avoidability of circular formulas abstract clark has defined the notion of n avoidance basis which contains the avoidable formulas with at most n variables that are closest to be unavoidable in some sense the family c i of circular formulas is such that c 1 a a c 2 a b a b a b c 3 a b c a b c a b c a b c and so on for every i n the n avoidance basis contains c i clark showed that the avoidability index of every circular formula and of every formula in the 3 avoidance basis and thus of every avoidable formula containing at most 3 variables is at most 4 we determine exactly the avoidability index of these formulas

multidimensional unrepetitive configurations note abstract we show that one can associate a letter of a finite alphabet with each lattice point of the n dimensional space so that an r th power free word corresponds to each straight line of the space this result is obtained as a consequence of the existence of bi infinite words on finite alphabets which are r th power free together with a certain class of their subwords an analogous property is obtained for some avoidable sets of patterns

on the least number of palindromes in two dimensional words abstract we investigate the least number of distinct palindromic sub arrays in two dimensional words over a finite alphabet a 1 a 2 a q for a given alphabet size q we discuss the case for both periodic as well as aperiodic words

semilinear copulas a family of copulas called semilinear is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square this family contains some other known families of copulas e g cuadras auge frechet and has a nice statistical interpretation several construction methods are provided especially concerning aggregation of semilinear copulas and a special form of ordinal sum construction is introduced some results about related families of quasi copulas and semicopulas are hence given

growth rate of binary words avoiding xxxr consider the set of those binary words with no non empty factors of the form x x x r du mousavi schaeffer and shallit asked whether this set of words grows polynomially or exponentially with length in this paper we demonstrate the existence of upper and lower bounds of the form n lg n o lg n on the number of such words of length n where lg n denotes the base 2 logarithm of n

solution of lyapunov equations by alternating direction implicit iteration abstract in this report a new procedure is presented for solving the lyapunov matrix equation first the system is reduced to tridiagonal form with gaussian similarity transformations then the resulting system is solved with alternating direction implicit adi iteration a matrix commutation property essential for model problem convergence of adi iteration applied to elliptic difference equations is not needed for this application all stable lyapunov matrix equations are model adi problems

minimal critical exponent of quasiperiodic words abstract a quasiperiodic word is a finite or infinite word that can be obtained by concatenations or overlaps of a given finite word we proved that the least critical exponent of a quasiperiodic binary word is 7 3 for larger alphabets given any 0 we provide a construction of a ternary quasiperiodic word with critical exponent 2

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

on dejean s conjecture over large alphabets the maximal exponent of a non empty finite word is the ratio of its length to its period dejean 1972 conjectured that for any n 5 there exists an infinite word over n letters with no factor of its exponent larger than n n 1 we prove that this conjecture is true for n 33

meta theorems on inequalities for scalar fuzzy set cardinalities we present meta theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification when adopting a scalar approach to fuzzy set cardinality the conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection in particular this conjunctor should fulfil a number of bell type inequalities the advantage of these meta theorems is that repetitious calculations can be avoided this is illustrated in the demonstration of the lukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the jaccard coefficient or equivalently the triangle inequality of the corresponding dissimilarity measures

growth rates of complexity of power free languages we present a new fast algorithm for calculating the growth rate of complexity for regular languages using this algorithm we develop a space and time efficient method to approximate growth rates of complexity of arbitrary power free languages over finite alphabets through extensive computer assisted studies we sufficiently improve all known upper bounds for growth rates of such languages obtain a lot of new bounds and discover some general regularities

repetition free words abstract we give a systematical treatment of repetition free words and repetition free homomorphisms we study the global structure of the set of repetition free infinite words and the behaviour of the density function of repetition free words generally we consider finite or infinite words which do not contain given words as subwords

analytic perturbation of sylvester matrix equations we consider an analytic perturbation of the sylvester matrix equation we are mainly interested in the singular case that is when the null space of the unperturbed sylvester operator is not trivial but the perturbed equation has a unique solution in this case the solution of the perturbed equation can be given in terms of a laurent series we provide a necessary and sufficient condition for the existence of a laurent series with a first order pole a recursive procedure for the calculation of the laurent series coefficients is given

on the compositional characterization of complete fuzzy pre orders complete pre orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations in this paper we investigate this characterization in a fuzzy setting we consider two types of completeness weak completeness and strong completeness and decompose a fuzzy pre order by means of an indifference generator in particular a frank t norm in the weakly complete case we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them in the strongly complete case we lay bare a stronger type of transitivity of the strict preference relation we conclude the paper with a rather negative result there is no hope to obtain a compositional characterization of weakly complete fuzzy pre orders and hence also not of fuzzy pre orders in general

factoring fuzzy transitivity transitivity of a precise weak preference relation implies that its asymmetric component strict preference and its symmetric component indifference are transitive and conversely the transitivity of strict preference and indifference implies that the underlying weak preference relation is transitive this paper examines to what extent analogous results may be obtained for the different transitivity conditions and different methods for deriving strict preference and indifference being used in the literature of choice with fuzzy preferences

conic aggregation functions inspired by the notion of conic t norms we introduce conic aggregation functions such aggregation functions are completely characterized by their zero set special classes of binary conic aggregation functions such as conic quasi copulas and conic copulas are considered we provide the necessary and sufficient conditions on the boundary curve of the zero set of a conic aggregation function to obtain a conic quasi copula and conclude that the class of conic copulas is a proper subclass of the class of conic quasi copulas moreover we characterize the class of singular conic copulas we investigate some aggregations of conic quasi copulas some examples are also provided

on an idempotent transformation of aggregation functions and its application on absolutely continuous archimedean copulas in this paper we isolate an interesting transformation of binary aggregation functions from a result on the propagation of transitivity in additive preference structures this transformation is based on a projection technique and results in a smaller or equal binary aggregation function the transformation is idempotent and acts internally on the classes of conjunctors semi copulas and quasi copulas hence also copulas are transformed into quasi copulas but in general not into copulas remarkably the frank copulas are mapped to new copulas

a generalized adi iterative method the adi iterative method for the solution of sylvester s equationax xb c proceeds by strictly alternating between the solution of the two equations begin gathered left a delta k 1 i right x 2k 1 x 2k left b delta k 1 i right c hfill x 2k 2 left b tau k 1 i right left a tau k 1 i right x 2k 1 c hfill end gathered fork 0 1 2 herex o is a given initial approximate solution and the k and k are real or complex parameters chosen so that the computed approximate solutionsx k converge rapidly to the solutionx of the sylvester equation ask increases this paper discusses the possibility of solving one of the equations in the adi iterative method more often than the other one i e relaxing the strict alternation requirement in order to achieve a higher rate of convergence our analysis based on potential theory shows that this generalization of the adi iterative method can give faster convergence than when strict alternation is required

a proof of dejean s conjecture we prove dejean s conjecture specifically we show that dejean s conjecture holds for the last remaining open values of n namely 15 n 26

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

every real number greater than 1 is a critical exponent we prove that for every real number a 1 there exists an infinite word w over a finite alphabet such that a is the critical exponent of w

exponential lower bounds for the number of words of uniform length avoiding a pattern we study words on a finite alphabet avoiding a finite collection of patterns given a pattern p in which every letter that occurs in p occurs at least twice we show that the number of words of length n on a finite alphabet that avoid p grows exponentially with n as long as the alphabet has at least four letters moreover we give lower bounds describing this exponential growth in terms of the size of the alphabet and the number of letters occurring in p we also obtain analogous results for the number of words avoiding a finite collection of patterns we conclude by giving some questions

square free words obtained from prefixes by permutations an infinite square free word w over a three letter alphabet t is said to have a k stem s if w sw 1w 2 where for each i there exists a permutation p i of t which extended to a morphism gives w i p i s we show that there exists an infinite k stem word for k 1 2 3 9 and 13 k 19 but not for 4 k 8 and 10 k 12 the problem whether k stem words exist for each k 20 remains open

bounds for the generalized repetition threshold the notion of the repetition threshold which is the object of dejean s conjecture 1972 was generalized by ilie et al 2005 8 to include the lengths of the avoided words we give a lower and an upper bound on this generalized repetition threshold

dejean s conjecture holds for n 30 abstract we extend carpi s results by showing that dejean s conjecture holds for n 30

avoidance bases for formulas with reversal abstract in the interest of studying formulas with reversal of high avoidability index we find n avoidance bases for formulas with reversal for n 1 2 3 we demonstrate that there is a unique formula with reversal in each of these three bases of highest avoidability index n 2 these formulas are xx x y x y r and x y z x y r z r which belong to an infinite family of formulas with reversal that has been the subject of recent study by the authors

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

a characterization of power free morphisms abstract a word is called kth power free if it does not contain any non empty factor uk a morphism is kth power free if it preserves kth power free words a morphism is power free if it is kth power free for every k 1 we show that it is decidable whether a morphism is power free more precisely we prove that a morphism h is power free iff h is a square free morphism and for each letter a the image h a2 is cube free

polynomial growth in semigroup varieties abstract in 1989 m sapir posed the problem of describing all semigroup varieties where every finitely generated f g semigroup has polynomial growth here we find the solution of this problem for the case of an arbitrary nonperiodic semigroup variety defined by a system of identities over a finite set of variables we also show that there exists an algorithm to decide whether or not the given finite system of homogeneous semigroup identities defines a variety where every f g semigroup has polynomial growth

the number of binary words avoiding abelian fourth powers grows exponentially we show that the number of binary words of length n avoiding abelian fourth powers grows faster than rn where r 21 16

an analysis of block successive overrelaxation for a class of matrices with complex spectra optimal block successive overrelaxation bsor parameters are derived for a special class of matrices having complex spectra these matrices arise from the central difference approximation to singularly perturbed convection diffusion equations using a coarse grid and thus have large cell reynolds number a simple problem is examined in detail and it is shown that bsor is an excellent method for cell reynolds numbers near 1 and that the rate of convergence is invariant to the cell reynolds number for large cell reynolds number

sensor array signal processing via a procrustes rotations based eigenanalysis of the esprit data pencil the authors propose the procrustes rotation based esprit algorithm pro esprit for estimating characteristic parameters associated with a scene of radiating sources given data derived from two identical sensor arrays the x and y arrays which are displaced relative to one another pro esprit uses the inherent redundancy built into the esprit array system to compensate for deviations from the ideal array model this redundancy is manifested by two fundamental properties of the noiseless x and y data matrices comprising the esprit data pencil a they have the same row space the signal subspace and 2 they have the same column space the source subspace these properties are utilized by invoking two primary tools 1 the solution to the procrustes problem for optimally approximating an invariant subspace rotation and 2 the total least squares method for perturbing each of two estimates of a common subspace in a minimal fashion until the two perturbed spaces are the same

a note on an idempotent transformation of absolutely continuous archimedean copulas abstract in de baets et al 6 the authors define for a copula c an idempotent transformation d c which in some cases generates new copulas in this note we give a negative answer to a question posed in 6 showing that d c is not necessarily a copula when c is an absolutely continuous archimedean copula

relatively free semigroups of intermediate growth abstract the problem of calculating the growth of a finitely generated f g semigroup satisfying the given system of identities is considered examples of relatively free semigroups having intermediate growth new growth criteria and constructions are given the main ideas employed in the proofs are based on some new points of view on growth our method is also useful for constructing various examples of f g nilpotent and nil semigroups having intermediate growth and arbitrarily small gelfand kirillov superdimension

square free shuffles of words let u v denote the set of all shuffles of the words u and v it is shown that for each integer n 3 there exists a square free ternary word u of length n such that u u contains a square free word this property is then shown to also hold for infinite words i e there exists an infinite square free word u on three letters such that u can be shuffled with itself to produce an infinite square free word w u u

on infinite words obtained by selective substitution grammars abstract we present a method of generating infinite words from selective substitution grammars introduced by rozenberg 1977 comparison is made with some of the well known limiting processes and limit language families and certain closure properties are examined a technique is given for obtaining infinite non repetitive words several decidability results are established

freeness of partial words the paper approaches the classical combinatorial problem of freeness of words in the more general case of partial words first we propose an algorithm that tests efficiently whether a partial word is k free or not for a given k then we show that there exist arbitrarily many k free infinite partial words over a binary alphabet containing an infinite number of holes for k 3 moreover we present an efficient algorithm for the construction of a cube free partial word with a given number of holes over a binary alphabet in the final section of the paper we show that there exists an infinite word over a four symbol alphabet in which we can substitute randomly one symbol with a hole and still obtain a cube free word we show that such a word does not exist for alphabets with fewer symbols further we prove that in this word we can replace arbitrarily many symbols with holes such that each two consecutive holes are separated by at least two symbols and obtain a cube free partial word this result seems interesting because any partial word containing two holes with less than two symbols between them is not cube free finally we modify the previously presented algorithm to construct over a four symbol alphabet a cube free partial word with exactly n holes having minimal length among all the possible cube free partial words with at least n holes

solution of underdetermined sylvester equations in sensor array signal processing abstract we propose efficient solution of an underdetermined matrix equation the matrix equations studied here appear in the study of subspace based high resolution multiparameter direction of arrival estimation the proposed gradient based technique is at least an order of magnitude computational improvement over the existing approaches simulation results are provided to illustrated the proposed technique

additive decomposition of fuzzy pre orders fuzzy pre orders reflexive and min transitive fuzzy relations constitute an important class of fuzzy relations by means of an indifference generator a fuzzy pre order can be decomposed additively into two parts an indifference relation and a strict preference relation when using a frank t norm as indifference generator we fully characterize the transitivity of these parts only in case the minimum operator is used as generator both parts are min transitive the transitivity of the indifference relation is determined by the frank t norms while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t norm

some further results on squarefree arithmetic progressions in infinite words abstract in a recent paper one of us posed three open problems concerning squarefree arithmetic progressions in infinite words in this paper we solve these problems and prove some additional results for instance among other things we show that there exists a squarefree word w over a ternary alphabet such that for every p 3 the subsequence of w indexed by the multiples of p contains a square

fuzzy preference structures without incomparability abstract in this paper we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability this study is carried out for the fuzzy preference structures introduced recently by de baets van de walle and kerre a set of remarkable theorems gives detailed insight in the relationships between the sup t transitivity of the fuzzy preference and indifference relations and the sup t transitivity of the fuzzy large preference relation several paths of thought involving t norms with or without zero divisors are explored and where required illustrative counterexamples confirm the falsity of certain implications finally we introduce the t n ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this ferrers property

growth problems for avoidable words abstract a word w is said to avoid a word u if no block subword factor of w is the image of u under a homomorphism of free semigroups without unit the theory of words avoiding xx square free words has been much studied the word u is said to be avoidable on n letters if there are arbitrarily long words on an n letter alphabet that avoid u if u is avoidable on n letters for some n let u be the minimum possible such n we show that u has a linear bound in terms of the alphabet size of u we further show that there exists a word that is avoidable on four letters but not on three letters moreover if u is this word the number of words of length l on a u letter alphabet that avoid u has a polynomial bound in terms of l so that the question of the existence of such an example is resolved in the affirmative in contrast for xx the bound is known to be exponential

sor for ax xb c we consider a new approach to the block sor method applied to linear systems of equations which can be written as a matrix equation ax xb c such systems arise for example from finite difference discretizations of seperable elliptic boundary value problems on rectangular domains on one hand this gives us an iterative method for the solution of such matrix equations e g lyapunov s matrix equation where b at and on the other hand the problem of choosing appropriate parameters for the block sor method can be written in a more compact form which may be helpful especially for non self adjoint problems i e if a and b are nonsymmetric using this technique we determine under more general assumptions than those of chin and manteuffel the optimal sor parameters for the model problem of a convection diffusion equation

shift invariant binary aggregation operators when applying an information fusion method usually the output results are required to be a scalar value in multiple situations the application of the fusion method to input values shifted by a constant should result in an output that is shifted likewise in this paper we present a study of the shift invariance property of aggregation operators the relationship between shift invariant binary aggregation operators and 1 lipschitz aggregation operators is shown and a full description of shift invariant aggregation operators is given

biconic aggregation functions we introduce a new method to construct aggregation functions these aggregation functions are called biconic aggregation functions with a given diagonal resp opposite diagonal section and their construction is based on linear interpolation on segments connecting the diagonal resp opposite diagonal of the unit square to the points 0 1 and 1 0 resp 0 0 and 1 1 subclasses of biconic aggregation functions such as biconic semi copulas biconic quasi copulas and biconic copulas are studied in detail

on the subword complexity of square free dol languages abstract the subword complexity of a language k is the function which to every positive integer n assigns the number of different subwords of length n occuring in words of k a language k is square free if no word in it contains a subword of the form xx where x is a nonempty word the best upper and lower bounds on the subword complexity of infinite square free dol languages are established

problems in between words and abelian words k abelian avoidability we consider a recently defined notion of k abelian equivalence of words in connection with avoidability problems this equivalence relation for a fixed natural number k takes into account the numbers of occurrences of the different factors of length k and the prefix and the suffix of length k 1 we search for the smallest alphabet in which k abelian squares and cubes can be avoided respectively for 2 abelian squares this is four as in the case of abelian words while for 2 abelian cubes we have only strong evidence that the size is two as it is in the case of words however we are able to prove this optimal value only for 8 abelian cubes

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

parallel vlsi matrix pencil algorithm for high resolution direction finding algorithms to find the directions of arrival doas of multiple signals from measurements on an array of antenna doublets esprit method and their parallel vlsi implementation are discussed in particular algorithms that allow large scale pipelining and use only robust unitary transformations are identified this problem is solved by a matrix pencil approach in which the generalized eigenvalues of a pair of data matrices are determined a modified stewart jacobi approach is used for which convergence is improved and parameter computations are simplified the resulting architecture is a two layer jacobi array that can handle all the subproblems two qr factorizations two svds and a single generalized schur decomposition the mapping of the subproblems on a single parallel array of cordic processors is considered

some results on k power free morphisms one way to generate infinite k power free words is to iterate a k power free morphism that is a morphism that preserves finite k power free words we first prove that the monoid of k power free endomorphisms on an alphabet containing at least three letters is not finitely generated test sets for k power free morphisms that is the sets t such that a morphism f is k power free if and only if f t is k power free give characterizations of these morphisms in the case of binary morphisms and k 3 we prove that a set t of cube free words is a test set for cube freeness if and only if it contains 12 particular factors consequently a morphism f on a b is cube free if and only if f aabbababbabbaabaababaabb is cube free length 24 is optimal another consequence is an unpublished result of leconte a binary morphism is cube free if and only if the images of all cube free words of length 7 are cube free when k3 we show that no finite test set exists for morphisms defined on an alphabet containing at least three letters in the last part we show that to generate an infinite cube free word by iterating a morphism we do not necessarily need a cube free morphism we give a new characterization of some morphisms that generate infinite cube free words

avoidable patterns on two letters abstract we examine avoidable patterns unavoidable in the sense of bean ehrenfeucht mcnulty 1979 we prove that each pattern on two letters of length at least 13 is avoidable on an alphabet with two letters the proof is based essentially on two facts first each pattern containing an overlapping factor is avoidable by the infinite word of thue morse secondly each pattern without overlapping factor is avoidable by the infinite word of fibonacci we further discuss the minimal alphabet on which very short patterns on a 2 letter and a 3 letter alphabet are avoidable

a powerful abelian square free substitution over 4 letters in 1961 paul erdos posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet an abelian square is a non empty word uv where u and v are permutations anagrams of each other the case of the four letter alphabet s 4 a b c d turned out to be the most challenging and remained open until 1992 when the author presented an abelian square free a 2 free endomorphism g 8 5 of s 4 the size of this g 8 5 i e g 8 5 abcd is equal to 4x85 uniform modulus until recently all known methods for constructing arbitrarily long a 2 free words on s 4 have been based on the structure of g 8 5 and on the endomorphism g 9 8 of s 4 found in 2002 in this paper a great many new a 2 free endomorphisms of s 4 are reported the sizes of these endomorphisms range from 4x102 to 4x115 importantly twelve of the new a 2 free endomorphisms of modulus m 109 can be used to construct an a 2 free commutatively functional substitution s 1 0 9 of s 4 with 12 image words for each letter the properties of s 1 0 9 lead to a considerable improvement for the lower bound of the exponential growth of c n i e of the number of a 2 free words over 4 letters of length n it is obtained that c n b 5 email protected n with b 12 1 m 1 02306 originally in 1998 carpi established the exponential growth of c n by showing that c n b email protected n with b 2 1 9 t 2 1 9 8 5 3 8 5 1 000021 where t 85 3 85 is the modulus of the substitution that he constructs starting from g 8 5

on abelian squares and substitutions we study substitutions preserving abelian square free words and the more general notion of substitutions with bounded abelian squares in particular we prove the existence of algorithms deciding whether a substitution mapping each letter into a set of commutatively equivalent words belongs to one of these classes

orthogonal grid constructions of copulas we establish a general framework for constructing copulas that can be regarded as a patchwork like assembly of arbitrary copulas with nonoverlapping rectangles as patches we derive a family of construction methods that require the choice of a single background copula when this background copula is the greatest copula we retrieve the well known ordinal sum construction while in the case of the smallest copula we obtain a construction method that is dual to the ordinal sum construction also nonsingular background copulas lead to suitable construction methods

uniformly growing k th power free homomorphisms a string is called kth power free if it does not have xk as a nonempty substring for all nonnegative rational numbers k kth power free strings and kth power free homomorphisms are investigated and the shortest uniformly growing square free k 2 and cube free k 3 homomorphisms mapping into least alphabets with three and two letters are introduced it is shown that there exist exponentially many square free and cube free strings of each length over these alphabets sharpening the kth power freeness to the repetitive threshold rt n of n letter alphabets we provide arguments for the nonexistence of various rt n th power free homomorphisms

characterization of some binary words with few squares thue proved that the factors occurring infinitely many times in square free words over 0 1 2 avoiding the factors in 010 212 are the factors of the fixed point of the morphism 0 012 1 02 2 1 he similarly characterized square free words avoiding 010 020 and 121 212 as the factors of two morphic words in this paper we exhibit smaller morphisms to define these two square free morphic words and we give such characterizations for six types of binary words containing few distinct squares

on repetition free binary words of minimal density we study the minimal proportion density of one letter in n th power free binary words first we introduce and analyse a general notion of minimal letter density for any infinite set of words which don t contain a specified set of prohibited subwords we then prove that for n th power free binary words the density function is frac 1 n frac 1 n 3 frac 1 n 4 cal o frac 1 n 5 we also consider a generalization of n th power free words for fractional powers exponents a word is x th power free for a real x if it does not contain subwords of exponent x or more we study the minimal proportion of one letter in x th power free binary words as a function of x and prove in particular that this function is discontinuous at frac 7 3 as well as at all integer points n geq 3 finally we give an estimate of the size of the jumps

growth unavoidable words and m sapir s conjecture for semigroup varieties let r be a nonperiodic semigroup variety satisfying the nontrivial identity zn w where zn is the nth element of the so called zimin s sequence z1 u1 zn 1 znun 1zn n 1 2 of unavoidable words it is shown that every finitely generated f g semigroup s r has subexponential growth and every uniformly recurrent infinite word which is geodesic and lexicographically reduced relative to s is periodic furthermore the length of the period is bounded above by a constant which depends only on the number n and on the number of generators of s the first of these results gives a positive answer to the problem posed by m sapir some applications including examples of f g relatively free semigroups having intermediate growth and maximal equals 1 superdimension are considered