test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

on binary equality sets and a solution to the test set conjecture in the binary case abstract k culik ii and j karhumaki rairo inform theor 14 1980 349 369 conjectured that the equality set of two injective morphisms over a binary alphabet is of the form u v for some possibly empty words u and v here we show that such an equality set is always either of the above form or of the form uw v for some words u w and v as an application we give a simple proof for the test set conjecture in the binary case cf k culik ii and a salomaa j comput system sci 20 1980 379 396 in fact we show that a test set can always be chosen to contain no more than three words

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

divergence free wavelet bases on the hypercube free slip boundary conditions and applications for solving the instationary stokes equations we construct wavelet riesz bases for the usual sobolev spaces of divergence free functions on that have vanishing normals at the boundary we give a simultaneous space time variational formulation of the instationary stokes equations that defines a boundedly invertible mapping between a bochner space and the dual of another bochner space by equipping these bochner spaces by tensor products of temporal and divergence free spatial wavelets the stokes problem is rewritten as an equivalent well posed bi infinite matrix vector equation this equation can be solved with an adaptive wavelet method in linear complexity with best possible rate that under some mild besov smoothness conditions is nearly independent of the space dimension for proving one of the intermediate results we construct an eigenfunction basis of the stationary stokes operator

the ehrenfeucht conjecture a algebra framework for its proof abstract we describe an algebraic framework including in particular the affine monoid of a given ring within this framework it is a simple matter to deduce ehrenfeucht s conjecture from hilbert s basis theorem

multiple factorizations of words and defect effect we prove that if x is a finite prefix set and w is a non periodic bi infinite word possessing 3 disjoint x factorizations then the combinatorial rank of x is at most card x gamma 2 this is one of the rare cases when a cumulative defect effect is known to hold finally connections to the critical factorization theorem are discussed

on the equivalence problem of compositions of morphisms and inverse morphisms on it is shown that the equivalence problem of h 1h on the family of context free languages is decidable where h is the family of morphisms and h 1 is the family of inverse morphisms

on perturbations of solutions to the navier stokes equations with large initial data and their dynamics abstract we prove a theorem on stability of a strong solution of the navier stokes equation with respect to perturbation of the initial velocity in the norm of d a 1 4 where a is the stokes operator and also with respect to certain perturbations of the acting body force the theorem is applied to obtain new results on the dynamics of solutions of the navier stokes equations

analyticity of the stokes semigroup in spaces of bounded functions the analyticity of the stokes semigroup with the dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the helmholtz decomposition the proof is based on a blow up argument this is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years

on l3 stability of strong solutions of the navier stokes equations with the navier type boundary conditions abstract we show that a strong solution u of the navier stokes initial boundary value problem on the time interval 0 t where 0 t is stable in the l 3 norm provided u l 2 0 t w 2 2 l 0 t w 1 2 the problem is considered with the so called navier type boundary conditions

on the asymptotic limit of the navier stokes system on domains with rough boundaries abstract we study the asymptotic behavior of solutions to the incompressible navier stokes system considered on a sequence of spatial domains whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non degenerate meaning not oriented in a single direction

a note on decidability questions on presentations of word semigroups we apply automata theoretic tools and some recently established compactness properties in the study of f semigroups that is subsemigroups of free semigroups with each f semigroup we associate an f presentation which turns out to be finite for all finitely generated f semigroups connections between f presentations and ordinary presentations of semigroups are also pointed out it is also shown that it is undecidable whether two finitely generated f semigroups satisfy a common relation in their f presentation

a compactness property of the k abelian monoids abstract the k abelian equivalence of words counting the numbers of occurrences of factors of length at most k has been analyzed in recent years from several different directions we continue this analysis the k abelian equivalence classes are known to constitute a monoid hence equations over these monoids are well defined we show that these monoids satisfy a compactness property each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems we give two proofs for this compactness result one is based the fact that the monoid can be embedded into the multiplicative monoid of matrices and the other directly on linear algebra the former method allows the application of hilbert s basis theorem the latter one in turn allows to conclude an upper bound for the size of the finite subsystem

on the analyticity of the semigroup generated by the stokes operator with neumann type boundary conditions on lipschitz subdomains of riemannian manifolds we study the analyticity of the semigroup generated by the stokes operator equipped with neumann type boundary conditions on l p spaces in lipschitz domains our strategy is to regularize this operator by considering the hodge laplacian which has the additional property that it commutes with the leray projection

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

systemes entiers d equations sur un alphabet fini et conjecture d ehrenfeucht the following conjecture has been posed by a ehrenfeucht for each language l over a finite alphabet there exists a finite subset of f of l such that for each pair of morphisms of into a free monoid the equality is true for all in l if and only if it is true for all in f such a finite subset f of l is called a test set of l a system s of equations over the alphabet is called entire if there exists a morphism of into a free monoid such that s 1 we have introduced for each morphism of in a free monoid such that is finite and 1 1 1 a generalized automaton m called the shuffle automaton of which recognizes a rational language k over a finite alphabet and a pair of morphisms of into such that 1 k nivat s 1966 characterization of rational relations is a motivation for this construction and implies that all entire systems over a finite alphabet are rational culik ii and salomaa 1978 have proved that every rational language over a finite alphabet admits a test set culik ii and karhumaki 1983 have shown that if s is a system of equations over a finite alphabet and if there exists a language l over a finite alphabet and morphisms and from into such that s l then if l admits a test set f then t f is a finite equivalent subsystem of s this gives an algorithm which determines for each entire system of equations over a finite alphabet a finite subsystem this result can be improved by using the following theorem spehner 1984 each finitely generated submonoid m of a free monoid has a finite m presentation i e there exists a finite alphabet and a finite part of such that m is isomorphic to the quotient monoid of by the smallest malcev congruence containing is a malcev congruence of if the quotient monoid is embeddable in a group the algorithm for determining the shuffle automaton m and the finite equivalent subsystem of s 1 works with a maximum efficiency because m can be embedded in the minimal automaton of the language k

reducing slip boundary value problems from the half to the whole space applications to inviscid limits and to non newtonian fluids abstract the study of a very large class of linear and non linear stationary and evolutive partial differential problems in the half space or similar under the slip boundary condition is reduced here to the much simpler study of the corresponding results for the same problem in the whole space the approach is particularly suitable for proving new results in strong norms to determine whether this extension is available turns out to be a simple exercise the verification depends on a few general features of the functional space x related to the space variables hence we present an approach as much as possible independent of the particular space x we appeal to a reflection technique hence a crucial assumption is to be in the presence of flat boundaries see below instead of stating general theorems we rather prefer to illustrate how to apply our results by considering a couple of interesting problems as a main example we show that sharp vanishing viscosity limit results that hold for the evolution navier stokes equations in the whole space can be extended to the slip boundary value problem in the half space we also show some applications to non newtonian fluid problems

resolvent estimates in l p for the stokes operator in lipschitz domains we establish the l p resolvent estimates for the stokes operator in lipschitz domains in mathbb r d d geqq 3 for frac 1 p frac 1 2 frac 1 2d varepsilon the result implies that the stokes operator in a three dimensional lipschitz domain generates a bounded analytic semigroup in l p for 3 2 e p 3 e this gives an affirmative answer to a conjecture of m taylor progr nonlinear differential equations appl vol 42 pp 320 334

algebraic properties of word equations abstract the question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words recently aleksi saarela has introduced a new approach to the problem that is based on linear algebraic properties of polynomials encoding the equations and their solutions in this paper we develop further this approach and take into account other algebraic properties of polynomials namely their factorization this in particular allows to improve the bound for the number of independent equations with maximal rank from quadratic to linear

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

on test sets for checking morphism equivalence on languages with fair distribution of letters abstract a test set for a language l is a finite subset t of l with the property that each pair of morphisms that agrees on t also agrees on l some results concerning test sets for languages with fair distribution of letters are presented the first result is that every d0l language with fair distribution of letters has a test set the second result shows that every language l with fair distribution has a test set relative to morphisms g h which have bounded balance on l these results are generalizations of results of culik ii and karhumaki 1983

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

boundary characteristic point regularity for navier stokes equations blow up scaling and petrovskii type criterion a formal approach the three dimensional 3d navier stokes equations less thanbrgreater than less thanbrgreater thanu t u del u del p delta u divu 0 in q 0 0 1 less thanbrgreater than less thanbrgr

lp stability for the strong solutions of the navier stokes equations in the whole space abstract we consider the motion of a viscous fluid filling the whole space r3 governed by the classical navier stokes equations 1 existence of global in time regular solutions for that system of non linear partial differential equations is still an open problem from either the mathematical and the physical point of view an interesting property is the stability or not of the eventual global regular solutions here we assume that v1 t x is a solution with initial data a1 x for small perturbations of a1 we want the solution v1 t x being slightly perturbed too due to viscosity it is even expected that the perturbed solution v2 t x approaches the unperturbed one as time goes to infinity this is just the result proved in this paper to measure the distance between v1 t x and v2 t x at each time t suitable norms are introduced lp norms for fluids filling a bounded vessel exponential decay of the above distance is expected such a strong result is not reasonable for fluids filling the entire space

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

boundary behavior of viscous fluids influence of wall roughness and friction driven boundary conditions we consider a family of solutions to the evolutionary navier stokes system supplemented with the complete slip boundary conditions on domains with rough boundaries we give a complete description of the asymptotic limit by means of convergence arguments and identify a general class of boundary conditions

many aspects of defect theorems we give a survey and a unified presentation of the defect theorem its generalizations and recent aspects of interest in its basic form the defect theorem states that if a set of n words satisfies a nontrivial relation then these words can be expressed simultaneously as products of at most n 1 words in other words dependency of words causes a defect effect there does not exist just one defect theorem but several ones depending on the restrictions that are put to the n 1 words the defect theorem is closely related to equations of words and in this way to the compactness theorem for systems of word equations

equations over finite sets of words and equivalence problems in automata theory abstract several observations concerning equations over monoids of finite sets of words are made monoids considered are the monoid of finite prefix codes the monoid of all finite sets and the monoid of all finite multisets as well as some others as an application some known decidability results in automata theory are extended finally several open questions are raised

viscous boundary layers for the navier stokes equations with the navier slip conditions we tackle the issue of the inviscid limit of the incompressible navier stokes equations when the navier slip with friction conditions are prescribed on impermeable boundaries we justify an asymptotic expansion which involves a weak amplitude boundary layer with the same thickness as in prandtl s theory and a linear behavior this analysis holds for general regular domains in both dimensions two and three

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

test sets for finite substitutions abstract ehrenfeucht s conjecture that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s has recently been verified in this paper we extend this result from endomorphisms to certain types of finite substitutions by applying konig s lemma and embeddings of the monoid into a free algebra

very weak solutions and large uniqueness classes of stationary navier stokes equations in bounded domains of r2 extending the notion of very weak solutions developed recently in the three dimensional case to bounded domains r2 we obtain a new class of unique solutions u in lq q 2 to the stationary navier stokes system u u u p f divu k u g with data f k g of low regularity as a main consequence we obtain a new uniqueness class also for classical weak or strong solutions indeed such a solution is unique if its lq norm is sufficiently small or the data satisfy the uniqueness condition of a very weak solution

lp theory for stokes and navier stokes equations with navier boundary condition abstract this work was intended as an attempt at studying stationary stokes and navier stokes problem with navier boundary conditions 1 3 we wish to investigate some results of existence uniqueness and regularity of solutions in hilbert case and in l p theory

a class of solutions to stationary stokes and navier stokes equations with boundary data in w 1 q q we develop a theory for a general class of very weak solutions to stationary stokes and navier stokes equations in a bounded domain with boundary of class c2 1 corresponding to boundary data in the distribution space w 1 q q 1 q these solutions exist and are unique for small data in the nonlinear case in their class of existence and satisfy a correponding estimate in terms of the data moreover they become regular if the data are regular to our knowledge the only existence result for solutions attaining such boundary data is due to giga 16 proposition 2 2 for the stokes case however the methods and the approach used in the present paper are different than giga s and cover more general issues including the nonlinear navier stokes equations and the precise way in which the boundary data are attained by the solutions we also introduce in the last section a further generalization of the solution class

spectral discretization of the vorticity velocity and pressure formulation of the stokes problem we consider the stokes problem in a square or a cube provided with nonstandard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity we write a variational formulation of this problem with three independent unknowns the vorticity the velocity and the pressure next we propose a discretization by spectral methods which relies on this formulation and since it leads to an inf sup condition on the pressure in a natural way we prove optimal error estimates for the three unknowns we present numerical experiments which are in perfect coherence with the analysis

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

the expressibility of languages and relations by word equations classically several properties and relations of words such as being a power of the same word can be expressed by using word equations this paper is devoted to a general study of the expressive power of word equations as main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations in particular the primitiveness and the equal length are such properties as well as being any word over a proper subalphabet

on systems of word equations with simple loop sets consider the infinite system s of word equations x 0u 1 ix 1u 2 ix 2 u m ix m y 0v 1 iy 1v 2 iy 2 v n iy n email protected n for each email protected n let t k be the subsystem of s given by email protected k k 1 k 2 we prove two properties of the above system 1 let k 1 if f is a solution of t k such that primitive roots of f u 1 f u 2 f u m are of equal length as well as primitive roots of f v 1 f v 2 f v n then f is a solution of the whole s 2 if n 1 then for any k 2 a solution f of t k is also a solution of s

initial and boundary values for lαq lp solution of the navier stokes equations in the half space abstract in this paper we study the initial and boundary value problem of the navier stokes equations in the half space we prove the existence of weak solution u l q 0 l p r n 1 2 1 n p 2 q 0 n p with u l loc q 2 0 l loc p 2 r n for the solenoidal initial data h b p q 1 n p r n and the boundary data g l q 0 b p p 1 p r n 1 when h b p q 1 n p r n g l q 0 b p p 1 p r n 1 is small enough moreover the solution is unique in the class l q 0 t l p r n for any t if 0 and for some t if 0

learning probabilistic automata a study in state distinguishability known algorithms for learning pdfa can only be shown to run in time polynomial in the so called distinguishability m of the target machine besides the number of states and the usual accuracy and confidence parameters we show that the dependence on m is necessary in the worst case for every algorithm whose structure resembles existing ones as a technical tool a new variant of statistical queries termed l queries is defined we show how to simulate l queries using classical statistical queries and show that known pac algorithms for learning pdfa are in fact statistical query algorithms our results include a lower bound every algorithm to learn pdfa with queries using a reasonable tolerance must make w 1 m 1 c queries for every c 0 finally an adaptive algorithm that pac learns w r t another measure of complexity is described this yields better efficiency in many cases while retaining the same inevitable worst case behavior our algorithm requires fewer input parameters than previously existing ones and has a better sample bound

the regularity of the stokes operator and the fujita kato approach to the navier stokes initial value problem in lipschitz domains abstract we study the regularity of the navier stokes equations in arbitrary lipschitz domains

learning decision trees using the fourier spectrum this work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution this algorithm uses membership queries the decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node i e summation of a subset of the input variables over gf 2 this paper shows how to learn in polynomial time any function that can be approximated in norm l 2 by a polynomially sparse function i e a function with only polynomially many nonzero fourier coefficients the authors demonstrate that any function f whose l 1 norm i e the sum of absolute value of the fourier coefficients is polynomial can be approximated by a polynomially sparse function and prove that boolean decision trees with linear operations are a subset of this class of functions moreover it is shown that the functions with polynomial l 1 norm can be learned deterministically the algorithm can also exactly identi

uniform regularity for the navier stokes equation with navier boundary condition we prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the navier stokes equation with the navier boundary condition has a strong solution this solution is uniformly bounded in a conormal sobolev space and has only one normal derivative bounded in l this allows us to obtain the vanishing viscosity limit to the incompressible euler system from a strong compactness argument

multiplicities a deterministic view of nondeterminism abstract we discuss four important problems in automata theory each of which has three natural variants a deterministic one a nondeterministic one and as an in between case a nondeterministic one with multiplicities our goal is to support a general view that the third variant is much closer to the first one than to the second one

local and global cyclicity in free semigroups it is shown that the system of equations x1r xnr s x1s xns r r s n is equivalent to its two element subset x1a xna b x1b xnb a x1a xna c x1c xnc a whenever a b c are integers such that 1 1

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

efficient constructions of test sets for regular and context free languages we present a simple construction of linear size test sets for regular languages and of single exponential test sets for context free languages in the case of regular sets the size of our test set is exactly the number of transitions of the automaton this improves the best known upper bounds exponential for regular and doubly exponential for context free languages we give also an o n log n time algorithm for the morphism equivalence and an o n3log n time algorithm to test the gsm equivalence on a regular language an o n2log n time algorithm is given to test the equivalence of two deterministic gsm s as well as that of two deterministic finite transducers

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

relevance of the slip condition for fluid flows near an irregular boundary we consider the navier stokes equation in a domain with a rough boundary the roughness is modeled by a small amplitude and small wavelength oscillation with typical scale varepsilon ll 1 for periodic oscillation it is well known that the best homogenized that is regular in varepsilon boundary condition is of navier type such result still holds for random stationary irregularities as shown recently by the first author 5 15 we study here arbitrary irregularity patterns

from the boltzmann equation to the stokes fourier system in a bounded domain we prove that the renormalized solutions of the boltzmann equation considered in a bounded domain with different types of kinetic boundary conditions converge to the stokes fourier system with different types of fluid boundary conditions when the main free path goes to zero this extends the work of f golse and d levermore 9 to the case of a bounded domain 2003 wiley periodicals inc

on the system of word equations x o u i i x 1 u i 2 x 2 u i 3 x 3 y 0 v i 1 y 1 v i 2 y 2 v i 3 y 3 i 0 1 2 l in a free monoid abstract we prove that if x 0 u 1 i x 1 u 2 i x 2 y 0 v 1 i y 1 v 2 i y 2 holds for all i 0 1 2 3 then it is true for each i 0 1 2 and that if x 0 u 1 i x 1 u 2 i x 2 u 3 i x 3 y 0 v 1 i y 1 v 2 i y 2 v 3 i y 3 holds for all i 0 1 2 3 4 5 then it is true for each i 0 1 2

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

on three element codes abstract we show that three element codes have some special properties which do not hold even for four element codes firstly for each three element code a if u and v are words in pref xa pref ya with x y a x y then one of them is a prefix of the other i e among the words which can be covered in two different ways from left to right there exists a unique maximal possibly infinite element secondly each three element code has a bounded delay in at least one direction

on the equivalence problem for binary dol systems it is shown that to test whether two dol sequences in the binary case coincide it is enough to test whether four first words of these sequences are the same the result is optimal

exact controllability for the magnetohydrodynamic equations we study the local exact controllability of the steady state solutions of the magnetohydrodynamic equations the main result of the paper asserts that the steady state solutions of these equations are locally controllable if they are smooth enough we reduce the local exact controllability of the steady state solutions of the magnetohydrodynamic equations to the global exact controllability of the null solution of the linearized magnetohydrodynamic system via a x ed point argument the treatment of the reduced problem relies on two carleman type inequalities for the backward adjoint system c 2003 wiley periodicals inc

space time regularity of the koch tataru solutions to navier stokes equations abstract in koch tataru 2001 the authors have proved local and global well posedness of the navier stokes equations with small initial data u 0 b m o 1 and then the spatial analyticity of the koch tataru solution stated as t k 2 k u x t for any positive integer k has been presented by germain pavlovic staffilani 2007 where x t is the space defined by koch tataru 2001 see also definition 1 2 in this paper we shall present the space time regularity more precisely for any positive integer m k we have t m k 2 t m k u x t

on regularity of a weak solution to the navier stokes equation with generalized impermeability boundary conditions abstract we consider the 3d navier stokes equation with generalized impermeability boundary conditions as auxiliary results we prove the local in time existence of a strong solution strong in a limited sense and a theorem on structure then taking advantage of the boundary conditions we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor finally we apply these general results to the case of an axially symmetric flow with zero angular velocity

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

well posedness for the navier stokes nernst planck poisson system in triebel lizorkin space and besov space with negative indices abstract this paper is concerned with the well posedness of the navier stokes nerst planck poisson system nsnpp let s p 2 n p we prove that the nsnpp has a unique local solution u v w e u t e v t e v t for u 0 v 0 w 0 in a subspace i e v u 1 v v 1 v v 1 of f 1 2 b p s p b p s p with u 0 0 we also prove that there exists a unique small global solution u v w e u e v e v for any small initial data u 0 v 0 w 0 f 1 2 b p s p b p s p with u 0 0

pac learnability of probabilistic deterministic finite state automata we study the learnability of probabilistic deterministic finite state automata under a modified pac learning criterion we argue that it is necessary to add additional parameters to the sample complexity polynomial namely a bound on the expected length of strings generated from any state and a bound on the distinguishability between states with this we demonstrate that the class of pdfas is pac learnable using a variant of a standard state merging algorithm and the kullback leibler divergence as error function