systems of numeration and fractal functions relating to substitutions french abstract let a be a finite alphabet a substitution over a un n n a fixed point for and for each a a a a real number we establish under some assumptions an asymptotic formula concerning the sum s n i n u i n n this result generalizes some previous results from coquet or brillhart erdos and morton moreover relations with self affine functions in a sense which generalizes a definition from kamae are proved the calculi leave over systems of representation of integers and real numbers

another generalization of abelian equivalence the binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u based on this classical notion we introduce the m binomial equivalence of two words refining the abelian equivalence two words x and y are m binomially equivalent if for all words v of length at most m the binomial coefficients of x and v and respectively y and v are equal the m binomial complexity of an infinite word x maps an integer n to the number of m binomial equivalence classes of factors of length n occurring in x we study the first properties of m binomial equivalence we compute the m binomial complexity of two classes of words sturmian words and pure morphic words that are fixed points of parikh constant morphisms like the thue morse word i e images by the morphism of all the letters have the same parikh vector we prove that the frequency of each symbol of an infinite recurrent word with bounded 2 binomial complexity is rational

the asymptotic distribution of elements in automatic sequences in an automatic sequence an element need not have an asymptotic density in this paper a necessary and sufficient criterion is proved for the existence of the asymptotic density of a given element if it does not exist the asymptotic distribution of the element can be described in terms of a function h whose graph is self similar an algorithm is given to decide whether h is piecewise continuously differentiable and in this case it can be computed effectively finally it is shown that the h density of an element in an automatic sequence always exists and equals its logarithmic density

symmetrized β integers this article deals with b numeration systems which are numeration systems with a non integral base b 1 in this framework there exist elements which naturally play the role of integers which are called b integers the set of non negative b integers denoted by z b has various equivalent definitions which arise from different points of view nevertheless these definitions may be generalized on negative real numbers in a non unique way depending on the chosen framework we focus in this article on confluent parry units they are the positive roots of the polynomials x d i 1 d 1kx i 1 where the integers d and k satisfy d 2 and k 1 for any of these numbers we prove that there exists a discrete subset of r that we denote by z b s which is locally isomorphic to z b and such that z b s z b s moreover z b s is a model set and satisfies the inflation property there exists l 1 such that lz b s z b s l is then called the inflation factor for z b s finally we compute inflation factors for z b s of the form b i i being a positive integer

real analytic sets in complex spaces and cr maps if r is a real analytic set in mathbb c n viewed as mathbb r 2n then for any point p r there is a uniquely defined germ xp of the smallest complex analytic variety which contains rp the germ of r at p it is shown that if r is irreducible of constant dimension then the function p dim xp is constant on a dense open subset of r as an application it is proved that a continuous map from a real analytic cr manifold m into mathbb c n which is cr on some open subset of m and whose graph is a real analytic set in m times mathbb c n is necessarily cr everywhere on m

stepped surfaces and rauzy fractals induced from automorphisms on the free group of rank 2 from irreducible unimodular pisot substitutions of rank 2 we can construct the discrete approximations of the lines the so called stepped surfaces and moreover we obtain rauzy fractals as geometric realizations of the symbolic dynamical systems in this paper we study a particular infinite family of automorphisms on the free group of rank 2 we see that for every hyperbolic quadratic unit integer l there is a corresponding automorphism in the infinite family whose incidence matrix has eigenvalue l and the automorphism is conjugate to a substitution or an alternating substitution and we construct stepped surfaces and rauzy fractals induced from automorphisms in the family by using tiling substitutions we also discuss the domain exchange transformations on rauzy fractals

asymptotic expansions for linear homogeneous divide and conquer recurrences algebraic and analytic approaches collated among all sequences that satisfy a divide and conquer recurrence those which are rational with respect to a numeration system are certainly the most basic and the most essential nevertheless until recently this specific class of sequences has not been systematically studied from the asymptotic standpoint we recall how a mechanical process designed by the author permits to compute their asymptotic expansions the process is based on linear algebra and involves computing jordan normal forms joint spectral radii and solving dilation equations the main contribution of the present article is the comparison between our algebraic method and the classical analytic number theory approach moreover we develop new ways to compute the fourier series of the periodic functions involved in the expansion the article comes with an extended bibliography

multimodularity convexity and optimization properties in this paper we investigate the properties of multimodular functions in doing so we give elementary proofs for properties already established by hajek and we generalize some of his results in particular we extend the relation between convexity and multimodularity to some convex subsets of z m we also obtain general optimization results for average costs related to a sequence of multimodular functions rather than to a single function under this general context we show that the expected average cost problem is optimized by using regular sequences we finally illustrate the usefulness of this theory in admission control into a d d 1 queue with fixed batch arrivals with no state information we show that the regular policy minimizes the average queue length for the case of an infinite queue but not for the case of a finite queue when further adding a constraint on the losses it is shown that a regular policy is also optimal for the finite queue case

tree shifts of finite type a one sided resp two sided shift of finite type of dimension one can be described as the set of infinite resp bi infinite sequences of consecutive edges in a finite state automaton while the conjugacy of shifts of finite type is decidable for one sided shifts of finite type of dimension one the result is unknown in the two sided case in this paper we study the shifts of finite type defined by infinite ranked trees indeed infinite ranked trees have a natural structure of symbolic dynamical systems we prove a decomposition theorem for these tree shifts i e we show that a conjugacy between two tree shifts can be broken down into a finite sequence of elementary transformations called in splittings and in amalgamations we prove that the conjugacy problem is decidable for tree shifts of finite type this result makes the class of tree shifts closer to the class of one sided shifts of sequences than to the class of two sided ones our proof uses the notion of bottom up tree automata

diffusion for chaotic plane sections of 3 periodic surfaces we study chaotic plane sections of some particular family of triply periodic surfaces the question about possible behavior of such sections was posed by s p novikov we prove some estimations on the diffusion rate of these sections using the connection between novikov s problem and systems of isometries some natural generalization of interval exchange transformations using thermodynamical formalism we construct an invariant measure for systems of isometries of a special class called the rauzy gasket and investigate the main properties of the lyapunov spectrum of the corresponding suspension flow

a note on abelian returns in rotation words pursuing the study started by rigo salimov and vandomme we use elementary number theoretic techniques to characterize rotation words having a finite set of abelian returns to all prefixes we also make the connection between the three gap theorem and the number of semi abelian returns for sturmian words simplifying some arguments developed by puzynina and zamboni

lattices and multi dimensional words in the present paper we develop a formalism to generate multi dimensional words using lattices which generalizes the construction of real numbers one dimensional words from a sequence of partial quotients using continued fractions the construction was introduced in a special case by simpson and tijdeman in order to derive a multi dimensional generalization of the theorem of fine and wilf we show that the produced multi dimensional words are intrinsically connected with k dimensional sturmian words

limit theorems for self similar tilings we study deviation of ergodic averages for dynamical systems given by self similar tilings on the plane and in higher dimensions the main object of our paper is a special family of finitely additive measures for our systems an asymptotic formula is given for ergodic integrals in terms of these finitely additive measures and as a corollary limit theorems are obtained for dynamical systems given by self similar tilings

on the pathwise optimal bernoulli routing policy for homogeneous parallel servers a long standing conjecture on the optimal bernoulli routing policy is proven to be true for the case of equal exponential service times it is shown that splitting equally among the queues minimizes the departure times in a stochastic pathwise sense a new technique is used showing that certain distributional properties related to schur convexity propagate forward in time

on nash approximation of complex analytic sets in runge domains abstract we prove that every complex analytic set x in a runge domain can be approximated by nash sets on any relatively compact subdomain 0 of moreover for every nash subset y of with y x the approximating sets can be chosen so that they contain y 0 as a consequence we derive a necessary and sufficient condition for a complex analytic set x to admit a nash approximation which coincides with x along its arbitrary given subset

balanced sequences and optimal routing the objective pursued in this paper is two fold the first part addresses the following combinatorial problem is it possible to construct an infinite sequence over n letters where each letter is distributed as evenly as possible and appears with a given rate the second objective of the paper is to use this construction in the framework of optimal routing in queuing networks we show under rather general assumptions that the optimal deterministic routing in stochastic event graphs is such a sequence

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

pattern spectra substring enumeration and automatic sequences abstract let s n n 0 be an infinite sequence on 1 1 in a previous paper morton and mourant 1989 showed how to expand s n n 0 uniquely as a possibly infinite termwise product of certain special infinite sequences on 1 1 called pattern sequences moreover they characterized those sequences for which the expansion or pattern spectrum is finite in this paper we first give the expansion of a subsequence of the prouhet thue morse sequence studied by newman and slater 1969 and 1975 and coquet 1983 then we characterize the sequences given by certain special infinite products next we prove a general theorem characterizing the pattern spectrum when s is an automatic sequence in the sense of cobham 1972 and christol 1980 we also show how to deduce this theorem as the consequence of a purely language theoretic result about enumeration of substrings finally we prove that no sequence can be its own pattern spectrum

tilings induced by a class of cubic rauzy fractals we study aperiodic and periodic tilings induced by the rauzy fractal and its subtiles associated with beta substitutions related to the polynomial x 3 ax 2 bx 1 for a b 1 in particular we compute the corresponding boundary graphs describing the adjacencies in the tilings these graphs are a valuable tool for more advanced studies of the topological properties of the rauzy fractals as an example we show that the rauzy fractals are not homeomorphic to a closed disc as soon as a 2b 4 the methods presented in this paper may be used to obtain similar results for other classes of substitutions

a class of cubic rauzy fractals in this paper we study arithmetical and topological properties for a class of rauzy fractals r a given by the polynomial x 3 a x 2 x 1 where a 2 is an integer in particular we prove the number of neighbors of r a in the periodic tiling is equal to 8 we also give explicitly an automaton that generates the boundary of r a as a consequence we prove that r 2 is homeomorphic to a topological disk

displacement rank and quasitriangular decomposition for r toeplitz matrices abstract an r toeplitz r t matrix can be regarded as a block toeplitz matrix of order mr from which an l shaped border of k r rows and columns is removed the concept of displacement rank is extended to r t matrices and it is shown that both an r t matrix and its inverse can be expressed as the sum of at most two products of r t matrices this result reduces to a standard lu decomposition in the case when an r t matrix becomes an arbitrary matrix an explicit expression for the inverse is given which contains three triangular matrices and one quasitriangular matrix furthermore the inverse of an r t matrix is completely determined by the elements in its first and last r rows and columns the results presented generalize known properties of toeplitz and block toeplitz matrices

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

toeplitz matrices with banded inverses abstract given a toeplitz matrix t with banded inverse i e t 1 ij 0 for j i p we show that the elements of t can be expressed in terms of the roots of a polynomial then using properties we have previously established we generalize this result appropriately to allow singular t and show that the converse also holds finally we give a sufficient condition for the decay of the elements of t as one moves away from the diagonal

on the abelian complexity of generalized thue morse sequences abstract in this paper we study the abelian complexity n a b t k of generalized thue morse sequences t k for every integer k 2 we obtain the exact value of n a b t k for every integer n k consequently n a b t k is ultimately periodic with period k moreover we show that the abelian complexities of a class of infinite sequences are automatic sequences

inversion of certain extensions of toeplitz matrices it is shown how an algorithm for inverting toeplitz matrices using o n2 operations can be modified to deal with a certain type of extension named conjugate toeplitz matrices the block partitioned case is also included

eigenvalues of a certain class of tridiagonal matrices we make a short comment on a paper involving the minimal eigenvalues of a specialized version of 2 toeplitz tridiagonal matrices

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

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

maximal pattern complexity of two dimensional words the maximal pattern complexity of one dimensional words has been studied in several papers t kamae l zamboni sequence entropy and the maximal pattern complexity of infinite words ergodic theory dynam systems 22 4 2002 1191 1199 t kamae l zamboni maximal pattern complexity for discrete systems ergodic theory dynam systems 22 4 2002 1201 1214 t kamae h rao pattern complexity over l letters e comb j to appear t kamae y m xue two dimensional word with 2k maximal pattern complexity osaka j math 41 2 2004 257 265 we study the maximal pattern complexity p k of two dimensional words a two dimensional version of the notion of strong recurrence is introduced it is shown that if is strongly recurrent then either is doubly periodic or p k 2k k 1 2 accordingly we define a two dimensional pattern sturmian word as a strongly recurrent word with p k 2k examples of pattern sturmian words are given

abelian properties of parry words abelian complexity of a word u is a function that counts the number of pairwise non abelian equivalent factors of u of length n we prove that for any c balanced parry word u the values of the abelian complexity function can be computed by a finite state automaton the proof is based on the notion of relative parikh vectors the approach works for any function f n that can be expressed in terms of the set of relative parikh vectors corresponding to the length n for example we show that the balance function of a c balanced parry word is computable by a finite state automaton as well

admission control in stochastic event graphs we first show that the expectation of convex increasing functions of the workload or waiting time in max linear systems under a single input sequence is multi modular this is done using a coupling argument and a vectorial version of lindley s equation next we use this result and the optimization theory based on multi modular costs to construct the optimal open loop admission control in general max linear systems under admission rate constraints this optimization result only requires stationarity assumptions on the arrival process and on the service times of the servers in the system

geometric theory of unimodular pisot substitutions we are concerned with the tiling flow t associated to a substitution over a finite alphabet our focus is on substitutions that are unimodular pisot i e their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the perron eigenvalue 1 the motivation is provided by the still open conjecture asserting that t has pure discrete spectrum for any such we develop a number of necessary and sufficient conditions for pure discrete spectrum including injectivity of the canonical torus map the geometric realization geometric coincidence condition partial commutation of t and the dual r d 1 action measure and tiling properties of rauzy fractals and concrete algorithms some of these are original and some have already appeared in the literature as sufficient conditions only but they all emerge from a unified approach based on the new device the strand space f of the proof of the necessity hinges on determination of the discrete spectrum of t as that of the associated kronecker toral flow

balancing and clustering of words in the burrows wheeler transform compression algorithms based on burrows wheeler transform bwt take advantage of the fact that the word output of bwt shows a local similarity and then turns out to be highly compressible the aim of the present paper is to study such clustering effect by using notions and methods from combinatorics on words the notion of balance of a word plays a central role in our investigation empirical observations suggest that balance is actually the combinatorial property of input word that ensure optimal bwt compression moreover it is reasonable to assume that the more balanced the input word is the more local similarity we have after bwt and therefore the better the compression is this hypothesis is here corroborated by experiments on real text by using local entropy as a measure of the degree of balance of a word in the setting of combinatorics on words a sound confirmation of previous hypothesis is given by a result of mantaci et al 2003 27 which states that in the case of a binary alphabet there is an equivalence between circularly balanced words words having a clusterized bwt and the conjugates of standard words in the case of alphabets of size greater than two there is no more equivalence the last section of the present paper is devoted to investigate the relationships between these notions and other related ones as for instance palindromic richness in the case of a general alphabet

complexity of trajectories in rectangular billiards to a trajectory of the billiard in a cube we assign its symbolic trajectory the sequence of numbers of coordinate planes to which the faces met by the trajectory are parallel the complexity of the trajectory is the number of different words of lengthn occurring in it we prove that for generic trajectories the complexity is well defined and calculate it confirming the conjecture of arnoux mauduit shiokawa and tamura amst

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

dual systems of algebraic iterated function systems abstract we study a class of graph directed iterated function systems on r with algebraic parameters which we call algebraic gifs we construct a dual ifs of an algebraic gifs and study the relations between the two systems we determine when a dual system satisfies the open set condition which is fundamental for feasible pisot systems we construct the left and right rauzy thurston tilings and study their multiplicities and decompositions we also investigate their relation with codings space domain exchange transformation and the pisot spectrum conjecture the dual ifs provides a unified and simple framework for rauzy fractals tilings and related studies and allows us gain better understanding

generating polynomials for matrices with toeplitz or conjugate toeplitz inverses abstract a complex matrix a a ij has been called conjugate toeplitz if a ij c i 1 a i j where c denotes conjugation a necessary and sufficient condition is derived for a matrix h to have a conjugate toeplitz inverse the elements of h are generated from the coefficients of certain polynomials the result is simplified either when h is to have a toeplitz inverse or when h is banded and is to have a conjugate toeplitz inverse each of these consequences of the main theorem is a different generalization of a previous result on banded matrices having a toeplitz inverse

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

on the holomorphic closure dimension of real analytic sets given a real analytic or more generally semianalytic set r in cn viewed as r2n there is for every p r a unique smallest complex analytic germ xp that contains the germ rp we call dimc xp the holomorphic closure dimension of r at p we show that the holomorphic closure dimension of an irreducible r is constant on the complement of a closed proper analytic subset of r and we discuss the relationship between this dimension and the cr dimension of r

periodic scheduling of marked graphs using balanced binary words this paper presents an algorithm to statically schedule live and strongly connected marked graphs the proposed algorithm computes the best execution where the execution rate is maximal and place sizes are minimal the proposed algorithm provides transition schedules represented as binary words these words are chosen to be balanced the contributions of this paper are the proposed algorithm itself along with the characterization of the best execution of any marked graph

on the average waiting time for regular routing to deterministic queues we consider a deterministic queueing system in whichn 2 servers of different speeds operate in parallel each service in queuei takes the deterministic times i identical customers arrive exactly one per time unit and it is desirable to route them to the queues so that the average waiting time we consider as waiting time the time a customer waits in the buffer of a queue and thus the service time is not included in this is minimized we provide an algorithm to compute lower and upper bounds on this quantity the upper bound is found by showing that there is a periodic policy for which the average waiting time is no greater than the lower bound plus n 2 1 thus the bounds coincide whenn 2 for obtaining the lower bound we give explicit formulae for the average waiting time in case of regular routing to a deterministic queue

additive and multiplicative properties of point sets based on beta integers to each number 1 correspond abelian groups in rd of the form i 1d z ei which obey the set z of beta integers is a countable set of numbers it is precisely the set of real numbers which are polynomial in when they are written in basis and z z when n we prove here a list of arithmetic properties of z addition multiplication relation with integers when is a quadratic pisot vijayaraghavan unit quasicrystallographic inflation factors are particular examples we also consider the case of a cubic pisot vijayaraghavan unit associated with the seven fold cyclotomic ring at the end we show how the point sets are vertices of d dimensional tilings

extremal splittings of point processes the sequence with nth term defined by n 1 p np is an extremal zero one valued sequence of asymptotic mean p in the following sense for example if a fraction p of customers from a point process with iid interarrival times is sent to an exponential server queue according to a prespecified splitting sequence then the long term average queue size is minimized when the above sequence is used the proof involves consideration of the lower convex envelope j which is a function on rm of a function j on zm an explicit representation is given for j in terms of j for j in a broad class of functions which we call multimodular the expected queue size just before an arrival considered as a function of the zero one splitting sequence is shown to belong to this class

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

tameness of holomorphic closure dimension in a semialgebraic set given a semianalytic set s in mathbb c n and a point p in bar s there is a unique smallest complex analytic germ x p which contains s p called the holomorphic closure of s p we show that if s is semialgebraic then x p is a nash germ for every p and s admits a semialgebraic filtration by the holomorphic closure dimension as a consequence every semialgebraic subset of a complex vector space admits a semialgebraic stratification into cr manifolds

mellin transforms and asymptotics digital sums abstract arithmetic functions related to number representation systems exhibit various periodicity phenomena for instance a well known theorem of delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function we show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory namely the mellin perron formulae this approach yields naturally the fourier series involved in the expansions of a variety of digital sums related to number representation systems

properties of words and recognizability of fixed points of a substitution resume soit u un mot infini non periodique point fixe d une substitution primitive nous montrons que pour n assez grand u ne contient aucune puissance n eme de mot nous donnons ensuite une condition necessaire et suffisante pour que soit reconnaissable au sens de host et queffelec et nous montrons qu ill suffit de bilateraliser cette definition pour obtenir une propriete verifiee cette fois par toutes les substitutions primitives a point fixe non periodique

the ring of k regular sequences ii in this paper we continue our study of k regular sequences begun in 1992 we prove some new results give many new examples from the literature and state some open problems

power and exponential sums for generalized coding systems by a measure theoretic approach a measure theoretic approach to study the power and the exponential sums for the usual coding system has been developed since the 1990s in this paper we first introduce a new coding system and then give explicit formulas for the power and the exponential sums for the coding system by the measure theoretic approach an expression for the power sum using the generalized takagi function will also be given

joint spectral radius dilation equations and asymptotic behavior of radix rational sequences abstract radix rational sequences are solutions of systems of recurrence equations based on the radix representation of the index for each radix rational sequence with complex values we provide an asymptotic expansion essentially in the scale n log l n the precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence of first order differences the coefficients are holderian functions obtained through some dilation equations which are usual in the domains of wavelets and refinement schemes the proofs are ultimately based on elementary linear algebra

central sets and substitutive dynamical systems abstract in this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of pisot type central sets first introduced by furstenberg using notions from topological dynamics constitute a special class of subsets of n possessing strong combinatorial properties each central set contains arbitrarily long arithmetic progressions and solutions to all partition regular systems of homogeneous linear equations we give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the stone cech compactification n this provides a new arithmetical approach to an outstanding conjecture in tiling theory the pisot substitution conjecture the results in this paper rely on interactions between different areas of mathematics some of which had not previously been directly linked they include the general theory of combinatorics on words abstract numeration systems tilings topological dynamics and the algebraic topological properties of stone cech compactification of n

the determination of companion matrices characterizing toeplitz and r toeplitz matrices abstract it is shown that the unknown elements in the companion matrix used to characterize a nonsingular toeplitz matrix are easily found from the elements of the inverse of the toeplitz matrix similar formulae are obtained for the two companion like matrices characterizing an r toeplitz matrix this case includes the result for a block toeplitz matrix

a note on conjugate toeplitz matrices abstract two properties of conjugate toeplitz matrices are given 1 an expression for the elements inside the inverse matrix of an r toeplitz and of a conjugate toeplitz matrix is obtained in a direct way from elementary results about arbitrary partitioned matrices 2 the necessary conditions to apply the generalized trench algorithm for conjugate toeplitz matrices are discussed it is shown that there exist strongly invertible conjugate toeplitz matrices to which the algorithm may not be applied

free choice petri nets an algebraic approach in this paper we give evolution equations for free choice petri nets which generalize the max algebraic setting already known for event graphs these evolution equations can be seen as a coupling of two linear systems a min linear system and a quasi x linear one this leads to new methods and algorithms to 1 in the untimed case check liveness and several other basic logical properties 2 in the timed case establish various conservation and monotonicity properties and 3 in the stochastic case check stability i e the fact that the marking remains bounded in probability and constructs minimal stationary regimes the main tools for proving these properties are graph theory idempotent algebras and ergodic theory

the eigenproblem of a tridiagonal 2 toeplitz matrix abstract the characteristic polynomial of a tridiagonal 2 toeplitz matrix is shown to be closely connected to polynomials which satisfy the three point chebyshev recurrence relationship this is an extension of the well known result for a tridiagonal toeplitz matrix when the order of the matrix is odd the eigenvalues are found explicitly in terms of the chebyshev zeros the eigenvectors are found in terms of the polynomials satisfying the three point recurrence relationship

balances for fixed points of primitive substitutions an infinite word defined over a finite alphabet a is balanced if for any pair of factors of the same length and for any letter a in the alphabet a a 1 where a denotes the number of occurrences of the letter a in the word in this paper we generalize this notion and introduce a measure of balance for an infinite sequence in the case of fixed points of primitive substitutions we show that the asymptotic behaviour of this measure is in part ruled by the spectrum of the incidence matrix associated with the substitution connections with frequencies of letters and other balance properties are also discussed

characterisation and properties of r toeplitz matrices abstract an r toeplitz rt matrix can be regarded as a block toeplitz matrix with order some multiple of r from which a bottom border and a right hand border have been removed it is shown that some known properties of toeplitz matrices extend to this new class in particular a necessary and sufficient condition is given in terms of a linear matrix equation for a nonsingular matrix to have rt form from which it follows that the elements of the inverse are completely determined by those in its first and last r rows and columns in addition some results are presented for cases when the inverse of an rt matrix has certain banded forms

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

cyclic displacements and decompositions of inverse matrices for cupl toeplitz matrices abstract in this paper we deal mainly with a class of cupl toeplitz matrices 32 without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor row first plus last right circulants furthermore the inverses of the cupl hankel matrices can be also obtained which benefits from the relation between the cupl toeplitz matrices and the cupl hankel matrices

on the modulus of continuity for spectral measures in substitution dynamics abstract the paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms which yield information about the fractal structure of these measures the main results are first a holder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function second a log holder estimate for self similar suspension flows and third a holder asymptotic expansion of the spectral measure at zero for such flows our second result implies log holder estimates for the spectral measures of translation flows along stable foliations of pseudo anosov automorphisms a key technical tool in the proof of the second result is an arithmetic diophantine proposition which has other applications in appendix a this proposition is used to derive new decay estimates for the fourier transforms of bernoulli convolutions

two dimensional iterated morphisms and discrete planes iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters by words the aim of this paper is to introduce a formalism for a notion of two dimensional morphisms we show that they can be iterated by using local rules and that they generate two dimensional patterns related to discrete approximations of irrational planes with algebraic parameters we associate such a two dimensional morphism with any usual pisot unimodular one dimensional iterated morphism over a three letter alphabet

abelian complexity and abelian co decomposition we propose a technique for exploring the abelian complexity of recurrent infinite words focusing particularly on infinite words associated with parry numbers using that technique we give an affirmative answer to the open question posed by richomme saari and zamboni whether the abelian complexity of the tribonacci word attains each value in 4 5 6 infinitely many times