improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

low complexity soft decoding algorithms for reed solomon codes part ii soft input soft output iterative decoding in this paper we present a practical approach to the iterative decoding of reed solomon rs codes the presented methodology utilizes an architecture in which the output produced by steps of belief propagation bp is successively applied to a legacy decoding algorithm due to the suboptimal performance of bp conducted on the inherently dense rs parity check matrix a method is first provided for the construction of reduced density binary parity check equations iterative decoding is then conducted utilizing a subset of a redundant set of parity check equations to minimize the number of connections into the least reliable bits simulation results show that performance comparable to and exceeding the best known practical rs decoding techniques is achievable with the presented methodology the complexity of the proposed algorithm is significantly lower than these existing procedures and permits a practical implementation in hardware

efficient computation of maximal anti exponent in palindrome free strings a palindrome is a string x a 1 a n which is equal to its reversal x a n a 1 we consider gapped palindromes which are strings of the form u v u where u v are strings v 2 and u is the reversal of u replicating the standard notion of string exponent we define the anti exponent of a gapped palindrome u v u as the quotient of u v u by u v to get an efficient computation of maximal anti exponent of factors in a palindrome free string we apply techniques based on the suffix automaton and the reversed lempel ziv factorisation our algorithm runs in o n time on a fixed size alphabet or o n log on a large alphabet which dramatically outperforms the naive cubic time solution

on the number of α power free binary words for 2 α 7 3 we study the number u a n of a power free binary words of length n and the asymptotics of this number when n tends to infinity for a fixed rational number a in 2 7 3 for any such a we prove a structure result that allows us to describe constructively the sequence u a n as a 2 regular sequence this provides an algorithm that computes the number u a n in logarithmic time for fixed a then generalizing recent results on 2 free words we describe the asymptotic behaviour of u a n in terms of joint spectral quantities of a pair of matrices that one can efficiently construct given a rational number a for a 7 3 we compute the automaton and give sharp estimates for the asymptotic behaviour of u a n

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

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

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

rich square free words abstract a word w is rich if it has w 1 distinct palindromic factors including the empty word a word is square free if it does not have a factor uu where u is a non empty word pelantova and starosta 2013 16 proved that every infinite rich word contains a square we will give another proof for that result pelantova and starosta marked with r n the length of a longest rich square free word on an alphabet of size n the exact value of r n was left as an open question we will give an upper and a lower bound for r n the lower bound is conjectured to be exact but it is not explicit we will also generalize the notion of repetition threshold for a limited class of infinite words the repetition thresholds for episturmian and rich words are left as an open question

pseudorandom construction of low density parity check codes using linear congruential sequences we consider maximal length linear congruential sequences generated using a simple recursion to generate the bipartite graph of a low density parity check ldpc code the main advantage is that the graph structure of the codes edge connections can be generated using a recursion rather than having to store the graph connections in memory which facilitates hardware implementation of the decoder for this class of codes sufficient conditions on the recursion parameters are derived such that regular ldpc codes can be constructed with no cycles of length four or less simulation results show that these codes provide almost the same performance of a constrained pseudorandom construction that explicitly avoids cycles of length less than or equal to four

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

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

subword complexity and power avoidance abstract we begin a systematic study of the relations between subword complexity of infinite words and their power avoidance among other things we show that the thue morse word has the minimum possible subword complexity over all overlap free binary words and all 7 3 power free binary words but not over all 7 3 power free binary words the twisted thue morse word has the maximum possible subword complexity over all overlap free binary words but no word has the maximum subword complexity over all 7 3 power free binary words if some word attains the minimum possible subword complexity over all square free ternary words then one such word is the ternary thue word the recently constructed 1 2 bonacci word has the minimum possible subword complexity over all symmetric square free ternary words

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

repetiveness of languages generated by morphisms abstract we study the repetition of subwords in languages generated by morphisms fundamental to our approach is the notion of quasi repetitive elements using these elements we present a new characterization for repetitive morphisms from which we derive a simple proof for the fact that a d0l language is repetitive if and only if it is strongly repetitive ehrenfeucht and rozenberg inform and control 59 1983 13 35 from this proof we obtain a structurally simple polynomial time algorithm for deciding whether such a language is repetitive from further results on quasi repetitive elements we obtain as a consequence a complete characterization for all those morphisms on a two letter alphabet that are repetitive a result which is closely related to a result of seebold bull eatcs 36 1988 137 151 on the d0l periodicity problem finally we characterize those morphisms f on a two letter alphabet for which the language l f generated by f or the language sl f of subwords of l f are context free or even regular

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

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

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

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

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

study of predictor corrector block method via multiple shooting to blasius and sakiadis flow abstract in this paper a predictor corrector two point block method is proposed to solve the well know blasius and sakiadis flow numerically the blasius and sakiadis flow will be modeled by a third order boundary value problem the main motivation of this study is to provide a new method that can solve the higher order bvp directly without reducing it to a system of first order equation two approximate solutions will be obtained simultaneously in a single step by using predictor corrector two point block method able to solve the third order boundary value problem directly the proposed direct predictor corrector two point block method will be adapted with multiple shooting techniques via a three step iterative method the advantage of the proposed code is that the multiple shooting will converge faster than the shooting method that has been implemented in other software the developed code will automatically choose the guessing values in order to solve the given problems some numerical results are presented and a comparison to the existing methods has been included to show the performance of the proposed method for solving blasius and sakiadis flow

new family of iterative methods for nonlinear equations in this paper we suggest and analyze a new family of iterative methods for solving nonlinear equations using the system of coupled equations coupled with decomposition technique several numerical examples are given to illustrate the efficiency and performance of the new methods these new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations

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

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

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

repetition avoidance in products of factors we consider a variation on a classical avoidance problem from combinatorics on words that has been introduced by mousavi and shallit at dlt 2013 let texttt pexp i w be the supremum of the exponent over the products of i factors of the word w the repetition threshold texttt rt i k is then the infimum of texttt pexp i w over all words w in sigma omega k moussavi and shallit obtained that texttt rt i 2 2i and texttt rt 2 3 tfrac 13 4 we show that texttt rt i 3 tfrac 3i 2 tfrac14 if i is even and texttt rt i 3 tfrac 3i 2 tfrac16 if i is odd and i ge3

three step iterative methods for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of the new method

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

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

fewest repetitions versus maximal exponent powers in infinite binary words a square is the concatenation of a nonempty word with itself a word has period p if its letters at distance p match the exponent of a nonempty word is the quotient of its length over its smallest period in this article we give some new results on the trade off between the number of squares and the number of maximal exponent powers in infinite binary words in the three cases where the maximal exponent is 7 3 5 2 and 3 these are the only threshold values related to the question

algebraic soft decision decoding of reed solomon codes using bit level soft information the performance of algebraic soft decision decoding of reed solomon codes using bit level soft information is investigated optimal multiplicity assignment strategies for algebraic soft decision decoding sdd with infinite cost are first studied over erasure channels and the binary symmetric channel the corresponding decoding radii are calculated in closed forms and tight bounds on the error probability are derived the multiplicity assignment strategy and the corresponding performance analysis are then generalized to characterize the decoding region of algebraic sdd over a mixed error and bit level erasure channel the bit level decoding region of the proposed multiplicity assignment strategy is shown to be significantly larger than that of conventional berlekamp massey decoding as an application a bit level generalized minimum distance decoding algorithm is proposed the proposed decoding compares favorably with many other reed solomon sdd algorithms over various channels moreover owing to the simplicity of the proposed bit level generalized minimum distance decoding its performance can be tightly bounded using order statistics

a note on some quadrature based three step iterative methods for non linear equations in a recent paper n a mir t zaman some quadrature based three step iterative methods for non linear equations appl math comput 193 2007 366 373 some new three step iterative methods for non linear equations have been proposed in this note we show that the algorithm 2 2 and algorithm 2 3 given by the authors have twelfth order and ninth order convergence respectively not seventh order one as claimed in their work

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

proof of dejean s conjecture for alphabets with 5 6 7 8 9 10 and 11 letters abstract axel thue proved that overlapping factors could be avoided in arbitrarily long words on a two letter alphabet while on the same alphabet square factors always occur in words longer than 3 francoise dejean stated an analogous result for three letter alphabets every long enough word has a factor which is a fractional power with an exponent at least 7 4 and there exist arbitrary long words in which no factor is a fractional power with an exponent strictly greater than 7 4 the number 7 4 is called the repetition threshold of the three letter alphabets thereafter she proposed the following conjecture the repetition threshold of the k letter alphabets is equal to k k 1 except in the particular cases k 3 where this threshold is 7 4 and k 4 where it is 7 5 for k 4 this conjecture was proved by j j pansiot 1984 in this paper we give a computer aided proof of dejean s conjecture for several other values 5 6 7 8 9 10 and 11

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

higher order iterative schemes for nonlinear equations using decomposition technique in this paper we suggest and analyze some unique recurrence relations which can generate different classes of iterative methods for solving nonlinear equations using the system of coupled equations together with decomposition technique by using an auxiliary function various numerical examples are given to illustrate the efficiency and performance of the newly suggested methods these new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations

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

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

a generalization of thue freeness for partial words this paper approaches the combinatorial problem of thue freeness for partial words partial words are sequences over a finite alphabet that may contain a number of holes first we give an infinite word over a three letter alphabet which avoids squares of length greater than two even after we replace an infinite number of positions with holes then we give an infinite word over an eight letter alphabet that avoids longer squares even after an arbitrary selection of its positions are replaced with holes and show that the alphabet size is optimal we find similar results for overlap free partial words

iterative soft input soft output decoding of reed solomon codes by adapting the parity check matrix an iterative algorithm is presented for soft input soft output siso decoding of reed solomon rs codes the proposed iterative algorithm uses the sum product algorithm spa in conjunction with a binary parity check matrix of the rs code the novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the spa is applied at each iteration the proposed algorithm can be geometrically interpreted as a two stage gradient descent with an adaptive potential function this adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and therefore significantly improves the performance simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding hdd and compare favorably with other popular soft decision decoding methods

last cases of dejean s conjecture dejean conjectured that the repetition threshold for a k letter alphabet is kk 1 when k 5 dejean s conjecture has already been proved for k 14 and for k 27 we present here a proof for 8 k 38 the same technique is also applied to prove ochem s stronger version of the conjecture for 9 k 38

on the number of dejean words over alphabets of 5 6 7 8 9 and 10 letters we give lower bounds on the growth rate of dejean words i e minimally repetitive words over a k letter alphabet for 5 k 10 put together with the known upper bounds we estimate these growth rates with the precision of 0 005 as a consequence we establish the exponential growth of the number of dejean words over a k letter alphabet for 5 k 10

an iterative method for solving nonlinear functional equations an iterative method for solving nonlinear functional equations viz nonlinear volterra integral equations algebraic equations and systems of ordinary differential equation nonlinear algebraic equations and fractional differential equations has been discussed

low complexity soft decoding algorithms for reed solomon codes part i an algebraic soft in hard out chase decoder in this paper we present an algebraic methodology for implementing low complexity chase type decoding of reed solomon rs codes of length n in such a set of 2 test vectors that are equivalent on all except n coordinate positions is first produced the similarity of the test vectors is utilized to reduce the complexity of interpolation the process of constructing a set of polynomials that obey constraints imposed by each test vector by first considering the equivalent indices a polynomial common to all test vectors is constructed the required set of polynomials is then produced by interpolating the final dissimilar indices utilizing a binary tree structure in the second decoding step factorization a candidate message is extracted from each interpolation polynomial such that one may be chosen as the decoded message although an expression for the direct evaluation of each candidate message is provided carrying out this computation for each polynomial is extremely complex thus a novel reduced complexity methodology is also given although suboptimal simulation results affirm that the loss in performance incurred by this procedure is decreasing with increasing code length n and negligible for long n 100 codes significant coding gains are shown to be achievable over traditional hard in hard out decoding procedures e g berlekamp massey at an equivalent and in some cases lower computational complexity furthermore these gains are shown to be similar to the recently proposed soft in hard out algebraic techniques e g sudan ko tter vardy that bear significantly more complex implementations than the proposed algorithm

soft decision decoding of reed solomon codes this paper presents a maximum likelihood decoding mld and a suboptimum decoding algorithm for reed solomon rs codes the proposed algorithms are based on the algebraic structure of the binary images of rs codes theoretical bounds on the performance are derived and shown to be consistent with simulation results the proposed suboptimum algorithm achieves near mld performance with significantly lower decoding complexity it is also shown that the proposed suboptimum algorithm has better performance compared with generalized minimum distance decoding while the proposed mld algorithm has significantly lower decoding complexity than the well known vardy be ery 1991 mld algorithm

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

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

on iterative methods for nonlinear equations several iterative methods have been suggested and analyzed for solving nonlinear equation f x 0 all these methods can be classified as one step and two step methods in this paper we have suggested and analyzed several two step methods for solving nonlinear equations we have also shown that these two step methods are quadratically convergent several numerical examples are given to illustrate this comparison our results can be viewed as important refinement and improvement of the previously known results

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

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

a note on three step iterative method for nonlinear equations abstract noor and noor m a noor k i noor three step iterative methods for nonlinear equations appl math comput 183 2006 322 327 proposed a three step iterative method for solving nonlinear equations and they showed that the method has third order convergence in this paper we propose a new three step iterative method for solving nonlinear equations which is a significant improvement of the method proposed by noor and noor it is shown that this iterative method has fourth order convergence not third order convergence some numerical experiments are given to illustrate the performance of the three step iterative method

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

decoding by local optimization corresp the maximum likelihood decoding problem for linear binary n k codes is reformulated as a continuous optimization problem in a k dimensional solid cube we obtain a near optimum solution of this problem by use of a simple gradient local optimization algorithm computer simulation results are presented for the 21 11 projective geometry code and the 47 23 quadratic residue code