on the existence of local strong solutions to chemotaxis shallow water system with large data and vacuum abstract in this paper motivated by the chemotaxis navier stokes system arising from mathematical biology 43 a modified shallow water type chemotactic model is derived for large initial data allowing vacuum the local existence of strong solutions together with the blow up criterion is established

partial words and a theorem of fine and wilf a partial word is a word that is a partial mapping into an alphabet we prove a variant of fine and wilf s theorem for partial words and give extensions of some general combinatorial properties of words

on the control of stochastic systems abstract a general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the iterative method of the authors or its more recent adaptations for stochastic operator equations it is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations the method therefore is applicable to the control problem application is made to the state space equation x ax bu c where a b c are stochastic matrices corresponding to stochastic operators i e involving randomly time varying elements e g a ij t a t t f a p s it is emphasized that the processes are arbitrary stochatic processes with known statistics no assumption is made of wiener or markov behavior or of smallness of fluctuations and no closure approximations are necessary the method differs in interesting aspects from volterra series expansions used by wiener and others and has advantages over the other methods because of recent progress in the solution of the nonlinear case it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

multiple source shortest paths in embedded graphs let g be a directed graph with n vertices and nonnegative weights in its directed edges embedded on a surface of genus g and let f be an arbitrary face of g we describe a randomized algorithm to preprocess the graph in o gn log n time with high probability so that the shortest path distance from any vertex on the boundary of f to any other vertex in g can be retrieved in o log n time our result directly generalizes the o n log n time algorithm of klein proceedings of the 16th annual acm siam symposium on discrete algorithms 2005 for multiple source shortest paths in planar graphs intuitively our preprocessing algorithm maintains a shortest path tree as its source point moves continuously around the boundary of f as an application of our algorithm we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded undirected graphs in o g 2 n log n time with high probability our high probability time bounds hold in the worst case for g

regular languages of partial words we initiate a study of languages of partial words related to regular languages of full words first we investigate the possibility of expressing a regular language of full words as the image of a partial words language through a substitution that only replaces the hole symbols of the partial words by a finite set of letters results regarding the structure uniqueness and succinctness of such a representation as well as a series of related decidability and computational hardness results are presented finally we introduce a hierarchy of classes of languages of partial words by grouping together languages that can be connected in various strong ways to regular languages and derive their closure properties with respect to several regular operations

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

monge properties of sequence alignment alignment is an important sequence comparison measure algorithms that compute alignments have a wide range of applications namely in bioinformatic tools alignments can be computed as maximum scoring paths in alignment dags in this paper we study the properties of matrices that contain alignment scores between a string and all the sub strings of another string we focus on the fact that these matrices have the monge property and are sparse in some sense related studies were recently presented for hsm and dist matrices leading to o nlogn procedure for multiplying those matrices where o n bounds the sizes of the strings our results strictly generalize previous solutions we measure the sparseness of the matrices with variable d and present an algorithm for matrix multiplication in o email protected log 3 email protected time which we improve to o email protected log 2 email protected within the same space we discuss applications of this algorithm namely fully incremental alignment and alignment update we study experimentally the performance of the methods we propose

strong solutions of the navier stokes equations for isentropic compressible fluids we study strong solutions of the isentropic compressible navier stokes equations in a domain r3 we first prove the local existence of unique strong solutions provided that the initial data 0 and u0 satisfy a natural compatibility condition the important point in this paper is that we allow the initial vacuum the initial density may vanish in an open subset of we then prove a new uniqueness result and stability result our results are valid for unbounded domains as well as bounded ones

discovering instances of poetic allusion from anthologies of classical japanese poems waka is a form of traditional japanese poetry with a 1300 year history in this paper we attempt to semi automatically discover instances of poetic allusion or more generally to find similar poems in anthologies of waka poems one reasonable approach would be to arrange all possible pairs of poems in two anthologies in decreasing order of similarity values and to scrutinize high ranked pairs by human effort the means of defining similarity between waka poems plays a key role in this approach in this paper we generalize existing dis similarity measures into a uniform framework called string resemblance systems and using this framework we develop new similarity measures suitable for finding similar poems using the measures we report successful results in finding instances of poetic allusion between two anthologies kokin shu and shin kokin shu most interestingly we have found an instance of poetic allusion that has never before been pointed out in the long history of waka research

incremental string comparison the problem of comparing two sequences a and b to determine their longest common subsequence lcs or the edit distance between them has been much studied in this paper we consider the following incremental version of these problems given an appropriate encoding of a comparison between a and b can one incrementally compute the answer for a and bb and the answer for a and bb with equal efficiency where b is an additional symbol our main result is a theorem exposing a surprising relationship between the dynamic programming solutions for two such adjacent problems given a threshold k on the number of differences to be permitted in an alignment the theorem leads directly to an o k algorithm for incrementally computing a new solution from an old one as contrasts the o k2 time required to compute a solution from scratch we further show with a series of applications that this algorithm is indeed more powerful than its nonincremental counterpart we show this by solving the applications with greater asymptotic efficiency than heretofore possible for example we obtain o nk algorithms for the longest prefix approximate match problem the approximate overlap problem and cyclic string comparison

decay estimates for isentropic compressible navier stokes equations in bounded domain abstract in this paper under the hypothesis that is upper bounded we construct a lyapunov functional for the multidimensional isentropic compressible navier stokes equations and show that the weak solutions decay exponentially to the equilibrium state in l 2 norm this can be regarded as a generalization of matsumura and nishida s results a matsumura t nishida in computing methods in applied sciences and engineering vol v 1982 pp 389 406 since our analysis is done in the framework of lions p l lions oxford science publication 1998 and feireisl et al e fereisl a novotny h petzeltova j math fluid mech 3 2001 358 392 the higher regularity of u and the uniformly positive lower bound of are not necessary in our analysis and vacuum may be admitted indeed the upper bound of the density plays the essential role in our proof

avoiding large squares in infinite binary words we consider three aspects of avoiding large squares in infinite binary words first we construct an infinite binary word avoiding both cubes xxx and squares yy with y 4 our construction is somewhat simpler than the original construction of dekking second we construct an infinite binary word avoiding all squares except 02 12 and 01 2 our construction is somewhat simpler than the original construction of fraenkel and simpson in both cases we also show how to modify our construction to obtain exponentially many words of length n with the given avoidance properties finally we answer an open question of prodinger and urbanek from 1979 by demonstrating the existence of two infinite binary words each avoiding arbitrarily large squares such that their perfect shuffle has arbitrarily large squares

on large time behavior of solutions to the compressible navier stokes equations in the half space in r3 the navier stokes equation for compressible viscous fluid is considered on the half space in r3 under the zero dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum time decay properties in l2 norms for solutions of the linearized problem are investigated to obtain the rate of convergence in l2 norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in some lower bounds are derived for solutions to the linearized problem one of which indicates a nonlinear phenomenon not appearing in the case of the cauchy problem on the whole space

large time behavior of solutions to an outflow problem for a shallow water model abstract we investigate the existence and the time asymptotic stability of boundary layer solutions for a one dimensional shallow water model under the subsonic condition on the far field equilibrium state we show that there exists a one parameter family of boundary layer solutions satisfying the outflow constant flux boundary condition we also show that there is a unique time asymptotic boundary layer profile for a given initial data and prove its stability using standard energy methods provided that the initial perturbation is sufficiently small our stability result is obtained without the zero mass condition nor artificial condition on the profile strength rather the time asymptotic profile is determined alone by the initial perturbation from the end state so that it satisfies the zero mass condition and its amplitude is appropriately small simultaneously

new complexity results for the k covers problem the k covers problem kcp asks us to compute a minimum cardinality set of strings of given length k 1 that covers a given string it was shown in a recent paper by reduction to 3 sat that the k covers problem is np complete in this paper we introduce a new problem that we call the k bounded relaxed vertex cover problem rvcp k which we show is equivalent to k bounded set cover scp k we show further that kcp is a special case of rvcp k restricted to certain classes g x k of graphs that represent all strings x thus a minimum k cover can be approximated to within a factor k in polynomial time we discuss approximate solutions of kcp and we state a number of conjectures and open problems related to kcp and g x k

application of the decomposition method to inversion of matrices abstract the decomposition method g adomian stochastic systems academic press new york london 1983 is applied to inversion of matrices by approximation

longest common subsequence problem for unoriented and cyclic strings given a finite set of strings x the longest common subsequence problem lcs consists in finding a subsequence common to all strings in x that is of maximal length lcs is a central problem in stringology and finds broad applications in text compression conception of error detecting codes or biological sequence comparison however in numerous contexts words represent cyclic or unoriented sequences of symbols and lcs must be generalized to consider both orientations and or all cyclic shifts of the strings involved this occurs especially in computational biology when genetic material is sequenced from circular dna or rna molecules in this work we define three variants of lcs when the input words are unoriented and or cyclic we show that these problems are np hard and w 1 hard if parameterized in the number of input strings these results still hold even if the three lcs variants are restricted to input languages over a binary alphabet we also settle the parameterized complexity of our problems for most relevant parameters moreover we study the approximability of these problems we discuss the existence of approximation bounds depending on the cardinality of the alphabet on the length of the shortest sequence and on the number of input sequences for this we prove that maximum independent set in r uniform hypergraphs is w 1 hard if parameterized in the cardinality of the sought independent set and at least as hard to approximate as maximum independent set in graphs

periodicity algorithms and a conjecture on overlaps in partial words we propose an algorithm that given as input a full word w of length n and positive integers p and d outputs if any exists a maximal p periodic partial word contained in w with the property that no two holes are within distance d so called d valid our algorithm runs in o nd time and is used for the study of repetition freeness of partial words furthermore we construct an infinite word over a five letter alphabet that is overlap free even after holes are inserted in arbitrary 2 valid positions answering affirmatively a conjecture from blanchet sadri mercas and scott

existence of global solutions for 2 dimensional viscous compressible flows abstract we prove the existence of global weak solutions for 2 dimensional unsteady viscous compressible flows of an ideal isothermal gas whatever the size of the data may be this is achieved by a suitable coupling of the method of hopf 17 with standard techniques in orlicz spaces

on the aperiodic avoidability of binary patterns with variables and reversals in this work we present a characterisation of the avoidability of all unary and binary patterns that do not only contain variables but also reversals of their instances with respect to aperiodic infinite words these types of patterns were studied recently in either more general or particular cases

asymptotic behavior of solutions of the compressible navier stokes equations on the half space asymptotic behavior of solutions of the compressible navier stokes equations on the half space r n n 2 is considered around a given constant equilibrium a solution formula for the linearized problem is derived and l p estimates for solutions of the linearized problem are obtained for 2 p it is shown that as in the case of the cauchy problem the leading part of the solution of the linearized problem is decomposed into two parts one that behaves like diffusion waves and the other one purely diffusively there however are some aspects different from the cauchy problem especially in considering spatial derivatives it is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary stokes problem in some l p spaces and as a result a solution formula for the nonstationary stokes problem is obtained large time behavior of solutions of the nonlinear problem is then investigated in l p spaces for 2 p by applying the results on the linearized analysis and the weighted energy method the results indicate that there may be some nonlinear interaction phenomena not appearing in the cauchy problem

sticky particles and scalar conservation laws one dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line which stick under collision with conservation of mass and momentum

sparse dynamic programming ii convex and concave cost functions dynamic programming solutions to two recurrence equations used to compute a sequence alignment from a set of matching fragments between two strings and to predict rna secondary structure are considered these recurrences are defined over a number of points that is quadratic in the input size however only a sparse set matters for the result efficient algorithms are given for solving these problems when the cost of a gap in the alignment or a loop in the secondary structure is taken as a convex or concave function of the gap or loop length the time complexity of our algorithms depends almost linearly on the number of points that need to be considered when the problems are sparse this results in a substantial speed up over known algorithms

codes orderings and partial words codes play an important role in the study of the combinatorics of words in this paper we introduce pcodes that play a role in the study of combinatorics of partial words partial words are strings over a finite alphabet that may contain a number of do not know symbols pcodes are defined in terms of the compatibility relation that considers two strings over the same alphabet that are equal except for a number of insertions and or deletions of symbols we describe various ways of defining and analyzing pcodes in particular many pcodes can be obtained as antichains with respect to certain partial orderings using a technique related to dominoes we show that the pcode property is decidable

all highest scoring paths in weighted grid graphs and their application to finding all approximate repeats in strings weighted paths in directed grid graphs of dimension m x n can be used to model the string edit problem which consists of obtaining optimal weighted alignments between substrings of a a m and substrings of b b n we build a data structure in o mn log m time that supports o log m time queries about the weight of any of the o m2n best paths from the vertices in column 0 of the graph to all other vertices using these techniques we present a simple o n2 log n time and theta n 2 space algorithm to find all the locally optimal approximate tandem or nontandem repeats xy within a string of size n this improves by a factor of log n upon several previous algorithms for this problem and is the first algorithm to find all locally optimal repeats for edit graphs with weights in 0 1 1 a slight modification of our techniques yields an o n2 algorithm for the cyclic string comparison problem as compared to o n2 log n for the case of general weights

relational codes of words we consider words i e strings over a finite alphabet together with a similarity relation induced by a compatibility relation on letters this notion generalizes that of partial words the theory of codes on combinatorics on words is revisited by defining r s codes for arbitrary similarity relations r and s we describe an algorithm to test whether or not a finite set of words is an r s code coding properties of finite sets of words are explored by finding maximal and minimal relations with respect to relational codes

approximate periodicity finding an approximate period in a given string s of length n is defined as follows let s be a periodic string closest to s under some distance metric find the smallest period of s this period is called an approximate period of s under the given metric let the distance between the input string s and a closest periodic string under the hamming distance s be k we develop algorithms that construct an approximate period of s under the hamming distance in time o n k log log n and under the swap distance in time o n 2 finally we show an o n log n algorithm for finite alphabets and an o n log 3 n algorithm for infinite alphabets that approximate the minimum number of mismatches between the input string and a closest periodic string under the hamming distance

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

a subquadratic sequence alignment algorithm for unrestricted scoring matrices given two strings of size n over a constant alphabet the classical algorithm for computing the similarity between two sequences d sankoff and j b kruskal eds time warps string edits and macromolecules addison wesley reading ma 1983 t f smith and m s waterman j molec biol 147 1981 pp 195 197 uses a dynamic programming matrix and compares the two strings in o n2 time we address the challenge of computing the similarity of two strings in subquadratic time for metrics which use a scoring matrix of unrestricted weights our algorithm applies to both local and global similarity computations the speed up is achieved by dividing the dynamic programming matrix into variable sized blocks as induced by lempel ziv parsing of both strings and utilizing the inherent periodic nature of both strings this leads to an o n 2 log n algorithm for an input of constant alphabet size for most texts the time complexity is actually o h n 2 log n where h le 1 is the entropy of the text we also present an algorithm for comparing two run length encoded strings of length m and n compressed into m and n runs respectively in o m n n m complexity this result extends to all distance or similarity scoring schemes that use an additive gap penalty

suffix arrays a new method for on line string searches a new and conceptually simple data structure called a suffix array for on line string searches is introduced in this paper constructing and querying suffix arrays is reduced to a sort and search paradigm that employs novel algorithms the main advantage of suffix arrays over suffix trees is that in practice they use three to five times less space from a complexity standpoint suffix arrays permit on line string searches of the type is w a substring of a to be answered in time o p log n where p is the length of w and n is the length of a which is competitive with and in some cases slightly better than suffix trees the only drawback is that in those instances where the underlying alphabet is finite and small suffix trees can be constructed in o n time in the worst case versus o n log n time for suffix arrays however an augmented algorithm is given that regardless of the alphabet size constructs suffix arrays in o n expected time albeit with lesser space efficiency it is

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

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

sparse dynamic programming i linear cost functions dynamic programming solutions to a number of different recurrence equations for sequence comparison and for rna secondary structure prediction are considered these recurrences are defined over a number of points that is quadratic in the input size however only a sparse set matters for the result efficient algorithms for these problems are given when the weight functions used in the recurrences are taken to be linear the time complexity of the algorithms depends almost linearly on the number of points that need to be considered when the problems are sparse this results in a substantial speed up over known algorithms

efficient all path score computations on grid graphs we study the integer weighted grid all paths scores igaps problem which is given a grid graph to compute the maximum weights of paths between every pair of a vertex on the first row of the graph and a vertex on the last row of the graph we also consider a variant of this problem periodic igaps where the input grid graph is periodic and infinite for these problems we consider both the general dense and the sparse cases for the sparse periodic igaps problem with 0 1 weights we give an o rlog 3 n 2 r time algorithm where r is the number of diagonal edges of weight 1 our result improves upon the previous o nr result by krusche and tiskin for this problem for the periodic igaps problem we give an o cn 2 time algorithm where c is the maximum weight of an edge this improves upon the previous o c 2n 2 algorithm of tiskin we also show a reduction from periodic igaps to igaps this reduction yields o n 2 algorithms for this problem

approximate periods of strings the study of approximately periodic strings is relevant to diverse applications such as molecular biology data compression and computer assisted music analysis here we study different forms of approximate periodicity under a variety of distance functions we consider three related problems for two of which we derive polynomial time algorithms we then show that the third problem is np complete

overlap freeness in infinite partial words we prove that there exist infinitely many infinite overlap free binary partial words containing at least one hole moreover we show that these words cannot contain more than one hole and the only hole must occur either in the first or in the second position we define that a partial word is k overlap free if it does not contain a factor of the form xyxyx where the length of x is at least k we prove that there exist infinitely many 2 overlap free binary partial words containing an infinite number of holes

compressible euler equations with vacuum abstract in this paper we first show that the regular solutions of compressible euler equations in r 3 with damping will not be global if the initial density function has compact support this implies that c 2 cannot be smooth across the boundary separating the gas and the vacuum after a finite time where c is the speed of sound then we study the local existence of solutions for isentropic gas flow in r when c 0 1 is smooth across using the energy method and the characteristics method

blowup of smooth solutions to the compressible navier stokes equation with compact density we present a sufficient condition on the blowup of smooth solutions to the compressible navier stokes equations in arbitrary space dimensions with initial density of compact support as an immediate application it is shown that any smooth solutions to the compressible navier stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support and an upper bound which depends only on the initial data on the blowup time follows from our elementary analysis immediately another implication is that there is no global small decay in time or even bounded in the case that all the viscosity coefficients are positive smooth solutions to the compressible navier stokes equations for polytropic fluids no matter how small the initial data are as long as the initial density is of compact support this is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum the blowup of smooth solutions to the compressible euler system with initial density and velocity of compact support is a simple consequence of our argument 1998 john wiley sons inc

the failure on continuous dependence on initial data for the navier stokes equations of compressible flow it is shown that physical solutions of the navier stokes equations for one dimensional compressible flow need not depend continuously on their initial data at least when vacuum states are allowed specifically two fluid regions initially separated by a third region of very low density 6 are considered it is shown that as delta to 0 the unique solutions corresponding to delta 0 do not in fact converge to a physical solution but rather to a nonphysical weak solution in which the two fluids cannot collide independent of their initial velocities and whose separate momenta need not be conserved a particular consequence is that solutions of the cavity problem delta 0 are not unique

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

how to play thue games a famous theorem of thue asserts that there exist arbitrarily long words without repetitions identical adjacent blocks over a 3 letter alphabet in this paper we study a game theoretic variant of this result where a word is constructed jointly by two players who alternately append letters to the end of existing word one of the players ann takes care of avoiding repetitions while the other one ben tries to force them we provide an explicit strategy for ann to avoid non trivial repetitions over a 9 letter alphabet this solves an open problem posed by pegden 13 we also consider a similar game for overlaps in words identical overlapping blocks we give an explicit strategy for ann to avoid overlaps over a 4 letter alphabet even if the players are placing letters on arbitrarily chosen positions in constructed word provided its final length is even additionally we prove that the bound of 4 in this result is optimal

repetitive perhaps but certainly not boring in this paper some of the work done on repetitions in strings is surveyed especially that of an algorithmic nature several open problems are described and conjectures formulated about some of them

the cauchy problem for the pressureless euler isentropic navier stokes equations we present a new hydrodynamic model consisting of the pressureless euler equations and the isentropic compressible navier stokes equations where the coupling of two systems is through the drag force this coupled system can be derived in the hydrodynamic limit from the particle fluid equations that are frequently used to study the medical sprays aerosols and sedimentation problems for the proposed system we first construct the local in time classical solutions in an appropriate l2l2 sobolev space we also establish the a priori large time behavior estimate by constructing a lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities and using this together with the bootstrapping argument we obtain the global classical solution the large time behavior estimate asserts that the velocity functions of the pressureless euler and the compressible navier stokes equations are aligned exponentially fast as time tends to infinity

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

stabilization of a stochastic nonlinear economy abstract methodology for dynamical systems described by nonlinear stochastic differential equations is applied to economic systems particularly the phillips turnovsky model generalizing the model further to nonlinear terms and stochastic coefficients and inputs to determine expectations and correlation matrices for the variables which represent solution processes of a set of coupled equations the theory is quite general and will apply if the model is changed to use different nonlinear forms e g for taxes as a function of income or different statistics for fluctuations which need not be stationary or gaussian or small or to include retarded effects

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

avoiding large squares in partial words well known results on the avoidance of large squares in full words include the following 1 fraenkel and simpson showed that we can construct an infinite binary word containing at most three distinct squares 2 entringer jackson and schatz showed that there exists an infinite binary word avoiding all squares of the form xx such that x 3 and that the bound 3 is optimal 3 dekking showed that there exists an infinite cube free binary word that avoids all squares xx with x 4 and that the bound of 4 is best possible in this paper we investigate these avoidance results in the context of partial words or sequences that may have some undefined symbols called holes here a square has the form uv with u and v compatible and consequently such a square is compatible with a number of full words that are squares over the given alphabet we show that 1 holds for partial words with at most two holes we prove that 2 extends to partial words having infinitely many holes regarding 3 we show that there exist binary partial words with infinitely many holes that avoid cubes and have only eleven full word squares compatible with factors of it moreover this number is optimal and all such squares xx satisfy x 4

partial words and a theorem of fine and wilf revisited a word of length n over a finite alphabet a is a map from 0 n 1 into a a partial word of length n over a is a partial map from 0 n 1 into a in the latter case elements of 0 n1 without image are called holes a word is just a partial word without holes in this paper we extend a fundamental periodicity result on words due to fine and wilf to partial words with two or three holes this study was initiated by berstel and boasson for partial words with one hole partial words are motivated by molecular biolog

an algorithm for locating nonoverlapping regions of maximum alignment score in this paper we present an o n 2 log 2 n algorithm for finding the two nonoverlapping substrings of a given string of length n which have the highest scoring alignment between them this significantly improves the previously best known bound of o n 3 for the worst case complexity of this problem one of the central ideas in the design of this algorithm is that of partitioning a matrix into pieces in such a way that all submatrices of interest for this problem can be put together as the union of very few of these pieces other ideas include the use of candidate lists an application of the ideas of apostolico et al siam j comput 19 1990 pp 968 988 to our problem domain and divide and conquer techniques

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

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

non existence of classical solutions with finite energy to the cauchy problem of the compressible navier stokes equations the well posedness of classical solutions with finite energy to the compressible navier stokes equations cns subject to arbitrarily large and smooth initial data is a challenging problem in the case when the fluid density is away from vacuum strictly positive this problem was first solved for the cns in either one dimension for general smooth initial data or multi dimension for smooth initial data near some equilibrium state that is small perturbation antontsev et al in boundary value problems in mechanics of nonhomogeneous fluids north holland publishing co amsterdam 1990 kazhikhov in sibirsk mat zh 23 60 64 1982 kazhikhov et al in prikl mat meh 41 282 291 1977 matsumura and nishida in proc jpn acad ser a math sci 55 337 342 1979 j math kyoto univ 20 67 104 1980 commun math phys 89 445 464 1983 in the case that the flow density may contain a vacuum the density can be zero at some space time point it seems to be a rather subtle problem to deal with the well posedness problem for cns the local well posedness of classical solutions containing a vacuum was shown in homogeneous sobolev space without the information of velocity in l2 norm for general regular initial data with some compatibility conditions being satisfied initially cho et al in j math pures appl 9 83 243 275 2004 cho and kim in j differ equ 228 377 411 2006 manuscr math 120 91 129 2006 choe and kim in j differ equ 190 504 523 2003 and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations huang et al in commun pure appl math 65 549 585 2012 however it was shown that any classical solutions to the compressible navier stokes equations in finite energy inhomogeneous sobolev space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported xin in commun pure appl math 51 229 240 1998 in this paper we investigate the well posedess of classical solutions to the cauchy problem of navier stokes equations and prove that the classical solution with finite energy does not exist in the inhomogeneous sobolev space for any short time under some natural assumptions on initial data near the vacuum this implies in particular that the homogeneous sobolev space is as crucial as studying the well posedness for the cauchy problem of compressible navier stokes equations in the presence of a vacuum at far fields even locally in time