multidimensional unrepetitive configurations note abstract we show that one can associate a letter of a finite alphabet with each lattice point of the n dimensional space so that an r th power free word corresponds to each straight line of the space this result is obtained as a consequence of the existence of bi infinite words on finite alphabets which are r th power free together with a certain class of their subwords an analogous property is obtained for some avoidable sets of patterns

growth rate of binary words avoiding xxxr consider the set of those binary words with no non empty factors of the form x x x r du mousavi schaeffer and shallit asked whether this set of words grows polynomially or exponentially with length in this paper we demonstrate the existence of upper and lower bounds of the form n lg n o lg n on the number of such words of length n where lg n denotes the base 2 logarithm of n

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

on the h s theory of hydrostatic euler equations in this paper we study the two dimensional hydrostatic euler equations in a periodic channel we prove the local existence and uniqueness of hs solutions under the local rayleigh condition this extends brenier s nonlinearity 12 3 495 512 1999 existence result by removing an artificial condition and proving uniqueness in addition we prove weak strong uniqueness mathematical justification of the formal derivation and stability of the hydrostatic euler equations these results are based on weighted hs a priori estimates which come from a new type of nonlinear cancellation between velocity and vorticity

relative entropies for kinetic equations in bounded domains irreversibility stationary solutions uniqueness the relative entropy method describes the irreversibility of the vlasov poisson and vlasov boltzmann poisson systems in bounded domains with incoming boundary conditions uniform in time estimates are deduced from the entropy in some cases these estimates are sufficient to prove the convergence of the solution to a unique stationary solution as time goes to infinity the method is also used to analyse other types of boundary conditions such as mass and energy preserving diffuse reflection boundary conditions and to prove the uniqueness of stationary solutions for some special collision terms

exponential lower bounds for the number of words of uniform length avoiding a pattern we study words on a finite alphabet avoiding a finite collection of patterns given a pattern p in which every letter that occurs in p occurs at least twice we show that the number of words of length n on a finite alphabet that avoid p grows exponentially with n as long as the alphabet has at least four letters moreover we give lower bounds describing this exponential growth in terms of the size of the alphabet and the number of letters occurring in p we also obtain analogous results for the number of words avoiding a finite collection of patterns we conclude by giving some questions

every real number greater than 1 is a critical exponent we prove that for every real number a 1 there exists an infinite word w over a finite alphabet such that a is the critical exponent of w

local in time existence and uniqueness of solutions to the prandtl equations by energy methods we prove local existence and uniqueness for the two dimensional prandtl system in weighted sobolev spaces under oleinik s monotonicity assumption in particular we do not use the crocco transform or any change of variables our proof is based on a new nonlinear energy estimate for the prandtl system this new energy estimate is based on a cancellation property that is valid under the monotonicity assumption to construct the solution we use a regularization of the system that preserves this nonlinear structure this new nonlinear structure may give some insight into the convergence properties from the navier stokes system to the euler system when the viscosity goes to 0 2015 wiley periodicals inc

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

ill posedness of the hydrostatic euler and singular vlasov equations in this paper we develop an abstract framework to establish ill posedness in the sense of hadamard for some nonlocal pdes displaying unbounded unstable spectra we apply this to prove the ill posedness for the hydrostatic euler equations as well as for the kinetic incompressible euler equations and the vlasov dirac benney system

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

on landau damping going beyond the linearized study has been a longstanding problem in the theory of landau damping in this paper we establish exponential landau damping in analytic regularity the damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables rather than exchanges of energy phase mixing is the driving mechanism the analysis involves new families of analytic norms measuring regularity by comparison with solutions of the free transport equation new functional inequalities a control of non linear echoes sharp deflection estimates and a newton approximation scheme our results hold for any potential no more singular than coulomb or newton interaction the limit cases are included with specific technical effort as a side result the stability of homogeneous equilibria of the non linear vlasov equation is established under sharp assumptions we point out the strong analogy with the kam theory and discuss physical implications finally we extend these results to some gevrey non analytic distribution functions

asymptotic preserving particle in cell method for the vlasov poisson system near quasineutrality this paper deals with the numerical resolution of the vlasov poisson system in a nearly quasineutral regime by particle in cell pic methods in this regime classical pic methods are subject to stability constraints on the time and space steps related to the small debye length and large plasma frequency here we propose an asymptotic preserving pic scheme which is not subjected to these limitations additionally when the plasma period and debye length are small compared to the time and space steps this method provides a consistent pic discretization of the quasineutral vlasov equation we perform several one dimensional numerical experiments which provide a solid validation of the method and its underlying concepts and compare the method with classical pic and direct implicit methods

pattern avoidance by palindromes we show that every avoidable pattern can be avoided by an infinite sequence of palindromes over a fixed finite alphabet

a note on prandtl boundary layers this note concerns nonlinear ill posedness of the prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary our analysis is built upon recent remarkable linear illposedness results established by gerard varet and dormy and an analysis by guo and tice we show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in sobolev spaces we also introduce a notion of weak well posedness and prove that the nonlinear prandtl equation is not well posed in this sense near nonstationary and nonmonotonic shear flows on the other hand we are able to verify that oleinik s monotonic solutions are well posed 2011 wiley periodicals inc

spectral theory of the linearized vlasov poisson equation etude de la theorie spectrale de l equation de vlasov poisson linearisee dont la solution se comporte comme une somme d ondes planes pour de grandes valeurs du temps prolongement analytique de la resolvante de l equation

perturbed kernel approximation on homogeneous manifolds current methods for interpolation and approximation within a native space rely heavily on the strict positive definiteness of the underlying kernels if the domains of approximation are the unit spheres in euclidean spaces then zonal kernels kernels that are invariant under the orthogonal group action are strongly favoured in the implementation of these methods to handle real world problems however some or all of the symmetries and positive definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution perturbation analysis is therefore needed to address the stability problem encountered in this paper we study two kinds of perturbations of positive definite kernels small random perturbations and perturbations by dunkl s intertwining operators c dunkl y xu orthogonal polynomials of several variables encyclopedia of mathematics and its applications vol 81 cambridge university press cambridge 2001 we show that with some reasonable assumptions a small random perturbation of a strictly positive definite kernel can still provide vehicles for interpolation and enjoy the same error estimates we examine the actions of the dunkl intertwining operators on zonal strictly positive definite kernels on spheres we show that the resulted kernels are strictly positive definite on spheres of lower dimensions

stability issues in the quasineutral limit of the one dimensional vlasov poisson equation this work is concerned with the quasineutral limit of the one dimensional vlasov poisson equation for initial data close to stationary homogeneous profiles our objective is threefold first we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the penrose instability criterion second we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition together with a symmetry condition we handle the case of well prepared as well as ill prepared data last we study a stationary boundary value problem for the formal limit the so called quasineutral vlasov equation we show the existence of numerous stationary states with a lot of freedom in the construction compared to that of bgk waves for vlasov poisson this illustrates the degeneracy of the limit equation

the number of binary words avoiding abelian fourth powers grows exponentially we show that the number of binary words of length n avoiding abelian fourth powers grows faster than rn where r 21 16

relatively free semigroups of intermediate growth abstract the problem of calculating the growth of a finitely generated f g semigroup satisfying the given system of identities is considered examples of relatively free semigroups having intermediate growth new growth criteria and constructions are given the main ideas employed in the proofs are based on some new points of view on growth our method is also useful for constructing various examples of f g nilpotent and nil semigroups having intermediate growth and arbitrarily small gelfand kirillov superdimension

new constructive aspects of the lovasz local lemma the lovasz local lemma lll is a powerful tool that gives sufficient conditions for avoiding all of a given set of bad events with positive probability a series of results have provided algorithms to efficiently construct structures whose existence is non constructively guaranteed by the lll culminating in the recent breakthrough of moser and tardos 2010 for the full asymmetric lll we show that the output distribution of the moser tardos algorithm well approximates the conditional lll distribution the distribution obtained by conditioning on all bad events being avoided we show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results we also show that when a lll application provides a small amount of slack the number of resamplings of the moser tardos algorithm is nearly linear in the number of underlying independent variables not events and can thus be used to give efficient constructions in cases where the underlying proof applies the lll to super polynomially many events even in cases where finding a bad event that holds is computationally hard we show that applying the algorithm to avoid a polynomial sized core subset of bad events leads to a desired outcome with high probability this is shown via a simple union bound over the probabilities of non core events in the conditional lll distribution and automatically leads to simple and efficient monte carlo and in most cases rnc algorithms we demonstrate this idea on several applications we give the first constant factor approximation algorithm for the santa claus problem by making a lll based proof of feige constructive we provide monte carlo algorithms for acyclic edge coloring nonrepetitive graph colorings and ramsey type graphs in all these applications the algorithm falls directly out of the non constructive lll based proof our algorithms are very simple often provide better bounds than previous algorithms and are in several cases the first efficient algorithms known as a second type of application we show that the properties of the conditional lll distribution can be used in cases beyond the critical dependency threshold of the lll avoiding all bad events is impossible in these cases as the first even nonconstructive result of this kind we show that by sampling a selected smaller core from the lll distribution we can avoid a fraction of bad events that is higher than the expectation max k sat is an illustrative example of this

square free shuffles of words let u v denote the set of all shuffles of the words u and v it is shown that for each integer n 3 there exists a square free ternary word u of length n such that u u contains a square free word this property is then shown to also hold for infinite words i e there exists an infinite square free word u on three letters such that u can be shuffled with itself to produce an infinite square free word w u u

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

growth problems for avoidable words abstract a word w is said to avoid a word u if no block subword factor of w is the image of u under a homomorphism of free semigroups without unit the theory of words avoiding xx square free words has been much studied the word u is said to be avoidable on n letters if there are arbitrarily long words on an n letter alphabet that avoid u if u is avoidable on n letters for some n let u be the minimum possible such n we show that u has a linear bound in terms of the alphabet size of u we further show that there exists a word that is avoidable on four letters but not on three letters moreover if u is this word the number of words of length l on a u letter alphabet that avoid u has a polynomial bound in terms of l so that the question of the existence of such an example is resolved in the affirmative in contrast for xx the bound is known to be exponential

overlap free words and finite automata abstract a method to represent certain words on a binary alphabet by shorter words on a larger alphabet is introduced we prove that overlap free words are represented by the words of a rational language several consequences are derived concerning the density function of the set of overlap free words on a binary alphabet and the prolongability of overlap free words in particular efficient algorithms are obtained computing the density function of the set of overlap free words on a binary alphabet testing whether a word on a binary alphabet is overlap free and computing its depth

some combinatorial properties of the thue morse sequence and a problem in semigroups abstract some combinatorial properties of the special factors of the thue morse sequence in a two letter alphabet are considered an application of this analysis to a problem in semigroups is given

quasineutral limit for vlasov poisson via wasserstein stability estimates in higher dimension abstract this work is concerned with the quasineutral limit of the vlasov poisson system in two and three dimensions we justify the formal limit for very small but rough perturbations of analytic initial data generalizing the results of 12 to higher dimension

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

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

on the subword complexity of square free dol languages abstract the subword complexity of a language k is the function which to every positive integer n assigns the number of different subwords of length n occuring in words of k a language k is square free if no word in it contains a subword of the form xx where x is a nonempty word the best upper and lower bounds on the subword complexity of infinite square free dol languages are established

problems in between words and abelian words k abelian avoidability we consider a recently defined notion of k abelian equivalence of words in connection with avoidability problems this equivalence relation for a fixed natural number k takes into account the numbers of occurrences of the different factors of length k and the prefix and the suffix of length k 1 we search for the smallest alphabet in which k abelian squares and cubes can be avoided respectively for 2 abelian squares this is four as in the case of abelian words while for 2 abelian cubes we have only strong evidence that the size is two as it is in the case of words however we are able to prove this optimal value only for 8 abelian cubes

improved error bounds for scattered data interpolation by radial basis functions if additional smoothness requirements and boundary conditions are met the well known approximation orders of scattered data interpolants by radial functions can roughly be doubled

on the ill posedness of the prandtl equation the concern of this paper is the cauchy problem for the prandtl equation this problem is known to be well posed for analytic data or for data with monotonicity properties we prove here that it is linearly ill posed in sobolev type spaces the key of the analysis is the construction at high tangential frequencies of unstable quasimodes for the linearization around solutions with non degenerate critical points interestingly the strong instability is due to vicosity which is coherent with well posedness results obtained for the inviscid version of the equation a numerical study of this instability is also provided

large time behavior of solutions to vlasov poisson fokker planck equations from evanescent collisions to diffusive limit the present contribution investigates the dynamics generated by the two dimensional vlasov poisson fokker planck equation for charged particles in a steady inhomogeneous background of opposite charges we provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted l 2 space and where dependencies on the mean free path tau and the debye length delta are made explicit in our analysis the mean free path covers the full range of possible values from the regime of evanescent collisions tau to infty to the strongly collisional regime tau to0 as a counterpart the largeness of the debye length that enforces a weakly nonlinear regime is used to close our nonlinear estimates accordingly we pay a special attention to relax as much as possible the tau dependent constraint on delta ensuring exponential decay with explicit tau dependent rates towards the stationary solution in the strongly collisional limit tau to0 we also examine all possible asymptotic regimes selected by a choice of observation time scale here also our emphasis is on strong convergence uniformity with respect to time and to initial data in bounded sets of a l 2 space our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive hypoelliptic estimates

strictly positive definite functions on spheres in euclidean spaces in this paper we study strictly positive definite functions on the unit sphere of the m dimensional euclidean space such functions can be used for solving a scattered data interpolation problem on spheres since positive definite functions on the sphere were already characterized by schoenberg some fifty years ago the issue here is to determine what kind of positive definite functions are actually strictly positive definite the study of this problem was initiated recently by xu and cheney proc amer math soc 116 1992 977 981 where certain sufficient conditions were derived a new approach which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres is presented here the relevant interpolation problem is subsequently analyzed by three different complementary methods the first is based on the de boor ron general least solution for the multivariate polynomial interpolation problem the second which is suitable only for m 2 is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials and reduces the problem to the structure of the integer zeros of bounded univariate exponentials finally the last method invokes the realization of harmonic polynomials as the polynomial kernel of the laplacian thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels

on abelian squares and substitutions we study substitutions preserving abelian square free words and the more general notion of substitutions with bounded abelian squares in particular we prove the existence of algorithms deciding whether a substitution mapping each letter into a set of commutatively equivalent words belongs to one of these classes

spherical spline interpolation basic theory and computational aspects abstract the purpose of the paper is to adapt to the spherical case the basic theory and the computational method known from surface spline interpolation in euclidean spaces spline functions are defined on the sphere the solution process is made simple and efficient for numerical computation in addition the convergence of the solution obtained by spherical spline interpolation is developed using estimates for legendre polynomials

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

on repetition free binary words of minimal density we study the minimal proportion density of one letter in n th power free binary words first we introduce and analyse a general notion of minimal letter density for any infinite set of words which don t contain a specified set of prohibited subwords we then prove that for n th power free binary words the density function is frac 1 n frac 1 n 3 frac 1 n 4 cal o frac 1 n 5 we also consider a generalization of n th power free words for fractional powers exponents a word is x th power free for a real x if it does not contain subwords of exponent x or more we study the minimal proportion of one letter in x th power free binary words as a function of x and prove in particular that this function is discontinuous at frac 7 3 as well as at all integer points n geq 3 finally we give an estimate of the size of the jumps

on the waterbag continuum the aim of this paper is to study the existence of a classical solution for the waterbag model with a continuum of waterbags which can been viewed as an infinite dimensional system of first order conservation laws the waterbag model which constitutes a special class of exact weak solution of the vlasov equation is well known in plasma physics and its applications in gyrokinetic theory and laser plasma interaction are very promising the proof of the existence of a continuum of regular waterbags relies on a generalized definition of hyperbolicity for an integrodifferential hyperbolic system of equations some results in singular integral operators theory and harmonic analysis riemann hilbert boundary value problems and energy estimates