fourier transform inequalities with measure weights abstract fourier transform norm inequalities f q p are proved for measure weights on moment subspaces of lp r n density theorems are established to extend the inequalities to all of lp r n in both cases the conditions for validity are computable for n 2 and are radial and the results are applied to prove spherical restriction theorems which include power weights t t n p 1 p n p 1

sharp ill posedness and well posedness results for the kdv burgers equation the periodic case we prove that the kdv burgers is globally well posed in h 1 t with a solution map that is analytic from h 1 t to c 0 t h 1 t whereas it is ill posed in h s t as soon as s 0 small enough in view of the result of kappeler and topalov for kdv it thus appears that even if the dissipation part of the kdv burgers equation allows to lower the c infty critical index with respect to the kdv equation it does not permit to improve the c 0 critical index

stability of two soliton collision for nonintegrable gkdv equations we continue our study of the collision of two solitons for the subcritical generalized kdv equations label kdvabs partial t u partial x partial x 2 u f u 0 quad quad 0 1 solitons are solutions of the type u t x q c 0 x x 0 c 0 t where c0 0 in 21 mainly devoted to the case f u u4 we have introduced a new framework to understand the collision of two solitons q c 1 q c 2 for 0 1 in the case c 2 ll c 1 or equivalently q c 2 h 1 ll q c 1 h 1 in this paper we consider the case of a general nonlinearity f u for which q c 1 q c 2 are nonlinearly stable in particular since f is general and c1 can be large the results are not perturbations of the ones for the power case in 21

global stabilization of the generalized korteweg de vries equation posed on a finite domain this paper is concerned with the internal stabilization of the generalized korteweg de vries equation on a bounded domain the global well posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval 1 4 the global exponential stability is obtained whatever the location where the damping is active confirming positively a conjecture of perla menzala vasconcellos and zuazua quart appl math 60 2002 pp 111 129

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

tests for no treatment effect in randomized clinical trials summary we propose a test of the null hypothesis of no treatment effect in a randomized clinical trial that is based on the randomization distribution of residuals these residuals result from regressing the response on covariates but not treatment in contrast to model based score tests this procedure maintains nominal size when the model is misspecified and in particular when relevant covariates are omitted from the regression the efficiency of the procedure is evaluated for regressions with some but not all required covariates for many generalized linear models and survival models conventional model based score tests are shown to have supranominal size when relevant covariates are omitted but logistic regression and the proportional hazards model are robust

right l groups geometric garside groups and solutions of the quantum yang baxter equation abstract right lattice ordered groups are introduced and studied as a general framework for garside groups and related groups without a garside element every right l group g has a two sided partially ordered subgroup n g which generalizes the quasi centre of an artin tits group the group n g splits into copies of z if g is noetherian the positive cone of a right l group is described as a structure that is known from algebraic logic a pair of left and right self similar hoops for noetherian right l groups g modularity of the lattice structure is characterized in terms of an operation on the set x g of atoms it is proved that modular garside groups are equivalent to finite projective spaces with a non degenerate labelling a concept of duality for x g is introduced and applied in the distributive case this gives a one to one correspondence between noetherian right l groups with duality and non degenerate unitary set theoretic solutions of the quantum yang baxter equation the description of garside groups via garside germs is extended to right l groups which yields a one sided enhancement and a new proof of dvurecenskij s non commutative extension of mundici s correspondence between abelian l groups and mv algebras

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

garside families and garside germs garside families have recently emerged as a relevant context for extending results involving garside monoids and groups which themselves extend the classical theory of generalized braid groups here we establish various characterizations of garside families that is equivalently various criteria for establishing the existence of normal decompositions of a certain type

artin groups of euclidean type this article resolves several long standing conjectures about artin groups of euclidean type specifically we prove that every irreducible euclidean artin group is a torsion free centerless group with a decidable word problem and a finite dimensional classifying space we do this by showing that each of these groups is isomorphic to a subgroup of a group with an infinite type garside structure the garside groups involved are introduced here for the first time they are constructed by applying semi standard procedures to crystallographic groups that contain euclidean coxeter groups but which need not be generated by the reflections they contain

description of two soliton collision for the quartic gkdv equation in this paper we give the rst description of the collision of two solitons for a nonintegrable equation in a special regime we consider solutions of the quartic gkdv equation tu x 2 xu u 4 0 which behave as t 1 like u t x qc1 x c1t qc2 x c2t t x where qc x ct is a soliton andk t kh1k qc2kh1k qc1kh1 the global behavior of u t is given by the following stability result for all t 2 r u t x qc1 t x y1 t qc2 t x y2 t t x where k t kh1k qc2kh1 and limt 1c1 t c limt 1c2 t c in the case where u t is a pure 2 soliton solution as t 1 i e limt 1 k t kh1 0 we obtain c c1 c 0 therefore in contrast with the integrable kdv equation or mkdv equation no global pure 2 soliton solution exists and the collision is inelastic a dierent notion of global 2 soliton is then proposed

infinite dimensional symplectic capacities and a squeezing theorem for hamiltonian pde s we study partial differential equations of hamiltonian form and treat them as infinite dimensional hamiltonian systems in a functional phase space ofx dependent functions in this phase space we construct an invariant symplectic capacity and prove a version of gromov s non squeezing theorem we give an interpretation of the theorem in terms of the energy transition to high frequencies problem

the cauchy problem for the schrodinger kdv system abstract in this paper we prove that in the general case i e not necessarily vanishing the cauchy problem for the schrodinger korteweg de vries system is locally well posed in l 2 h 3 4 and if 0 then it is locally well posed in h s h 3 4 with 3 16 s 1 4 these results improve the corresponding results of corcho and linares 2007 5 idea of the proof is to establish some bilinear and trilinear estimates in the space g s f s where g s and f s are dyadic bourgain type spaces related to the schrodinger operator i t x 2 and the airy operator t x 3 respectively but with a modification on f s in low frequency part of functions with a weaker structure related to the maximal function estimate of the airy operator

autonomous ovsyannikov theorem and applications to nonlocal evolution equations and systems abstract this work presents an ovsyannikov type theorem for an autonomous abstract cauchy problem in a scale of decreasing banach spaces which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution then using this theorem it studies the cauchy problem for camassa holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line which is the main goal of this paper finally it studies the continuity of the data to solution map in spaces of analytic functions

decomposition of garside groups and self similar l algebras abstract picantin s iterated crossed product representation of garside monoids is extended and reproved for a wide class of not necessarily noetherian partially ordered groups with a right invariant lattice structure it is shown that the tree like structure of such an iterated crossed product is equivalent to a partial cycle set closely related to a class of set theoretic solutions of the quantum yang baxter equation the decomposition of finite square free solutions is related to the crossed product representation of the corresponding structure group

a note on words in braid monoids abstract the purpose of this note is to give a self contained apart from simple facts about coxeter groups and we hope a bit shorter and more understandable account of some results of c1 c2 on normal forms of braids which are themselves based on the papers d1 t in particular a motivation was to give a proof of proposition 5 1 that we use in b m some proofs and results from section 2 onwards seem to be new i thank several people for improvements from earlier versions of the manuscript m geck for pointing out some errors f digne for pointing out that some results don t need the braid group to be of finite type and j y hee for suggesting and providing further improvements in that direction

scattering for the quartic generalised korteweg de vries equation abstract we show that the quartic generalised kdv equation u t u x x x u 4 x 0 is globally well posed for data in the critical scale invariant space h x 1 6 r with small norm and locally well posed for large norm improving a result of grunrock a grunrock a bilinear airy estimate with application to gkdv 3 differential integral equations 18 12 2005 1333 1339 as an application we obtain scattering results in h x 1 r h x 1 6 r for the radiation component of a perturbed soliton for this equation improving the asymptotic stability results of martel and merle y martel f merle asymptotic stability of solitons for subcritical generalized kdv equations arch ration mech anal 157 3 2001 219 254

on schrodinger maps we study the local well posedness theory for the schr odinger maps equation we work in n 1 dimensions for n geq 2 and prove a local well posedness for small initial data in h frac n 2 varepsilon

low regularity schrodinger maps ii global well posedness in dimensions d 3 in dimensions d 3 we prove that the schrodinger map initial value problem left begin array l partial ts s times delta x s hbox on mathbb r d times mathbb r s 0 s 0 end array right is globally well posed for small data s0 in the critical besov spaces dot b q d 2 mathbb r d mathbb s 2 q in mathbb s 2

frequency uniform decomposition method for the generalized bo kdv and nls equations abstract sufficient and necessary conditions for the embeddings between besov spaces b p q s 1 and modulation spaces m p q s 2 are obtained moreover using the frequency uniform decomposition method we study the cauchy problem for the generalized bo kdv and nls equations for which the global well posedness of solutions with the small rough data in certain modulation spaces m 2 1 s is shown

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

small data well posedness for derivative nonlinear schrodinger equations abstract we study the local and global solutions of the generalized derivative nonlinear schrodinger equation i t u u p u u x u x u where each monomial in p is of degree 3 or higher in low regularity sobolev spaces without using a gauge transformation instead we use a solution decomposition technique introduced in 4 during the perturbative argument to deal with the loss on derivative in nonlinearity it turns out that when each term in p contains only one derivative the equation is locally well posed in h 1 2 otherwise we have a local well posedness in h 3 2 if each monomial in p is of degree 5 or higher the solution can be extended globally by restricting to equations to the form i t u u x p u u with the quintic nonlinearity we were able to obtain the global well posedness in the critical sobolev space

global well posedness in l 2 for the periodic benjamin ono equation we prove that the benjamin ono equation is globally well posed in h s bbb t for s ge 0 moreover we show that the associated flow map is lipschitz on every bounded set of h s 0 bbb t s ge 0 and even real analytic in this space for small times this result is sharp in the sense that the flow map if it can be defined and coincides with the standard flow map on h 0 infty bbb t cannot be of class c 1 alpha alpha 0 from h 0 s bbb t into h 0 s bbb t as soon as s 0

asymptotic stability of solitons for subcritical generalized kdv equations we prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized korteweg de vries equations in the subcritical case this includes in particular the kdv equation and the modified kdv equation this result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation which has its own interest the proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized kdv equation

rough solutions of the fifth order kdv equations abstract we consider the cauchy problem of the fifth order equation arising from the korteweg de vries kdv hierarchy t u x 5 u c 1 x u x 2 u c 2 u x 3 u 0 x t r u 0 x u 0 x u 0 h s r we prove a priori bound of solutions for h s r with s 5 4 and the local well posedness for s 2 the method is a short time x s b space which was first developed by ionescu kenig and tataru 13 in the context of the kp i equation in addition we use a weight on localized x s b structures to reduce the contribution of high low frequency interaction where the low frequency has large modulation as an immediate result from a conservation law we obtain that the fifth order equation in the kdv hierarchy t u x 5 u 30 u 2 x u 20 x u x 2 u 10 u x 3 u 0 is globally well posed in the energy space h 2

global well posedness for schrodinger equation with derivative in h 1 2 r abstract in this paper we consider the cauchy problem of the cubic nonlinear schrodinger equation with derivative in h s r this equation was known to be the local well posedness for s 1 2 takaoka 1999 27 ill posedness for s 1 2 biagioni and linares 2001 1 etc and global well posedness for s 1 2 i team 2002 10 in this paper we show that it is global well posedness in the endpoint space h 1 2 r which remained open previously the main approach is the third generation i method combined with a new resonant decomposition technique the resonant decomposition is applied to control the singularity coming from the resonant interaction

pros and cons of permutation tests in clinical trials hypothesis testing in which the null hypothesis specifies no difference between treatment groups is an important tool in the assessment of new medical interventions for randomized clinical trials permutation tests that reflect the actual randomization are design based analyses for such hypotheses this means that only such design based permutation tests can ensure internal validity without which external validity is irrelevant however because of the conservatism of permutation tests the virtues of permutation tests continue to be debated in the literature and conclusions are generally of the type that permutation tests should always be used or permutation tests should never be used a better conclusion might be that there are situations in which permutation tests should be used and other situations in which permutation tests should not be used this approach opens the door to broader agreement but begs the obvious question of when to use permutation tests we consider this issue from a variety of perspectives and conclude that permutation tests are ideal to study efficacy in a randomized clinical trial which compares in a heterogeneous patient population two or more treatments each of which may be most effective in some patients when the primary analysis does not adjust for covariates we propose the p value interval as a novel measure of the conservatism of a permutation test that can be defined independently of the significance level this p value interval can be used to ensure that the permutation test have both good global power and an acceptable degree of conservatism copyright 2000 john wiley sons ltd

local well posedness for dispersion generalized benjamin ono equations in sobolev spaces abstract we prove that the cauchy problem for the dispersion generalized benjamin ono equation t u x 1 x u u u x 0 u x 0 u 0 x is locally well posed in the sobolev spaces h s for s 1 if 0 1 the new ingredient is that we generalize the methods of ionescu kenig and tataru 2008 13 to approach the problem in a less perturbative way in spite of the ill posedness results of molinet saut and tzvetkov 2001 21 moreover as a bi product we prove that if 0 1 the corresponding modified equation with the nonlinearity u u u x is locally well posed in h s for s 1 2 4

proof of the k pi 1 conjecture for affine artin groups we prove the k pi 1 conjecture for affine artin groups the complexified complement of an affine reflection arrangement is a classifying space this is a long standing problem due to arnol d pham and thom our proof is based on recent advancements in the theory of dual coxeter and artin groups as well as on several new results and constructions in particular we show that all affine noncrossing partition posets are el shellable we use these posets to construct finite classifying spaces for dual affine artin groups we introduce new cw models for the orbit configuration spaces associated with arbitrary coxeter groups we construct finite classifying spaces for the braided crystallographic groups introduced by mccammond and sulway

global well posedness and inviscid limit for the korteweg de vries burgers equation abstract considering the cauchy problem for the korteweg de vries burgers equation u t u x x x x 2 u u 2 x 0 u 0 where 0 1 and u is a real valued function we show that it is globally well posed in h s s s and uniformly globally well posed in h s s 3 4 for all 0 1 moreover we prove that for any t 0 its solution converges in c 0 t h s to that of the kdv equation if tends to 0

conditional monte carlo randomization tests for regression models we discuss the computation of randomization tests for clinical trials of two treatments when the primary outcome is based on a regression model we begin by revisiting the seminal paper of gail tan and piantadosi 1988 and then describe a method based on monte carlo generation of randomization sequences the tests based on this monte carlo procedure are design based in that they incorporate the particular randomization procedure used we discuss permuted block designs complete randomization and biased coin designs we also use a new technique by plamadeala and rosenberger 2012 for simple computation of conditional randomization tests like gail tan and piantadosi we focus on residuals from generalized linear models and martingale residuals from survival models such techniques do not apply to longitudinal data analysis and we introduce a method for computation of randomization tests based on the predicted rate of change from a generalized linear mixed model when outcomes are longitudinal we show by simulation that these randomization tests preserve the size and power well under model misspecification

a higher dispersion kdv equation on the line abstract the cauchy problem for a korteweg devries equation with dispersion of order m 2 j 1 where j is a positive integer kdvm is studied with data in sobolev and analytic spaces first optimal bilinear estimates in bourgain spaces are proved and using them well posedness in sobolev spaces h s s j 1 4 is established then well posedness in analytic gevrey spaces g s 0 is proved by using an analytic version of the bilinear estimates this implies that the uniform radius of analyticity persist for some time for the later times a lower bound for the radius of spacial analyticity is derived which is given by t c t with 4 3 e for any e 0 when j 1 and 1 when j 2 finally it is shown that the regularity of the solution in the time variable is gevrey of order m and this is optimal

asymptotic stability of solitons for mkdv abstract we prove a full asymptotic stability result for solitary wave solutions of the mkdv equation we consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the cauchy problem evolving from such data tend uniformly on the real line to another solitary wave as time goes to infinity we describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior this latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mkdv equation

continuity of the flow of kdv with regard to the wasserstein metrics and application to an invariant measure abstract in this paper we prove the continuity of the flow of kdv on spaces of probability measures with respect to a combination of wasserstein distances on h s s 0 and l 2 we are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined

well posedness for the cauchy problem associated to the hirota satsuma equation periodic case abstract we consider a system of korteweg de vries kdv equations coupled through nonlinear terms called the hirota satsuma system we study the initial value problem ivp associated to this system in the periodic case for given data in sobolev spaces h s h s 1 with regularity below the one given by the conservation laws using the fourier transform restriction norm method we prove local well posedness whenever s 1 2 also with some restriction on the parameters of the system we use the recent technique introduced by colliander et al called i method and almost conserved quantities to prove global well posedness for s 3 14

description of the inelastic collision of two solitary waves for the bbm equation we prove that the collision of two solitary waves of the bbm equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space we show precise estimates of the nonzero residue due to the collision moreover we give a precise description of the collision phenomenon change of size of the solitary waves and shifts in their trajectories to prove these results we extend the method introduced in martel and merle description of two soliton collision for the quartic gkdv equation submitted preprint http arxiv org abs 0709 2672 commun math phys 286 39 79 2009 for the generalized kdv equation in particular in the quartic case the main argument is the construction of an explicit approximate solution in the collision region

solitary waves of the generalized kdv equation with distributed delays abstract in this paper we study an integro differential equation based on the generalized kdv equation with a convolution term which introduces a time delay in the nonlinearity special attention is paid to the existence of solitary wave solutions motivated by m j ablowitz h seger soliton and inverse scattering transform siam philadelphia 1981 c k r t jones geometrical singular perturbation theory in r johnson ed dynamical systems in lecture notes in math vol 1609 springer new york 1995 t ogawa travelling wave solutions to perturbed korteweg de vries equations hiroshima math j 24 1994 401 422 we prove using the linear chain trick and geometric singular perturbation analysis that the solitary wave solutions persist when the average delay is suitably small for a special convolution kernel

small roots low elements and the weak order in coxeter groups abstract in this article we provide a new finite class of elements in any coxeter system w s called low elements they are defined from brink and howlett s small roots which are strongly linked to the automatic structure of w s our first main result is to show that they form a garside shadow in w s i e they contain s and are closed under join for the right weak order and by taking suffixes these low elements are the key to prove that all finitely generated artin tits groups have a finite garside family this result was announced in a note with p dehornoy in comptes rendus mathematiques 9 in which the present article was referred to under the following working title monotonicity of dominance depth on root systems and applications the proof is based on a fundamental property enjoyed by small roots and which is our second main result the set of small root is bipodal for a natural number n we define similarly n low elements from n small roots and conjecture that the set of n small roots is bipodal implying the set of n low elements is a garside shadow we prove this conjecture for affine coxeter groups and coxeter groups whose graph is labelled by 3 and to prove the latter we extend the root poset on positive roots to a weak order on the root system and define a bruhat order on the root system and study the paths in those orders in order to establish a criterion to prove bipodality involving only finite dihedral reflection subgroups

sequential monitoring with conditional randomization tests sequential monitoring in clinical trials is often employed to allow for early stopping and other interim decisions while maintaining the type i error rate however sequential monitoring is typically described only in the context of a population model we describe a computational method to implement sequential monitoring in a randomization based context in particular we discuss a new technique for the computation of approximate conditional tests following restricted randomization procedures and then apply this technique to approximate the joint distribution of sequentially computed conditional randomization tests we also describe the computation of a randomization based analog of the information fraction we apply these techniques to a restricted randomization procedure efron s biometrika 58 1971 403 417 biased coin design these techniques require derivation of certain conditional probabilities and conditional covariances of the randomization procedure we employ combinatoric techniques to derive these for the biased coin design 1 introduction sequential monitoring refers to analyzing data periodically during the course of a clinical trial with the purpose of detecting early evidence in support of or against a hypothesis a desirable feature of such a monitoring plan would be flexible inspections of the data that can occur at arbitrary time points at the same time sequentially tested hypotheses must maintain the overall probability of type i error at the prespecified level since repeated testing is known to inflate it the lan and demets 1983 error spending approach for sequential monitoring allows this the approach makes use of a type i error spending function which depends on the amount of statistical information available at the time of the interim inspection in

non crossing partition lattices in finite real reflection groups for a finite real reflection group w with coxeter element we give a case free proof that the closed interval i forms a lattice in the partial order on w induced by reflection length key to this is the construction of an isomorphic lattice of spherical simplicial complexes we also prove that the greatest element in this latter lattice embeds in the type w simplicial generalised associahedron and we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere

global well posedness of the energy critical nonlinear schrodinger equation with small initial data in h1 t3 a refined trilinear strichartz estimate for solutions to the schrodinger equation on the flat rational torus t3 is derived by a suitable modification of critical function space theory this is applied to prove a small data global well posedness result for the quintic nonlinear schrodinger equation in hs t3 for all s 1 this is the first energy critical global well posedness result in the setting of compact manifolds

finite complex reflection arrangements are k pi 1 let v be a finite dimensional complex vector space and w gl v be a finite complex reflection group let v reg be the complement in v of the reflecting hyperplanes we prove that v reg is a k 1 space this was predicted by a classical conjecture originally stated by brieskorn for complexified real reflection groups the complexified real case follows from a theorem of deligne and after contributions by nakamura and orliksolomon only six exceptional cases remained open in addition to solving these six cases our approach is applicable to most previously known cases including complexified real groups for which we obtain a new proof based on new geometric objects we also address a number of questions about 1 w v the braid group of w this includes a description of periodic elements in terms of a braid analog of springer s theory of regular elements

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

the power series method for nonlocal and nonlinear evolution equations abstract the initial value problem for a 4 parameter family of nonlocal and nonlinear evolution equations with data in a space of analytic functions is solved by using a power series method in abstract banach spaces in addition to determining the power series expansion of the solution this method also provides an estimate of the analytic lifespan expressed in terms of the norm of the initial data thus establishing an abstract cauchy kovalevsky type theorem for these equations

local well posedness for the dispersion generalized periodic kdv equation abstract in this paper we set up the local well posedness of the initial value problem for the dispersion generalized periodic kdv equation t u x d x u x u 2 u 0 for 2 s 4 and h s t and we show that the 4 is a lower endpoint to obtain the bilinear estimates 1 2 and 1 3 which are the crucial steps to obtain the local well posedness by picard iteration the case 2 was studied in kenig et al 1996 10

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

well posedness and ill posedness of kdv equation with higher dispersion abstract first the cauchy problem for kdv equation with 2 n 1 order dispersion is studied and the local well posedness result for the initial data in sobolev spaces h s r with s n 1 4 is established via the fourier restriction norm method second we prove that the kdv equation with 2 n 1 order dispersion is ill posed for the initial data in h s r with s n 1 4 n 2 n n if the flow map is c 2 differentiable at zero form h s r to c 0 t h s r finally we obtain the sharp regularity requirement for the kdv equation with 2 n 1 order dispersion s n 1 4

dual euclidean artin groups and the failure of the lattice property abstract the irreducible euclidean coxeter groups that naturally act geometrically on euclidean space are classified by the well known extended dynkin diagrams and these diagrams also encode the modified presentations that define the irreducible euclidean artin groups these artin groups have remained mysterious with some exceptions until very recently craig squier clarified the structure of the three examples with three generators more than twenty years ago and francois digne more recently proved that two of the infinite families can be understood by constructing a dual presentation for each of these groups and showing that it forms an infinite type garside structure in this article i establish that none of the remaining dual presentations for irreducible euclidean artin groups correspond to garside structures because their factorization posets fail to be lattices these are the first known examples of artin groups where all of their dual presentations fail to form garside structures nevertheless the results presented here about the cause of this failure form the foundation for a subsequent article in which the structure of euclidean artin groups is finally clarified

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

local and global well posedness for the 2d zakharov kuznetsov burgers equation in low regularity sobolev space abstract in the present paper we consider the cauchy problem of the 2d zakharov kuznetsov burgers zkb equation which has the dissipative term x 2 u this is known that the 2d zakharov kuznetsov equation is well posed in h s r 2 for s 1 2 and the 2d nonlinear parabolic equation with quadratic derivative nonlinearity is well posed in h s r 2 for s 0 by using the fourier restriction norm with dissipative effect we prove the well posedness for zkb equation in h s r 2 for s 1 2

set theoretic solutions of the yang baxter equation rc calculus and garside germs abstract building on a result by w rump we show how to exploit the right cyclic law x y x z y x y z in order to investigate the structure groups and monoids attached with involutive nondegenerate set theoretic solutions of the yang baxter equation we develop a sort of right cyclic calculus and use it to obtain short proofs for the existence both of the garside structure and of the i structure of such groups we describe finite quotients that play for the considered groups the role that coxeter groups play for artin tits groups

self spreading and strength of singularities for solutions to semilinear wave equations algebres microlocales propagation de la regularite solutions lineaires conduisant a l etalement auto etalement des singularites

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

quadratic dispersive generalized benjamin ono equation abstract this paper is concerned with quadratic dispersive generalized benjamin ono equation 1 1 1 2 we prove sharp local well posedness in h s s 1 4 with a solution map that is analytic from h s r to c 0 t h s r and the ill posed result in h s r as soon as s s in the sense that the flow map u 0 u t cannot be continuous

construction of solutions to the l 2 critical kdv equation with a given asymptotic behaviour we consider the critical korteweg de vries equation ut uxx u x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v h be with sufficient decay on the right that is 1 x0 v l for some 0 0 we construct a solution u t to the critical korteweg de vries equation such that lim t u t u t v n x j 1 rj t h1 0

gaussian groups and garside groups two generalisations of artin groups it is known that a number of algebraic properties of the braid groups extend to arbitrary finite coxeter type artin groups here we show how to extend the results to more general groups that we call garside groups define a gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths and where left and right lowest common multiples exist a garside monoid is a gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition a gaussian group is the group of fractions of a gaussian monoid and a garside group is the group of fractions of a garside monoid braid groups and more generally finite coxeter type artin groups are garside groups we determine algorithmic criteria in terms of presentations for recognizing gaussian and garside monoids and groups and exhibit infinite families of such groups we describe simple algorithms that solve the word problem in a gaussian group show that these algorithms have a quadratic complexity if the group is a garside group and prove that garside groups have quadratic isoperimetric inequalities we construct normal forms for gaussian groups and prove that in the case of a garside group the language of normal forms is regular symmetric and geodesic has the 5 fellow traveller property and has the uniqueness property this shows in particular that garside groups are geodesically fully biautomatic finally we consider an automorphism of a finite coxeter type artin group derived from an automorphism of its defining coxeter graph and prove that the subgroup of elements fixed by this automorphism is also a finite coxeter type artin group that can be explicitly determined 1991 mathematics subject classification primary 20f05 20f36 secondary 20b40 20m05

global well posedness results for the 2d schrodinger debye system abstract in this work we study the well posedness of the schrodinger debye system i t u 1 2 u u v t 0 x r 2 t v v u 2 0 1 u 0 x u 0 x v 0 x v 0 x in the context of sobolev spaces of type h k 1 r 2 h k 2 r 2 where k 1 k 2 are integers we prove some a priori estimates and therefore global well posedness using nonlinear interpolation of operators we prove some global well posedness results when u 0 v 0 h s r 2 h l r 2 where s l r are some convex combinations of k 1 and k 2

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

global well posedness of the one dimensional cubic nonlinear schrodinger equation in almost critical spaces abstract in this paper we first introduce a new function space m h p whose norm is given by the l p sum of modulated h norms of a given function in particular when 1 2 we show that the space m h p agrees with the modulation space m 2 p r on the real line and the fourier lebesgue space f l p t on the circle we use this equivalence of the norms and the galilean symmetry to adapt the conserved quantities constructed by killip visan zhang to the modulation space and fourier lebesgue space setting by applying the scaling symmetry we then prove global well posedness of the one dimensional cubic nonlinear schrodinger equation nls in almost critical spaces more precisely we show that the cubic nls on r is globally well posed in m 2 p r for any p while the renormalized cubic nls on t is globally well posed in f l p t for any p in appendix we also establish analogous global in time bounds for the modified kdv equation mkdv in the modulation spaces on the real line and in the fourier lebesgue spaces on the circle an additional key ingredient of the proof in this case is a galilean transform which converts the mkdv to the mkdv nls equation

periodic fourth order cubic nls local well posedness and non squeezing property abstract in this paper we consider the cubic fourth order nonlinear schrodinger equation 4nls under the periodic boundary condition we prove two results one is the local well posedness in h s t with 1 3 s 0 for the cauchy problem of the wick ordered 4nls the other one is the non squeezing property for the flow map of 4nls in the symplectic phase space l 2 t to prove the former we used the ideas introduced in 36 and 27 and to prove the latter we used the ideas in 8

parabolic deligne lusztig varieties abstract motivated by the broue conjecture on blocks with abelian defect groups for finite reductive groups we study parabolic deligne lusztig varieties and construct on those which occur in the broue conjecture an action of a braid monoid whose action on their l adic cohomology will conjecturally factor through a cyclotomic hecke algebra in order to construct this action we need to enlarge the set of varieties we consider to varieties attached to a ribbon category this category has a garside family which plays an important role in our constructions so we devote the first part of our paper to the necessary background on categories with garside families

inelastic interaction of nearly equal solitons for the quartic gkdv equation this paper describes the interaction of two solitons with nearly equal speeds for the quartic gkdv equation partial tu partial x partial x 2u u 4 0 quad t x in mathbb r 0 1 we call soliton a solution of 0 1 of the form u t x qc x ct y0 where c 0 y0 and q c q c 4 cq c since 0 1 is not an integrable model the general question of the collision of two given solitons q c 1 x c 1 t q c 2 x c 2 t with c1 c2 is an open problem

global well posedness and nonsqueezing property for the higher order kdv type flow abstract in this paper we prove that the periodic higher order kdv type equation t u 1 j 1 x 2 j 1 u 1 2 x u 2 0 t x r t u 0 x u 0 x u 0 h s t is globally well posed in h s for s j 2 j 3 the proof is based on i method introduced by colliander et al 4 we also prove the nonsqueezing property of the periodic higher order kdv type equation the proof relies on gromov s nonsqueezing theorem for the finite dimensional hamiltonian system and an approximation argument for the solution flow more precisely after taking the frequency truncation to the solution flow we apply the nonsqueezing theorem by using the approximation argument we extend this result to the infinite dimensional system this argument was introduced by kuksin 14 and made concretely by bourgain 2 for the 1d cubic nls flow and colliander et al 5 for the kdv flow one of our observations is that the higher order kdv type equation has the better modulation effect from the non resonant interaction than that the kdv equation has hence unlike the work of colliander et al 5 we can get the nonsqueezing property for the solution flow without the miura transform