been established a particularly nice characteristic property is that all complete returns to palindromes are palindromes in this note we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix

extensions of rich words a word w is rich if it has w 1 many distinct palindromic factors including the empty word this article contains several results about rich words particularly related to extending them a word w can be eventually extended richly in n ways if there exists a finite word u and n distinct letters a alph w such that wua is rich we will prove that every non unary rich word can be eventually extended richly in at least two different ways but not always in three or more ways we will also prove that every rich word can be extended to both periodic and aperiodic infinite rich words the defect of a finite word w is defined by d w w 1 pal w this concept has been studied in various papers here we will define a new concept infinite defect for a finite word w the definition is d w min d z zis an infinite word which has factorw we will show that the infinite defect of a finite word is always finite and give some upper bounds for it the difference between defect and infinite defect is also investigated we will also give an upper and a lower bound for the number of rich words a new class of words two dimensional rich words is also introduced

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

rauzy substitutions and multi dimensional sturmian words recently berthe and the author developed a method to construct a multi dimensional sturmian word as a limit of a sequence of words on lattice domains in the present paper the reach of the approach is studied in the special case of rauzy substitutions

liouville theorem for x elliptic operators abstract we prove a liouville type theorem for a class of degenerate elliptic operators of the form l u i j 1 n x i a i j x j u i 1 n b i x i u l is supposed to be x elliptic with respect to a family x x 1 x m of locally lipschitz continuous vector fields in the sense introduced in e lanconelli a e kogoj x elliptic operators and x control distances contributions in honor of the memory of ennio de giorgi ricerche di matematica 49 suppl 2000 223 243

liouville type theorems for some complex hypoelliptic operators it is shown that the natural analogue of liouville s theorem holds for the well known hypoelliptic operators l for n introduced and studied by folland and stein on the heisenberg group hn since these operators are non real for 0 the usual methods of potential theory fail and are replaced by an explicit use of the fundamental solution

powers in a class of a strict standard episturmian words this paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic sturmian words in particular we explicitly determine all integer powers occurring in such infinite words extending recent results of damanik and lenz d damanik d lenz powers in sturmian sequences european j combin 24 2003 377 390 doi 10 1016 s0195 6698 03 00026 x who studied powers in sturmian words the key tools in our analysis are canonical decompositions and a generalization of singular words which were originally defined for the ubiquitous fibonacci word our main results are demonstrated via some examples including the k bonacci word a generalization of the fibonacci word to a k letter alphabet k 2

covering theorems inequalities on metric spaces and applications to pde s we establish a covering lemma of besicovitch type for metric balls in the setting of holder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets for families of measurable functions we introduce the notions of power decay critical density and double ball property and with the aid of the covering theorem we show how these notions are related next we present an axiomatic procedure to establish harnack inequality that permits to handle both divergence and non divergence linear equations

dynamics for a stochastic degenerate parabolic equation abstract in this paper we consider the long time behavior of a class of stochastic degenerate parabolic equations involving an operator which is x elliptic with respect to a family of locally lipschitz continuous vector fields x x 1 x 2 x m the nonlinearity satisfies a dissipative condition with polynomial growth of arbitrary order p 2 the existence of the random attractor in l 2 o higher order integrability and continuity in h are established

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

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

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

more properties of the fibonacci word on an infinite alphabet abstract recently the fibonacci word w on an infinite alphabet was introduced by zhang et al 2017 13 as a fixed point of the morphism 2 i 2 i 2 i 1 2 i 1 2 i 2 over all i n in this paper we investigate the occurrence of squares palindromes and lyndon factors in this infinite word

a control condition for a weak harnack inequality abstract we introduce a new condition allowing to get a weak harnack inequality for non negative solutions to linear second order degenerate elliptic equations of x elliptic type roughly speaking our condition requires that the euclidean balls of small radius are representable by means of x controllable almost exponential maps

a characterization of fine words over a finite alphabet to any infinite word t over a finite alphabet a we can associate two infinite words min t and max t such that any prefix of min t resp max t is the lexicographically smallest resp greatest amongst the factors of t of the same length we say that an infinite word t over a is fine if there exists an infinite word s such that for any lexicographic order min t as where a min a in this paper we characterize fine words specifically we prove that an infinite word t is fine if and only if t is either a strict episturmian word or a strict skew episturmian word this characterization generalizes a recent result of g pirillo who proved that a fine word over a 2 letter alphabet is either an aperiodic sturmian word or an ultimately periodic but not periodic infinite word all of whose factors are finite sturmian

liouville properties and critical value of fully nonlinear elliptic operators abstract we prove some liouville properties for sub and supersolutions of fully nonlinear degenerate elliptic equations in the whole space our assumptions allow the coefficients of the first order terms to be large at infinity provided they have an appropriate sign as in ornstein uhlenbeck operators we give two applications the first is a stabilization property for large times of solutions to fully nonlinear parabolic equations the second is the solvability of an ergodic hamilton jacobi bellman equation that identifies a unique critical value of the operator

attractors met x elliptic operators abstract we consider degenerate parabolic and damped hyperbolic equations involving an operator l that is x elliptic with respect to a family of locally lipschitz continuous vector fields x x 1 x m the local well posedness is established under subcritical growth restrictions on the nonlinearity f which are determined by the geometry and functional setting naturally associated with the family of vector fields x assuming additionally that f is dissipative the global existence of solutions follows and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems

estimates of the green function for x elliptic operators in this note we establish two sided pointwise estimates near the pole of the green function for x elliptic operators in divergence form enjoying the doubling condition and the poincare inequality as a step towards this result we also prove a nonhomogeneous harnack inequality some representation formulas and a result on approximation by regular domains

on the number of episturmian palindromes episturmian words are a suitable generalization to arbitrary alphabets of sturmian words in this paper we are interested in the problem of enumerating the palindromes in all episturmian words over a k letter alphabet a k we give a formula for the map g k giving for any n the number of all palindromes of length n in all episturmian words over a k this formula extends to k 2 a similar result obtained for k 2 by the second and third authors in 2006 the map g k is expressed in terms of the map p k counting for each n the palindromic prefixes of all standard episturmian words epicentral words for any n 0 p 2 n f n 2 where f is the totient euler function the map p k plays an essential role also in the enumeration formula for the map l k counting for each n the finite episturmian words over a k similarly to euler s function the behavior of p k is quite irregular the first values of p k and of the related maps g k and l k for 3 k 6 have been calculated and reported in the paper some properties of p k are shown in particular broad upper and lower bounds for p k as well as for m 0 np k m and g k are determined finally some conjectures concerning the map p k are formulated

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

on different generalizations of episturmian words in this paper we study some classes of infinite words generalizing episturmian words and analyse the relations occurring among such classes in each case the reversal operator r is replaced by an arbitrary involutory antimorphism q of the free monoid a in particular we define the class of q words with seed whose standard elements q standard words with seed are constructed by an iterative q palindrome closure process starting from a finite word u 0 called the seed when the seed is empty one obtains q words episturmian words are exactly the r words one of the main theorems of the paper characterizes q words with seed as infinite words closed under q and having at most one left special factor of each length n n where n is some nonnegative integer depending on the word when n 0 we call such words q episturmian further results on the structure of q episturmian words are proved in particular some relationships between q words with or without seed and q episturmian words are shown

on a generalization of christoffel words epichristoffel words sturmian sequences are well known as the ones having minimal complexity over a 2 letter alphabet they are also the balanced sequences over a 2 letter alphabet and the sequences describing discrete lines they are famous and have been extensively studied since the 18th century one of the extensions of these sequences over a k letter alphabet with k 3 is the episturmian sequences which generalizes a construction of sturmian sequences using the palindromic closure operation there exists a finite version of the sturmian sequences called the christoffel words they have been known since the works of christoffel and have interested many mathematicians in this paper we introduce a generalization of christoffel words for an alphabet with 3 letters or more using the episturmian morphisms we call them the epichristoffel words we define this new class of finite words and show how some of the properties of the christoffel words can be generalized naturally or not for this class

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

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

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

liouville results for fully nonlinear equations modeled on hormander vector fields i the heisenberg group this paper studies liouville properties for viscosity sub and supersolutions of fully nonlinear degenerate elliptic pdes under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the hormander condition a general set of sufficient conditions is given such that all subsolutions bounded above are constant it includes the existence of a supersolution out of a big ball that explodes at infinity therefore for a large class of operators the problem is reduced to finding such a lyapunov like function this is done here for the vector fields that generate the heisenberg group giving explicit conditions on the sign and size of the first and zero th order terms in the equation the optimality of the conditions is shown via several examples a sequel of this paper applies the methods to other carnot groups and to grushin geometries

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

a generalized palindromization map in free monoids the palindromization map j in a free monoid a was introduced in 1997 by the first author in the case of a binary alphabet a and later extended by other authors to arbitrary alphabets acting on infinite words j generates the class of standard episturmian words including standard arnoux rauzy words in this paper we generalize the palindromization map starting with a given code x over a the new map j x maps x to the set pal of palindromes of a in this way some properties of j are lost and some are saved in a weak form when x has a finite deciphering delay one can extend j x to x w generating a class of infinite words much wider than standard episturmian words for a finite and maximal code x over a we give a suitable generalization of standard arnoux rauzy words called x ar words we prove that any x ar word is a morphic image of a standard arnoux rauzy word and we determine some suitable linear lower and upper bounds to its factor complexity for any code x we say that j x is conservative when j x x x we study conservative maps j x and conditions on x assuring that j x is conservative we also investigate the special case of morphic conservative maps j x i e maps such that email protected email protected email protected email protected email protected for an injective morphism f finally we generalize j x by replacing palindromic closure with q palindromic closure where q is any involutory antimorphism of a this yields an extension of the class of q standard words introduced by the authors in 2006

scaling characteristics of ocean wave height time series abstract fluctuations in the significant wave height can be quantified by using scaling statistics in this paper the scaling properties of the significant wave height were explored by using a large data set of hourly series from 25 monitoring stations located off the west coast of the us detrended fluctuation analysis dfa was used to investigate the scaling properties of the series dfa is a robust technique that can be used to detect long range correlations in nonstationary time series the significant wave height data was analyzed by using scales from hourly to monthly it was found that a common scaling behavior can be observed for all stations a breakpoint in the scaling region around 4 5 days was apparent spectral analysis confirms this result this breakpoint divided the scaling region into two distinct parts the first part was for finer scales up to 4 days which exhibited brown noise characteristics while the second one showed 1 f noise behavior at coarser scales 5 days to 1 month the first order and the second order dfa dfa1 and dfa2 were used to check the effect of seasonality it was found that there were no differences between dfa1 and dfa2 results indicating that there is no effect of trends in the wave height time series the resulting scaling coefficients range from 0 696 to 0 890 indicating that the wave height exhibits long term persistence there were no coherent spatial variations in the scaling coefficients

palindromes and sturmian words abstract an infinite word x over the alphabet a is sturmian if and only if g x n n 1 for any integer n where g x n is the number of distinct words of length n occurring in x a palindrome is a word that can be read indistinctly from left to right or from right to left we prove that x is sturmian if and only if h x n 1 n mod 2 for any integer n where h x n is the number of palindromes of length n occurring in x an infinite word x over the alphabet a is generated by a morphism f if there exists a letter c a such that lim n f n c x we prove the existence of a morphism that generates the palindromes of any infinite sturmian word generated by a morphism

palindrome complexity we study the palindrome complexity of infinite sequences on finite alphabets i e the number of palindromic factors blocks of given length occurring in a given sequence we survey the known results and obtain new results for some sequences in particular for rote sequences and for fixed points of primitive morphisms of constant length belonging to class p of hof knill simon we also give an upper bound for the palindrome complexity of a sequence in terms of its block complexity

on theta palindromic richness in this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism q it generalizes the notion of a palindrome into a q palindrome a word invariant under q for languages closed under q we give the relation email protected complexity and factor complexity we generalize the notion of richness email protected and we prove analogous characterizations of words that are q rich especially in the case of set of factors invariant under q a criterion for q richness of q episturmian words is given together with other examples of q rich words

quasiperiodic and lyndon episturmian words recently the second two authors characterized quasiperiodic sturmian words proving that a sturmian word is non quasiperiodic if and only if it is an infinite lyndon word here we extend this study to episturmian words a natural generalization of sturmian words by describing all the quasiperiods of an episturmian word which yields a characterization of quasiperiodic episturmian words in terms of their directive words even further we establish a complete characterization of all episturmian words that are lyndon words our main results show that unlike the sturmian case there is a much wider class of episturmian words that are non quasiperiodic besides those that are infinite lyndon words our key tools are morphisms and directive words in particular normalized directive words which we introduced in an earlier paper also of importance is the use of return words to characterize quasiperiodic episturmian words since such a method could be useful in other contexts

conjugacy and episturmian morphisms episturmian morphisms generalize sturmian morphisms here we study some intrinsic properties of these morphisms invertibility presentation cancellativity unitarity characterization by conjugacy most of them are generalizations of known properties of sturmian morphisms but we present also some results on episturmian morphisms that have not already been stated in the particular case of sturmian morphisms characterization of the episturmian morphisms that preserve palindromes new algorithms to compute conjugates we also study the conjugation of morphisms in the general case and show that the monoid of invertible morphisms on an alphabet containing at least three letters is not finitely generated

palindromic richness for languages invariant under more symmetries for a given finite group g consisting of morphisms and antimorphisms of a free monoid a we study infinite words with language closed under the group g we focus on the notion of g richness which describes words rich in generalized palindromic factors i e in factors w satisfying q w w for some antimorphism email protected g we give several equivalent descriptions which are generalizations of known characterizations of rich words in the terms of classical palindromes and show two examples of g rich words

proof of the brlek reutenauer conjecture brlek and reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2d u email protected n 0 t u n in which d u denotes the defect of u and t u n denotes c u n 1 c u n 2 p u n 1 p u n where c u and p u are the factor and palindromic complexity of u respectively this conjecture was verified for periodic words by brlek and reutenauer themselves using their results for periodic words we have recently proved the conjecture for uniformly recurrent words in the present article we prove the conjecture in its general version by a new method without exploiting the result for periodic words

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

palindrome pattern matching a palindrome is a string that reads the same forward and backward for a string x let pals x be the set of all maximal palindromes of x where each maximal palindrome in pals x is encoded by a pair c r of its center c and its radius r given a text t of length n and a pattern p of length m the palindrome pattern matching problem is to compute all positions i of t such that pals p pals t i i m 1 we present linear time algorithms to solve this problem

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