bifix codes and sturmian words we prove new results concerning the relation between bifix codes episturmian words and subgroups of free groups we study bifix codes in factorial sets of words we generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set f of words f maximal bifix codes in the case of bifix codes contained in sturmian sets of words we obtain several new results let f be a sturmian set of words defined as the set of factors of a strict episturmian word our results express the fact that an f maximal bifix code of degree d behaves just as the set of words of f of length d an f maximal bifix code of degree d in a sturmian set of words on an alphabet with k letters has k 1 d 1 elements this generalizes the fact that a sturmian set contains k 1 d 1 words of length d moreover given an infinite word x if there is a finite maximal bifix code x of degree d such that x has at most d factors of length d in x then x is ultimately periodic our main result states that any f maximal bifix code of degree d on the alphabet a is the basis of a subgroup of index d of the free group on a

saving comparisons in the crochemore perrin string matching algorithm crochemore and perrin discovered an elegant linear time constant space string matching algorithm that makes at most 2n m symbol comparisons this paper shows how to modify their algorithm to use fewer comparisons given any fixed e 0 the new algorithm takes linear time uses constant space and makes at most n 1 e2 n m symbol comparisons if o log m space is available then the algorithm makes at most n 12 n m symbol comparisons the pattern preprocessing step also takes linear time and uses constant space these are the first string matching algorithms that make fewer than 2n m symbol comparisons and use sub linear space

partial words and the critical factorization theorem revisited in this paper we consider one of the most fundamental results on the periodicity of words namely the critical factorization theorem given a word w and nonempty words u v satisfying w uv the minimal local period associated with the factorization u v is the length of the shortest square at position u 1 the critical factorization theorem shows that for any word there is always a factorization whose minimal local period is equal to the minimal period or global period of the word crochemore and perrin presented a linear time algorithm in the length of the word that finds a critical factorization from the computation of the maximal suffixes of the word with respect to two total orderings on words the lexicographic ordering related to a fixed total ordering on the alphabet and the lexicographic ordering obtained by reversing the order of letters in the alphabet here by refining crochemore and perrin s algorithm we give a version of the critical factorization theorem for partial words such sequences may contain do not know symbols or holes our proof provides an efficient algorithm which computes a critical factorization when one exists our results extend those of blanchet sadri and duncan for partial words with one hole a world wide web server interface at http www uncg edu mat research cft2 has been established for automated use of the program

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

on the density of lyndon roots in factors this work takes another look at the number of runs that a string may contain and provides an alternative proof for the bound we also propose another stronger conjecture that states the following for a fixed order on the alphabet within every factor of a word there are at most as many occurrences of lyndon roots corresponding to runs in the word as the length of the factor only first occurrences of roots in each run are considered

multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system compared with most of the existing tanh methods the jacobi elliptic function method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters the solutions obtained in this paper include a kink shaped and bell shaped soliton solutions b rational solutions c triangular periodic solutions and d jacobi and weierstrass doubly periodic wave solutions among them the jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition the efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper kdv mkdv ito s fifth mkdv hirota nizhnik novikov veselov broer kaup generalized coupled hirota satsuma coupled schrodinger kdv 2 1 dimensional dispersive long wave 2 1 dimensional davey stewartson equations in addition as an illustrative sample the properties of the soliton solutions and jacobi doubly periodic solutions for the hirota equation are shown by some figures the links among our proposed method the tanh method extended tanh method and the jacobi elliptic function method are clarified generally

verhulst lotka volterra vlv model of ideological struggle abstract a model for ideological struggles is formulated the underlying set is a closed one like a country but in which the population size is variable in time the dynamics of the struggle is described by model equations of verhulst lotka volterra kind several ideologies compete to increase their number of adepts such followers can be either converted from one ideology to another or become followers of an ideology though being previously ideologically free a reverse process is also allowed two kinds of conversions are considered unitary conversion e g by means of mass communication tools or binary conversion e g by means of interactions between people it is found that the steady state when it exists depends on the number of ideologies moreover when the number of ideologies increases some tension arises between them this tension can change in the course of time we propose to measure the ideology tensions through an appropriately defined scale index finally it is shown that a slight change in the conditions of the environment can prevent the extinction of some ideology after almost collapsing the ideology can spread again and can affect a significant part of the country s population two kinds of such resurrection effects are described as phoenix effects

periodicity and unbordered words a proof of the extended duval conjecture the relationship between the length of a word and the maximum length of its unbordered factors is investigated in this article consider a finite word w of length n we call a word bordered if it has a proper prefix which is also a suffix of that word let w denote the maximum length of all unbordered factors of w and let w denote the period of w clearly w w we establish that w e w if w has an unbordered prefix of length w and n 2 w 1 this bound is tight and solves the stronger version of an old conjecture by duval 1983 it follows from this result that in general n 3 w 3 implies w e w which gives an improved bound for the question raised by ehrenfeucht and silberger in 1979

upper bound on the heat transport in a layer of fluid of infinite prandtl number rigid lower boundary and stress free upper boundary we obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite prandtl number with rigid lower boundary and stress free upper boundary because of the asymmetric boundary conditions the solutions of the euler lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on the upper and lower boundary of the fluid the obtained bound on the convective heat transport and the corresponding wave number are between the values for a fluid layer with two rigid boundaries and a fluid layer with two stress free boundaries

new solitary wave solutions to the kuramoto sivashinsky and the kawahara equations abstract in this work the kuramoto sivashinsky equation and the kawahara equation are studied the tanh method and the extended tanh method are used for analytic treatment for these two equations by means of these methods new solitary wave solutions are determined for each equation the two approaches are reliable and manageable

compressed suffix arrays and suffix trees with applications to text indexing and string matching the proliferation of online text such as found on the world wide web and in online databases motivates the need for space efficient text indexing methods that support fast string searching we model this scenario as follows consider a text t consisting of n symbols drawn from a fixed alphabet sigma the text t can be represented in n lg sigma bits by encoding each symbol with lg sigma bits the goal is to support fast online queries for searching any string pattern p of m symbols with t being fully scanned only once namely when the index is created at preprocessing time the text indexing schemes published in the literature are greedy in terms of space usage they require omega n lg n additional bits of space in the worst case for example in the standard unit cost ram suffix trees and suffix arrays need omega n memory words each of omega lg n bits these indexes are larger than the text itself by a multiplicative factor of omega smash lg sigma n which is significant when sigma is of constant size such as in textsc ascii or textsc unicode on the other hand these indexes support fast searching either in o m lg sigma time or in o m lg n time plus an output sensitive cost o mathit occ for listing the mathit occ pattern occurrences we present a new text index that is based upon compressed representations of suffix arrays and suffix trees it achieves a fast smash o m lg sigma n lg sigma epsilon n search time in the worst case for any constant 0 epsilon leq 1 using at most smash bigl epsilon 1 o 1 bigr n lg sigma bits of storage our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case and for many scenarios the space is actually sublinear in practice as a concrete example the compressed suffix array for a typical 100 mb textsc ascii file can require 30 40 mb or less while the raw suffix array requires 500 mb our theoretical bounds improve emph both time and space of previous indexing schemes listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output sensitive cost giving o mathit occ smash lg sigma epsilon n time as a result when the patterns are sufficiently long we can use auxiliary data structures in o n lg sigma bits to obtain a total search bound of o m lg sigma n mathit occ time which is optimal

dirichlet inversion and lattice inversion problem abstract another application of dirichlet multiplication is considered in this note we show that dirichlet inversion in number theory plays an important role in lattice inversion problem with the help of this concept lattice inversion problem becomes straightforward

how the character comparison order shapes the shift function of on line pattern matching algorithms string matching is the problem of finding all occurrences of a string 0 m 1 of length m called a pattern in a longer string 0 n 1 of length n called a text several string matching algorithms have been designed to solve the problem in linear time most of them work in two steps called pattern preprocessing and text search step the paper addresses the definition and computation of the shift function in the pattern preprocessing step of on line string matching algorithms the shift function depends essentially on the order the pattern characters are compared with the corresponding text characters we consider a family of algorithms that do not change the character comparison order j during execution and we present a uniform definition of shift function j for such algorithms via a function iminj the definition allows one to compute j in o m log log m time in the worst case given iminj but sufficient conditions to compute j in o m time are provided computing iminj requires o m2 comparisons in general we introduce the class of compact orders which is the generalization of knuth morris pratt boyer moore and crochemore perrin character comparison orders and we give algorithms to compute both function iminj and shift function j in o m time for all compact orders we show that given the order j and the pattern there exists a set c of equivalent orders such that the function imink can be computed in linear time given iminj for all orders k c moreover we characterize two orders in the set c that respectively minimize and maximize the values of the shift function and we show that for both those orders the shift function can be computed in linear time given iminj

dirichlet multiplication and easily generated function analysis abstract following sine cosine functions sawtooth wave square wave triangular wave and trapezoidal wave become new easily generated periodic functions can a signal be considered to be a superposition of easily generated functions with different frequencies in order to answer this question we generalize fourier analysis to easily generated function analysis including easily generated functions series easily generated function transformation and discrete transformation for easily generated functions the results in this paper make it possible to represent a signal by use of easily generated functions a lot of techniques based on sine cosine functions can be translated into techniques based on easily generated functions because dirichlet multiplication in number theory plays a basic role in easily generated function analysis we briefly introduce this concept and present a related formula the main contents of this paper include dirichlet multiplication and a related formula relations between basic waveforms in electronics easily generated function series easily generated function transformation discrete transformation for easily generated functions and techniques of easily generated function analysis

two way string matching a new string matching algorithm is presented which can be viewed as an intermediate between the classical algorithms of knuth morris and pratt on the one hand and boyer and moore on the other hand the algorithm is linear in time and uses constant space as the algorithm of galil and seiferas it presents the advantage of being remarkably simple which consequently makes its analysis possible the algorithm relies on a previously known result in combinatorics on words called the critical factorization theorem which relates the global period of a word to its local repetitions of blocks

convective heat transport in a rotating fluid layer of infinite prandtl number optimum fields and upper bounds on nusselt number by means of the howard busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress free vertical boundaries rotating about a vertical axis we discuss the case of infinite prandtl number 1 ensuremath ensuremath alpha solution of the obtained variational problem and optimum fields possessing internal intermediate and boundary layers we investigate regions of rayleigh and taylor numbers r and ta where no analytical bounds can be derived and compare the analytical and numerical bounds for these regions of r and ta where such comparison is possible the increasing rotation has a different influence on the rescaled optimum fields of velocity w 1 temperature ensuremath theta 1 and the vertical component of the vorticity f 1 the increasing ta for fixed r leads to vanishing of the boundary layers of w 1 and ensuremath theta 1 opposite to this the increasing ta leads first to a formation of boundary layers of the field f 1 but further increasing the rotation causes vanishing of these boundary layers we obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying doering constantin method to the problems of rotating convection the wave number ensuremath alpha 1 corresponding to the optimum fields follows the asymptotic relationship ensuremath alpha 1 r 5 1 4 for intermediate rayleigh numbers however when r becomes large with respect to ta after a transition region the power law for ensuremath alpha 1 becomes close to the power law for the case without rotation the nusselt number nu is close to the nonrotational bound 0 32r 1 3 for the case of large r and small ta nu decreases with increasing taylor number thus the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed rayleigh number for the regions of rayleigh and taylor numbers where the numerical and asymptotic bounds on nu can be compared the numerical bounds are about 70 lower than the asymptotic bounds

simple real time constant space string matching string matching is the classical problem of finding all occurrences of a pattern in a text a real time string matching algorithm takes worst case constant time to check if a pattern occurrence ends at each text location we derive a real time variation of the elegant crochemore perrin constant space string matching algorithm that has a simple and efficient control structure we use observations about the locations of critical factorizations to deploy two tightly coupled simplified real time instances of the crochemore perrin algorithm that search for complementary parts of the pattern whose simultaneous occurrence indicates an occurrence of the complete pattern

a proof of the extended duval s conjecture we give a short and elementary proof of the following stronger version of duval s conjecture let u be an unbordered word and v a word of length u such that v is not a prefix of u then uv contains an unbordered word of length at least u 1

on maximal suffixes and constant space linear time versions of kmp algorithm constant space linear time string matching algorithms are usually very sophisticated most of them consist of two phases very technical preprocessing phase and searching phase an exception is one phase crochemore s algorithm theoret comput sci 92 1992 33 it is an on line version of knuth morris pratt algorithm kmp with on the fly computation of pattern shifts as approximate periods in this paper we explore further crochemore s approach and construct alternative algorithms which are differently structured in crochemore s algorithm the approximate period function is restarted from inside which means that several internal variables of this function are changing globally also crochemore s algorithm strongly depends on the concrete implementation of approximate periods computation we present a simple modification of kmp algorithm which works in o 1 space o n time for any function which computes periods or approximate periods in o 1 space and linear time the approximate period function can be treated as a black box we identify class of patterns self maximal words which are especially well suited for crochemore style string matching a new o 1 space string matching algorithm maxsuffix matching is proposed in the paper which gives yet another example of applicability of maximal suffixes

square wave analysis about two hundred years ago the french mathematician j b j fourier asserted that any function with period 2ir can be expressed as a trigonometric series this great idea has had an important influence upon many sciences including mathematics and physics in electronics fourier analysis has been playing an important role and a signal is often considered as a superposition of many sine and cosine waves with different frequencies however with the development of switching circuits and pulse technique it becomes very easy to generate and process square waves for example it s quite convenient to obtain a system of square waves with different frequencies from a common high frequency pulse by means of counters whereas it s complicated to do the same thing with sine and cosine functions again multipliers and summers for square waves are very simple

frequency analysis based on general periodic functions electrical waveforms may have various shapes and therefore it is necessary to study the frequency analysis based on general periodic functions in this chapter we introduce the frequency series and frequency transform based on general periodic functions considering the fact that the theory for fourier series in l2 and the theory for fourier transform in l2 are both simple and elegant we shall develop a theory for general frequency series in l2 and a theory for general frequency transform in l2 in the following we shall discuss when a frequency system is a complete system or an unconditional basis in l2 and when a frequency transform can be carried out in l2 we present a number of complete systems and unconditional bases formed by common waveforms almost every common waveform is considered as an example

inferring strings from lyndon factorization the lyndon factorization of a string w is a unique factorization ell 1 p 1 ldots ell m p m of w s t l1 l m is a sequence of lyndon words that is monotonically decreasing in lexicographic order in this paper we consider the reverse engineering problem on lyndon factorization given a sequence s s 1 p 1 s m p m of ordered pairs of positive integers find a string w whose lyndon factorization corresponds to the input sequence s i e the lyndon factorization of w is in a form of ell 1 p 1 ldots ell m p m with l i s i for all 1 i m firstly we show that there exists a simple o n time algorithm if the size of the alphabet is unbounded where n is the length of the output string secondly we present an o n time algorithm to compute a string over an alphabet of the smallest size thirdly we show how to compute only the size of the smallest alphabet in o m time fourthly we give an o m time algorithm to compute an o m size representation of a string over an alphabet of the smallest size finally we propose an efficient algorithm to enumerate all strings whose lyndon factorizations correspond to s

a multidimensional critical factorization theorem the critical factorization theorem is one of the principal results in combinatorics on words it relates local periodicities of a word to its global periodicity in this paper we give a multidimensional extension of it more precisely we give a new proof of the critical factorization theorem but in a weak form where the weakness is due to the fact that we loose the tightness of the local repetition order in exchange we gain the possibility of extending our proof to the multidimensional case indeed this new proof makes use of the theorem of fine and wilf that has several classical generalizations to the multidimensional case

transducers and repetitions the factor transducer of a word associates to each of its factors or subwords their first occurrence optimal bounds on the size of minimal factor transducers together with an algorithm for building them are given analogue results and a simple algorithm are given for the case of subsequential suffix transducers algorithms are applied to repetition searching in words le transducteur des facteurs d un mot associe a chacun de ses facteurs leur premiere occurrence on donne des bornes optimales sur la taille du transducteur minimal d un most ainsi qu un algorithme pour sa construction on donne des resultats analogues et un algorithme simple dans le cas du transducteur sous sequentiel des suffixes d un mot on donne une application a la detection de repetitions dans les mots

on the equivalence between a minimal codomain cardinality riesz basis construction a system of hadamard sylvester operators and a class of sparse binary optimization problems piecewise low order polynomial riesz basis families are constructed such that they share the same coefficient functionals of smoother orthonormal bases in a localized indexing subset it is shown that a minimal cardinality basis codomain can be realized by inducing sparsity via l1 regularization in the distributional derivatives of the basis functions and that the optimal construction can be found numerically by constrained binary optimization over a suitably large dictionary furthermore it is shown that a subset of these solutions are equivalent to a specific constrained analytical solution derived via sylvester type hadamard operators

the structure of subword graphs and suffix trees of fibonacci words we use automata theoretic approach to analyze properties of fibonacci words the directed acyclic subword graph dawg is a useful deterministic automaton accepting all suffixes of the word we show that dawg s of fibonacci words have particularly simple structure our main result is a unifying framework for a large collection of relatively simple properties of fibonacci words the simple structure of dawgs of fibonacci words gives in many cases simplified alternative proofs and new interpretation of several well known properties of fibonacci words in particular the structure of lengths of paths corresponds to a number theoretic characterization of occurrences of any subword using the structural properties of dawg s it can be easily shown that for a string we can check if is a subword of a fibonacci word in time o and o 1 space compact dawg s of fibonacci words show a very regular structure of their suffix trees and show how the suffix tree for the fibonacci word grows extending the leaves in a very simple way into the suffix tree for the next fibonacci word

the zooming method a recursive approach to time space efficient string matching a new approach to time space efficient string matching is presented the method is flexible its implementation depends whether or not the alphabet is linearly ordered the only known linear time constant space algorithm for string matching over nonordered alphabets is the galil seiferas algorithm see crochemore 1993 and galil 1983 which are rather complicated the zooming method gives probably the simplest string matching algorithm working in constant space and linear time for nonordered alphabets the novel feature of our algorithm is the application of the searching phase which is usually simpler than preprocessing in the preprocessing phase the preprocessing has a recursive structure similar to selection in linear time see aho 1974 for ordered alphabets the preprocessing part is much simpler its basic component is a simple and well known algorithm for finding the maximal suffix see duval 1983 hence we demonstrate a new application of this algorithm see also crochemore 1991 the idea of the zooming method was applied by crochemore et al 1995 to two dimensional patterns

on some solutions of the two dimensional sine gordon equation there exists an approach for finding of exact solutions of the two dimensional sine gordon equation using this approach three classes of solutions have been found one of the classes consists of running wave solutions which are a generalization of the solutions of the one dimensional sine gordon equation this class of solutions is studied here

space efficient search for maximal repetitions we study here a problem of finding all maximal repetitions in a string of length n we show that the problem can be solved in time o n log n in the presence of constant extra space and general unbounded alphabets subsequently we show that in the model with a constant size alphabet the problem can be solved in time o n with a help of o n extra space previously best known algorithms require linear additional space in both models

dynamical consequences of adaptation of the growth rates in a system of three competing populations we investigate the nonlinear dynamics of a system of populations competing for the same limited resource for the case where each of the populations adapts its growth rate to the total number of individuals in all populations we consider regions of parameter space where chaotic motion of the shilnikov kind exists and present results for two characteristic values of the growth ratio adaptation factor r r 0 15 and 5 negative r can lead to vanishing of regions of chaotic motion and to a stabilization of a fixed point of the studied model system of differential equations positive r lead to changes of the shape of the bifurcation diagrams in comparison with the bifurcation diagrams for the case without adaptation for the case r 5 we observe transition to chaos by period doubling bifurcations windows of periodic motion between the regions of chaotic motion and a region of transient chaos after the last window of periodic motion the lyapunov dimension for the chaotic attractors is close to two and the lyapunov spectrum has a structure which allows a topological analysis of the attractors of the investigated system

new class of running wave solutions of the 2 1 dimensional sine gordon equation a new class of running wave solutions of the 2 1 dimensional sine gordon equation is investigated the obtained waves require two spatial dimensions for their propagation i e they generalize solutions of the 2 0 dimensional sine gordon equation the parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency the obtained solutions describe a new class of josephson waves whose velocity is smaller than the swihart velocity if omega 0 the running waves are reduced to the self consistent phase current and magnetic field distributions in a large two dimensional josephson junction the self restriction coefficient for the josephson current corresponding to one of the structures is calculated

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

constant space string matching in sublinear average time given two strings pattern p of length m and text t of length n the string matching problem is to find all occurrences of the pattern p in the text t we present a string matching algorithms which works in o n average time and constant additional space for one dimensional texts and two dimensional arrays this is a first attempt to the small space string matching problem in which sublinear time algorithms are achieved we show that all occurrences of one or two dimensional patterns can be found in o n r average time with constant memory where r is the repetition size of p size of the longest repeated subword of p

linear time computation of local periods we present a linear time algorithm for computing all local periods of a given word this subsumes but is substantially more powerful than the computation of the global period of the word and on the other hand the computation of a critical factorization implied by the critical factorization theorem

recurrence and periodicity in infinite words from local periods we study recurrence and periodicity in infinite words by using local conditions in particular we give a characterization of recurrent periodic and eventually periodic infinite words in terms of local periods

on the correspondence between the self consistent 2d poisson boltzmann structures and the sine gordon waves by means of the transformation connecting the sine gordon equation and the two dimensional poisson boltzmann equation a correspondence between the self consistent two dimensional poisson boltzmann structures and the solutions of the sine gordon equation representing standing waves has been established in this way two new solutions of the sine gordon equation were obtained and studied and their application for description of waves into ferromagnetics was examined

application of the method of simplest equation for obtaining exact traveling wave solutions for the extended korteweg de vries equation and generalized camassa holm equation a version of the method of the simplest equation called modified method of simplest equation is applied to the extended korteweg de vries equation and to generalized camassa holm equation exact traveling wave solutions of these two nonlinear partial differential equations are obtained the equations of bernoulli riccati and the extended tanh equation are used as simplest equations some of the obtained solutions correspond to surface water waves

modified method of simplest equation and its application to nonlinear pdes abstract we search for traveling wave solutions of the class of pdes p 1 n 1 a p q p q t p r 2 n 2 b r q q t r s 1 n 3 c s q s q x s u 2 n 4 d u q q x u f q 0 where a p q b r q c s q d u q and f q are polynomials of q the basis of the investigation is a modification of the method of simplest equation the equations of bernoulli riccati and the extended tanh function equation are used as simplest equations the obtained general results are illustrated by obtaining exact solutions of versions of the generalized kuramoto sivashinsky equation reaction diffusion equation with density dependent diffusion and the reaction telegraph equation

finding the leftmost critical factorization on unordered alphabet we present a linear time and space algorithm computing the leftmost critical factorization of a given string on an unordered alphabet we consider algorithms computing the leftmost critical point on unordered alphabets we describe an o n log n time algorithm finding the leftmost critical point we investigate some combinatorial properties of strings related to our problem using found combinatorial facts we improve our first algorithm to o n algorithm

computing minimal and maximal suffixes of a substring we consider the problems of computing the maximal and the minimal non empty suffixes of substrings of a longer text of length n for the minimal suffix problem we show that for every 1 log n there exists a linear space data structure with o query time and o n log n preprocessing time as a sample application we show that this data structure can be used to compute the lyndon decomposition of any substring of the text in o k time where k is the number of distinct factors in the decomposition for the maximal suffix problem we give a linear space structure with o 1 query time and o n preprocessing time in other words we simultaneously achieve both the optimal query time and the optimal construction time

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations

characteristic sturmian words are extremal for the critical factorization theorem we prove that characteristic sturmian words are extremal for the critical factorization theorem cft in the following sense if p x n denotes the local period of an infinite word x at point n we prove that x is a characteristic sturmian word if and only if p x n is smaller than or equal to n 1 for all n 1 and it is equal to n 1 for infinitely many integers n this result is extremal with respect to the cft since a consequence of the cft is that for any infinite recurrent word x either the function p x is bounded and in such a case x is periodic or p x n n 1 for infinitely many integers n as a byproduct of the techniques used in the paper we extend a result of harju and nowotka 2002 in 18 stating that any finite fibonacci word f n n 5 has only one critical point indeed we determine the exact number of critical points in any finite standard sturmian word