minimal generating reflexive lattices of projections in finite von neumann algebras we show that the reflexive lattice generated by a double triangle lattice of projections in a finite von neumann algebra is topologically homeomorphic to the two dimensional sphere s 2 plus two distinct points corresponding to zero and i furthermore such a reflexive lattice is in general minimally generating for the von neumann algebra it generates as an application we show that if a reflexive lattice mathcal f generates the algebra m n mathbb c of all n n complex matrices for some n 3 then mathcal f setminus 0 i is connected if and only if it is homeomorphic to s 2

kadison singer algebras ii general case a new class of operator algebras kadison singer ks algebras is introduced these highly noncommutative non self adjoint algebras generalize triangular matrix algebras they are determined by certain minimally generating lattices of projections in the von neumann algebras corresponding to the commutant of the diagonals of the ks algebras it is shown that these lattices and their reduced forms are often homeomorphic to classical manifolds such as the sphere

order preserving pattern matching with k mismatches we consider a generalization of the recently introduced order preserving pattern matching the difference between the standard pattern matching and the order preserving variant is that instead of looking for an exact copy of the pattern we only require that the relative order between the elements is the same in our generalization we additionally allow up to k mismatches between the pattern of length m and the text of length n and the goal is to construct an efficient algorithm for small values of k our solution detects an order preserving occurrence with up to k mismatches in o n loglogm kloglogk time

string periods in the order preserving model abstract in the order preserving model two strings match if they share the same relative order between the characters at the corresponding positions this model is quite recent but it has already attracted significant attention because of its applications in data analysis we introduce several types of periods in this setting op periods then we give algorithms to compute these periods in time o n o n log log n o n log 2 log n log log log n o n log n depending on the type of periodicity in the most general variant the number of different op periods can be as big as n 2 and a compact representation is needed our algorithms require novel combinatorial insight into the properties of op periods in particular we characterize the fine wilf property for coprime op periods

a note on the diagonal maximality of operator algebras a subalgebra a of b h is said to be maximal with respect to its diagonal if it cannot be properly contained in any other subalgebra with the same diagonal in this paper we show that if t is a hyperreducible maximal triangular algebra with a totally atomic or nonatomic diagonal d then for a given separating vector of d the algebra of all operators t in t having as an eigenvector is maximal respect to its diagonal we also prove that each reflexive algebra defined by a double triangular lattice of projections in a matrix algebra has the diagonal maximality if the double triangular lattice of projections generates the whole matrix algebra

efficient 2 dimensional approximate matching of half rectangular figures efficient algorithms exist for the approximate two dimensional matching problem for rectangles this is the problem of finding all occurrences of an m m pattern in an n n text with no more than k mismatch insertion and deletion errors in computer vision it is important to generalize this problem to non rectangular figures we make progress towards this goal by defining half rectangular figures of height m and area a the approximate two dimensional matching problem for half rectangular patterns can be solved using a dynamic programming approach in time o an2 we show an o kn2formula formula k2n2 algorithm which combines convolutions with dynamic programming note that our algorithm is superior to previous known solutions for k m13 at the heart of the algorithm are the smaller matching problem and the k aligned ones with location problem these are interesting problems in their own right efficient algorithms to solve both these problems are presented

maximum number of distinct and nonequivalent nonstandard squares in a word the combinatorics of nonstandard squares in a word depends on how the equivalence of halves of the square is defined we consider abelian squares parameterized squares and order preserving squares the word uv is an abelian parameterized order preserving square if u and v are equivalent in the abelian parameterized order preserving sense the maximum number of ordinary squares in a word is known to be asymptotically linear but the exact bound is still investigated we present several results on the maximum number of distinct squares for nonstandard factor equivalence relations let sq abel n and sq abel n denote the maximum number of abelian squares in a word of length n over an alphabet of size which are distinct as words and which are nonequivalent in the abelian sense respectively for 2 we prove that sq abel n n 2 sq abel n n 3 2 and sq abel n o n 11 6 we also give linear bounds for parameterized and order preserving squares for alphabets of constant size sq param n o 1 n sq op n o 1 n the upper bounds have quadratic dependence on the alphabet size for order preserving squares and exponential dependence for parameterized squares as a side result we construct infinite words over the smallest alphabet which avoid nontrivial order preserving squares and nontrivial parameterized cubes nontrivial parameterized squares cannot be avoided in an infinite word a preliminary version of this paper was published at dlt 2014 24 in this full version we improve or extend the bounds on all three kinds of squares

dynamic ordered sets with exponential search trees we introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully dynamic linear space data structures this leads to an optimal bound of o slog n log log n for searching and updating a dynamic set x of n integer keys in linear space searching x for an integer y means finding the maximum key in x which is smaller than or equal to y this problem is equivalent to the standard text book problem of maintaining an ordered set the best previous deterministic linear space bound was o log n log log n due to fredman and willard from stoc 1990 no better deterministic search bound was known using polynomial space we also get the following worst case linear space trade offs between the number n the word length w and the maximal key u our results are generalized to finger searching and string searching providing optimal results for both in terms of n

the three squares lemma for partial words with one hole partial words or sequences over a finite alphabet that may have do not know symbols or holes have been recently the subject of much investigation several interesting combinatorial properties have been studied such as the periodic behavior and the counting of distinct squares in partial words in this paper we extend the three squares lemma on words to partial words with one hole this result provides special information about the squares in a partial word with at most one hole and puts restrictions on the positions at which periodic factors may occur which is in contrast with the well known periodicity lemma of fine and wilf

squares in partial words abstract we investigate the number of positions that do not start a square the number of square occurrences and the number of distinct squares in partial words i e sequences that may have undefined positions called holes we show that the limit of the ratio of the maximum number of positions not starting a square in a binary partial word with h holes over its length n is 15 31 and the limit of the ratio of the minimum number of square occurrences in a binary partial word with h holes over its length n is 103 187 provided the limit of h n is 0 both limits turn out to match with the known limits for binary full words those without holes we prove another surprising result that the maximal proportion of defined positions that are square free to the number of defined positions in a binary partial word with h holes of length n is 1 2 provided the limit of h n is in the interval 1 11 1 we also give a 2 k h tight bound on the number of rightmost occurrences of squares per position in a k ary partial word with h holes in addition we provide a more detailed analysis than earlier ones for the maximum number of distinct squares in a one hole partial word of length n over an alphabet of size k bound that is independent of k

on the number of frames in binary words abstract a frame is a square u u where u is an unbordered word let f n denote the maximum number of distinct frames in a binary word of length n we count this number for small values of n and show that f n is at most n 2 8 for all n and greater than 7 n 30 for any positive and infinitely many n we also show that fibonacci words which are known to contain plenty of distinct squares have only a few frames moreover by modifying the thue morse word we prove that the minimum number of occurrences of frames in a word of length n is n 2 2

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

efficient computations of l 1 and l rearrangement distances recently a new pattern matching paradigm was proposed pattern matching with address errors in this paradigm approximate string matching problems are studied where the content is unaltered and only the locations of the different entries may change specifically a broad class of problems was defined the class of rearrangement errors in this type of error the pattern is transformed through a sequence of rearrangement operations each with an associated cost the natural 1 and 2 rearrangement systems were considered the best algorithm presented for general patterns that may have repeating symbols is o nm in this paper we show that the problem can be approximated in linear time for general patterns another natural rearrangement system is considered in this paper the rearrangement distance for this new rearrangement system efficient exact solutions for different variants of the problem are provided as well as a faster approximation

derivations of a class of kadison singer algebras abstract let l be a double triangle lattice of projections in a finite von neumann algebra acting on a separable and complex hilbert space k we show that every derivation from the reflexive algebra determined by l into b k is quasi spatial and automatically continuous we also obtain that every local derivation on the reflexive algebra is a derivation

faster suffix tree construction with missing suffix links we consider suffix tree construction for situations with missing suffix links two examples of such situations are suffix trees for parameterized strings and suffix trees for two dimensional arrays these trees also have the property that the node degrees may be large we add a new back propagation component to mccreight s algorithm and also give a high probability hashing scheme for large degrees we show that these two features enable construction of suffix trees for general situations with missing suffix links in o n time with high probability this gives the first randomized linear time algorithm for constructing suffix trees for parameterized strings

abelian square rich words abstract an abelian square is the concatenation of two words that are anagrams of one another a word of length n can contain at most n 2 distinct factors and there exist words of length n containing n 2 distinct abelian square factors that is distinct factors that are abelian squares this motivates us to study infinite words such that the number of distinct abelian square factors of length n grows quadratically with n more precisely we say that an infinite word w is abelian square rich if for every n every factor of w of length n contains on average a number of distinct abelian square factors that is quadratic in n and uniformly abelian square rich if every factor of w contains a number of distinct abelian square factors that is proportional to the square of its length of course if a word is uniformly abelian square rich then it is abelian square rich but we show that the converse is not true in general we prove that the thue morse word is uniformly abelian square rich and that the function counting the number of distinct abelian square factors of length 2 n of the thue morse word is 2 regular as for sturmian words we prove that a sturmian word s of angle is uniformly abelian square rich if and only if the irrational has bounded partial quotients that is if and only if s has bounded exponent

distinct squares in run length encoded strings squares are strings of the form ww where w is any nonempty string two squares ww and w w are of different types if and only if w w fraenkel and simpson avieri s fraenkel jamie simpson how many squares can a string contain journal of combinatorial theory series a 82 1998 112 120 proved that the number of square types contained in a string of length n is bounded by o n the set of all different square types contained in a string is called the vocabulary of the string if a square can be obtained by a series of successive right rotations from another square then we say the latter covers the former a square is called a c square if no square with a smaller index can cover it and it is not a trivial square the set containing all c squares is called the covering set note that every string has a unique covering set furthermore the vocabulary of the covering set are called c vocabulary in this paper we prove that the cardinality of c vocabulary in a string is less than 143n where n is the number of runs in this string

abelian powers and repetitions in sturmian words abstract richomme saari and zamboni 2011 39 proved that at every position of a sturmian word starts an abelian power of exponent k for every k 0 we improve on this result by studying the maximum exponents of abelian powers and abelian repetitions an abelian repetition is an analogue of a fractional power in sturmian words we give a formula for computing the maximum exponent of an abelian power of abelian period m starting at a given position in any sturmian word of rotation angle by considering all possible abelian periods m we recover the result of richomme saari and zamboni as an analogue of the critical exponent we introduce the abelian critical exponent a s of a sturmian word s of angle as the quantity a s lim sup k m m lim sup k m m where k m resp k m denotes the maximum exponent of an abelian power resp of an abelian repetition of abelian period m the superior limits coincide for sturmian words we show that a s equals the lagrange constant of the number this yields a formula for computing a s in terms of the partial quotients of the continued fraction expansion of using this formula we prove that a s 5 and that the equality holds for the fibonacci word we further prove that a s is finite if and only if has bounded partial quotients that is if and only if s is power free for some real number concerning the infinite fibonacci word we prove that i the longest prefix that is an abelian repetition of period f j j 1 has length f j f j 1 f j 1 1 2 if j is even or f j f j 1 f j 1 2 if j is odd where f j is the j th fibonacci number ii the minimum abelian period of any factor is a fibonacci number further we derive a formula for the minimum abelian periods of the finite fibonacci words we prove that for j 3 the fibonacci word f j of length f j has minimum abelian period equal to f j 2 if j 0 1 2 mod 4 or to f 1 j 2 if j 3 mod 4

dynamic perfect hashing upper and lower bounds the dynamic dictionary problem is considered provide an algorithm for storing a dynamic set allowing the operations insert delete and lookup a dynamic perfect hashing strategy is given a randomized algorithm for the dynamic dictionary problem that takes o 1 worst case time for lookups and o 1 amortized expected time for insertions and deletions it uses space proportional to the size of the set stored furthermore lower bounds for the time complexity of a class of deterministic algorithms for the dictionary problem are proved this class encompasses realistic hashing based schemes that use linear space such algorithms have amortized worst case time complexity omega log n for a sequence of n insertions and lookups if the worst case lookup time is restricted to k then the lower bound becomes omega k cdot n 1 k

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

efficient counting of square substrings in a tree we give an algorithm which in o nlog 2n time counts all distinct squares in a labeled tree there are two main obstacles to overcome the first one is that the number of distinct squares in a tree is w n 4 3 see crochemore et al 2012 7 which differs substantially from the case of classical strings for which there are only linearly many distinct squares we overcome this obstacle by using a compact representation of all squares based on maximal cyclic shifts which requires only o nlogn space the second obstacle is lack of adequate algorithmic tools for labeled trees consequently we design several novel tools this is the most complex part of the paper in particular we extend to trees imre simon s compact representations of the failure table in pattern matching machines

the method of layer potentials for the heat equation in lipschitz cylinders pour l equation de la chaleur dans des cylindres de lipschitz on etudie des problemes aux valeurs limites d initiales avec conditions de neumann et de dirichlet

on the complexity of algebraic numbers ii continued fractions the continued fraction expansion of an irrational number alpha is eventually periodic if and only if alpha is a quadratic irrationality however very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three because of some numerical evidence and a belief that these numbers behave like most numbers in this respect it is often conjectured that their partial quotients form an unbounded sequence more modestly we may expect that if the sequence of partial quotients of an irrational number alpha is in some sense simple then alpha is either quadratic or transcendental the term simple can of course lead to many interpretations it may denote real numbers whose continued fraction expansion has some regularity or can be produced by a simple algorithm by a simple turing machine for example or arises from a simple dynamical system the aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity complexity for the continued fraction expansion of algebraic real numbers of degree at least three

order preserving indexing kubica et al 33 and kim et al 29 introduced order preserving pattern matching for a given text the goal is to find its factors having the same shape as a given pattern known results include a linear time algorithm for this problem in case of polynomially bounded alphabet and a generalization to multiple patterns we propose an index that enables order preserving pattern matching queries in time proportional to pattern length the index can be constructed in o n log log n expected time or in o n log 2 log n log log log n worst case time it is an incomplete order preserving suffix tree which may miss a single edge label at each branching node for most applications such incomplete suffix trees provide the same functional power as the complete ones we show a number of their applications including computation of longest common factors longest previously occurring factors and squares in a string in the order preserving setting we also give an o n log n time algorithm constructing complete order preserving suffix trees

order preserving matching we introduce a new string matching problem called order preserving matching on numeric strings where a pattern matches a text if the text contains a substring of values whose relative orders coincide with those of the pattern order preserving matching is applicable to many scenarios such as stock price analysis and musical melody matching in which the order relations should be matched instead of the strings themselves solving order preserving matching is closely related to the representation of order relations of a numeric string we define the prefix representation and the nearest neighbor representation of the pattern both of which lead to efficient algorithms for order preserving matching we present efficient algorithms for single and multiple pattern cases for the single pattern case we give an o nlogm time algorithm and optimize it further to obtain o n mlogm time for the multiple pattern case we give an o nlogm time algorithm

sequential prediction over hierarchical structures we study sequential compound decision problems in the context of sequential prediction of real valued sequences in particular we consider finite state fs predictors that are constructed based on a hierarchical structure such as the order preserving patterns of the sequence history we define hierarchical equivalence classes by tying certain models at a hierarchy level in a recursive manner in order to mitigate undertraining problems these equivalence classes defined on a hierarchical structure are then used to construct a super exponential number of sequential fs predictors based on their combinations and permutations we then introduce truly sequential algorithms with computational complexity only linear in the pattern length that 1 asymptotically achieve the performance of the best fs predictor or the best linear combination of all the fs predictors in an individual sequence manner without any stochastic assumptions over any data length n under a wide range of loss functions 2 achieve the mean square error of the best linear combination of all fs filters or predictors in the steady state for certain nonstationary models we illustrate the superior convergence and tracking capabilities of our algorithm with respect to several state of the art methods in the literature through simulations over synthetic and real benchmark data

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

parabolic regularity and dirichlet boundary value problems abstract we study the relationship between the regularity and dirichlet boundary value problems for parabolic equations of the form l u div a u u t 0 in lip 1 1 2 time varying cylinders where the coefficient matrix a a i j x t is uniformly elliptic and bounded we show that if the regularity problem r p for the equation l u 0 is solvable for some 1 p then the dirichlet problem d p for the adjoint equation l v 0 is also solvable where p p p 1 this result is an analogue of the result established in the elliptic case by kenig and pipher 1993 in the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by hofmann and lewis 1996 and nystrom 2006 for scalar parabolic systems in comparison our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains

l2 solvability of the dirichlet neumann and regularity problems for parabolic equations with time independent holder continuous coefficients we establish the l 2 solvability of dirichlet neumann and regularity problems for divergence form heat or diffusion equations with timeindependent holder continuous diffusion coefficients on bou

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

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

generalized pattern matching and periodicity under substring consistent equivalence relations let be a substring consistent equivalence relation scer on strings such that for any two strings x y x y implies that 1 x y and 2 x i j y i j for all 0 i j x examples of scer are parameterized pattern matching and order preserving pattern matching we present a generalized and efficient algorithm for pattern matching with scer also we show analogues of fine and wilf s periodicity lemma hold for scer

sturmian words and words with a critical exponent let s be a standard sturmian word that is a fixed point of a non trivial homomorphism associated to the infinite word s is a unique irrational number with 0 0 it contains a fractional power with exponent greater than e here is a constant that depends on the constant is given explicitly using these results we are able to give a short proof of mignosi s theorem and give an exact evaluation of the maximal power that can occur in a standard sturmian word

kadison singer algebras hyperfinite case a new class of operator algebras kadison singer algebras ks algebras is introduced these highly noncommutative non self adjoint algebras generalize triangular matrix algebras they are determined by certain minimally generating lattices of projections in the von neumann algebras corresponding to the commutant of the diagonals of the ks algebras a new invariant for the lattices is introduced to classify these algebras

finding patterns and periods in cartesian tree matching abstract we introduce a new metric of match called cartesian tree matching which means that two strings match if they have the same cartesian trees based on cartesian tree matching we define single pattern matching multiple pattern matching and indexing problems we propose a linear time algorithm for single pattern matching and randomized linear time algorithms for multiple pattern matching and indexing we also define three types of periods called full period initial period and general period we propose o n time o n log log n time and o n log n time algorithms for finding all the full periods initial periods and general periods of a string of length n respectively our efficient algorithms use a representation of the cartesian tree called parent distance representation

simple and flexible detection of contiguous repeats using a suffix tree we study the problem of detecting all occurrences of primitive tandem repeats and tandem arrays in a string we first give a simple time and space optimal algorithm to find all tandem repeats and then modify it to become a time and space optimal algorithm for finding only the primitive tandem repeats both of these algorithms are then extended to handle tandem arrays the contribution of this paper is both pedagogical and practical giving simple algorithms and implementations based on a suffix tree using only standard tree traversal techniques

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

on the average number of regularities in a word we study the average number of powers and runs occurring in a word of length n drawn from an alphabet of size s we show that a word contains email protected r 1 1 o n powers of exponent r at most email protected o n runs and also email protected n o n palindromes we also explore their abelian variants and prove that a binary word contains q n 3 2 abelian squares on the average while its number of abelian cubes is related to franel numbers finally as a consequence we show that a binary word has almost surely o n 3 2 abelian squares

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

a note on the number of squares in a word fraenkel and simpson a s fraenkel j simpson how many squares can a string contain j combin theory ser a 82 1998 112 120 proved that the number of squares in a word of length n is bounded by 2n in this note we improve this bound to 2n q logn based on the numerical evidence the conjectured bound is n

letter frequency in infinite repetition free words we estimate the extremal letter frequency in infinite words over a finite alphabet avoiding some repetitions for ternary square free words we improve the bounds of tarannikov on the minimal letter frequency and prove that the maximal letter frequency is 255653 kolpakov et al have studied the function r such that r x is the minimal letter frequency in an infinite binary x free word in particular they have shown that r is discontinuous at 73 and at every integer at least 3 we answer one of their questions by providing some other points of discontinuity for r finally we propose stronger versions of dejean s conjecture on repetition threshold in which unequal letter frequencies are required

on some generalizations of abelian power avoidability we prove that 2 abelian cubes are avoidable over a binary alphabet and that 3 abelian squares are avoidable over a ternary alphabet answering positively to two questions of karhumaki et al we also show the existence of infinite additive cube free words on several ternary alphabets to achieve this we give sufficient conditions for a morphism to be k abelian n power free resp additive n power free and then we give several morphisms which respect these conditions additionally all our constructions show that the number of such words grows exponen tially as a corollary we get a new lower bound of 3 1 19 1 059526 for the growth rate of abelian cube free words

on critical exponents in fixed points of non erasing morphisms let s be an alphabet of size t let f s s be a non erasing morphism let w be an infinite fixed point of f and let e w be the critical exponent of w we prove that if e w is finite then for a uniform f it is rational and for a non uniform f it lies in the field extension q r l 1 l where r l 1 l are the eigenvalues of the incidence matrix of f in particular e w is algebraic of degree at most t under certain conditions our proof implies an algorithm for computing e w