helicity is the only integral invariant of volume preserving transformations we prove that any regular integral invariant of volume preserving transformations is equivalent to the helicity specifically given a functional i defined on exact divergence free vector fields of class c 1 on a compact 3 manifold that is associated with a well behaved integral kernel we prove that i is invariant under arbitrary volume preserving diffeomorphisms if and only if it is a function of the helicity

global existence and blow up for the generalized sixth order boussinesq equation abstract in this paper we prove local well posedness in l 2 r and h 1 r for the generalized sixth order boussinesq equation u t t u x x u x x x x u x x x x x x u u x x our proof relies in the oscillatory integrals estimates introduced by kenig et al 1991 14 we also show that under suitable conditions a global solution for the initial value problem exists in addition we derive the sufficient conditions for the blow up of the solution to the problem

a numerical study of an ill posed boussinesq equation arising in water waves and nonlinear lattices filtering and regularization techniques1this research was supported by nsf grants no dms 9208061 and by the office of the vice president for research and associate provost for graduate studies at texas a m university 1 we consider an ill posed boussinesq equation which arises in shallow water waves and nonlinear lattices this equation has growing and decaying modes in the linear as well as nonlinear regimes and its linearized growth rate s for short waves of wavenumber k is given by s k 2 previous numerical studies have addressed numerical difficulties and construction of approximate solutions for ill posed problems with short wave instability up to s k e g kelvin helmholtz s k and rayleigh taylor s k instabilities these same issues are addressed and critically examined here for the present problem which has more severe short wave instability in order to develop numerical techniques for constructing good approximate solutions of this equation we use a finite difference scheme to investigate the effect of this short wave instability on the numerical accuracy of the exact solitary wave solution of this equation computational evidence is presented which indicates that numerical accuracy of the solutions is lost very quickly due to severe growth of numerical errors roundoff as well as truncation we use both filtering and regularization techniques to control growth of these errors and to provide better approximate solutions of this equation in the filtering technique numerical experiments with three types of spectral filters of increasing order of regularity are performed we examine the role of regularity of these filters on the accuracy of the numerical solutions numerical evidence is provided which indicates that the regularity of a filter plays an important role in improving the accuracy of the solutions in the regularization technique the ill posed equation is regularized by adding a higher order term to the equation two types of higher order terms are discussed i one that diminishes the growth rate of all modes below a cutoff wavenumber and sets the growth rate of all modes above it to zero and ii the other one diminishes the growth rate of all modes and the growth rate asymptotically approaches to zero as the wavenumber approaches infinity we have argued in favor of the first type of regularization and numerical results using a finite difference scheme are presented numerical evidence is provided which suggests that regularization in combination with the most regular c 2 here spectral filter for small values of the regularization parameter can provide good approximate solutions of the ill posed boussinesq equation for longer time than possible otherwise some of the ideas presented here can possibly be utilized for solving other ill posed problems with severe short wave instabilities and may have an important role to play in numerical studies of their solutions

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

asymptotic massey products induced currents and borromean torus links we introduce a class of currents which allows a new and very explicit form for the massey product of a third order link as a line integral the explicit form permits the introduction of an asymptotic massey product analogous to that introduced previously for gauss s integral by v arnold the average third order asymptotic massey product is shown to be equal to berger s third order helicity for divergence free vector fields in linked tori

existence of knotted vortex tubes in steady euler flows we prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible euler equation in mathbb r 3 more precisely given a finite collection of possibly linked and knotted disjoint thin tubes in mathbb r 3 we show that they can be transformed with a c m small diffeomorphism into a set of vortex tubes of a beltrami field that tends to zero at infinity the structure of the vortex lines in the tubes is extremely rich presenting a positive measure set of invariant tori and infinitely many periodic vortex lines the problem of the existence of steady knotted thin vortex tubes can be traced back to lord kelvin

hopf algebras of dimension pq ii abstract let h be a hopf algebra of dimension pq over an algebraically closed field of characteristic zero where p q are odd primes with p q 4 p 11 we prove that h is semisimple and thus isomorphic to a group algebra or the dual of a group algebra

on the even powers of the antipode of a finite dimensional hopf algebra abstract the trace of powers of the square of the antipode s 2 of a finite dimensional hopf algebra a over a field k is studied it is shown in many cases that the trace function vanishes on s 2 m when s 2 m 1 a finer properties of the antipode are related to this phenomenon

local well posedness for the sixth order boussinesq equation abstract in this work we study the local well posedness of the initial value problem for the nonlinear sixth order boussinesq equation u t t u x x u x x x x u x x x x x x u 2 x x where 1 we prove the local well posedness with initial data in non homogeneous sobolev spaces h s r for negative indices of s r

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

the set of types ofn dimensional semisimple and cosemisimple hopf algebras is finite in this paper we give a partial answer to the above statement proving that the set of types of semisimple and cosemisimple hopf algebras of a given dimension is finite in any characteristic the proof uses the formalism of deformation theory of bialgebras which is concerned from a geometrical point of view with the study of the affine algebraic set b v of all bialgebra structures which can be defined on a vector space v there are two basic facts regarding b v first of all the transport of structures defines an action of gl v on b v such that the orbit of a bialgebra a contains all structures isomorphic to a so the set of types of bialgebras on v is in a one to one correspondence with the set of orbits second we are led in a natural way to a cohomology theory for bialgebras studying the tangent spaces t o and t b v where o is the closure a a a a of the orbit through a bialgebra a one of the most important results of the deformation theory of bialgebras is that the orbit of a bialgebra is open if the cohomology vanishes therefore the number of orbits which correspond to bialgebras having trivial cohomology is finite we obtain our result by proving that the cohomology of a semisimple hopf algebra a vanishes if the characteristic of k is 0 or more generally if a is

global rough solutions to the sixth order boussinesq equation abstract we show that the cauchy problem for the sixth order boussinesq equation with cubic nonlinearity is globally well posed in h s r provided 3 2 s 2

well posedness for the cauchy problem associated to a periodic boussinesq equation abstract we consider the initial value problem associated to the nonlinear periodic sixth order boussinesq equation for given initial data in sobolev spaces h s t the local well posedness of this equation is established for s 1 2

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

signature asymptotique d un champ de vecteurs en dimension 3 consider a volume preserving vector field defined in some compact domain of 3space and tangent to its boundary a long piece of orbit can be made into a knot by connecting its endpoints by some arc whose length is less than the diameter of the domain in this paper we study the behaviour of the signatures of these knots as the lengths of the pieces of orbits go to infinity we relate this asymptotic signature to the asymptotic hopf invariant that has been studied by arnold

link invariants for flows in higher dimensions linking numbers in higher dimensions and their generalization including gauge fields are studied in the context of bf theories the linking numbers associated with n manifolds with smooth flows generated by divergence free p vector fields endowed with an invariant flow measure are computed in the context of quantum field theory they constitute invariants of smooth dynamical systems for nonsingular flows and generalize previous proposals of invariants in particular they generalize arnold s asymptotic hopf invariant from three to higher dimensions this invariant is generalized by coupling with a non abelian gauge flat connection with nontrivial holonomy the computation of the asymptotic jones witten invariants for flows is naturally extended to dimension n 2p 1 finally we give a possible interpretation and implementation of these issues in the context of 11 dimensional supergravity and string theory

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

some comments on chern simons gauge theory following m f atiyah and r bott ab and e witten w we consider the space of flat connections on the trivialsu 2 bundle over a surfacem modulo the space of gauge transformations we describe on this quotient space a natural hermitian line bundle with connection and prove that if the surfacem is now endowed with a complex structure this line bundle is isomorphic to the determinant bundle we show heuristically how path integral quantisation of the chern simons action yields holomorphic sections of this bundle

two exponential type integrators for the good boussinesq equation we introduce two exponential type integrators for the good boussinesq equation they are of orders one and two respectively and they require lower spatial regularity of the solution compared to classical exponential integrators for the first integrator we prove first order convergence in h r for solutions in h r 1 with r 1 2 this new integrator even converges with lower order in h r for solutions in h r i e without any additional smoothness assumptions for the second integrator we prove second order convergence in h r for solutions in h r 3 with r 1 2 and convergence in l 2 for solutions in h 3 numerical results are reported to illustrate the established error estimates the experiments clearly demonstrate that the new exponential type integrators are favorable over classical exponential integrators for initial data with low regularity

involutory hopf algebras in 1975 kaplansky conjectured that a finite dimensional semisimple hopf algebra is necessarily involutory twelve years later larson and radford proved the conjecture in characterisitic 0 and obtained significant partial results in positive characteristics the goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites no harpoons and gratifyingly few hits let h be a finite dimensional hopf algebra over a field k and let s h h denote its antipode then h is said to be involutory if s2 idh or equivalently if s is an involution on h in k kaplansky conjectured that a semisimple hopf algebra is necessarily involutory and in a fundamental paper lr2 larson and radford proved this assertion at least for fields of characteristic 0 furthermore they showed that if h is semisimple and cosemisimple and if chark p dimk h 2 then h must again be involutory a nice exposition of this material can be found in n while lr2 is reasonably brief it depelids heavily on certain earlier results on characters of hopf algebras as contained in l and lr 1 and the combined proof is substantial the goal of this paper is to present an efficient proof of the main results of lr2 requiring minimal prerequisites as will be apparent our proof follows the same basic outline as that of larson and radford and therefore we offer nothing new in this regard however by computing traces on certain tensor modules we are able to obtain the necessary character formulas quite quickly and this is where the simplifications occur in some sense the hero of this approach is the linear transformation t described below if v is a left h module then h 0 v is well known to be h isomorphic to h 0 vj where vj is equal to v but with the trivial action determined by the counit e this is proved quite simply by considering the map t h v h v given by t h ov hihi h2v for all h e h v e v h indeed one shows that t is invertible with inverse t 1 h x v e hz x s h2 v h received by the editors july 13 1993 and in revised form june 8 1994 originally communicaated to the proceedings of the ams by lance w small 1991 mathematics subject classification primary 1 6w30 research supported in part by nsf grants dms 9224662 and dms 9003374 1995 american mathematical society

kirby elements and quantum invariants we define the notion of a kirby element of a ribbon category mathcal c not necessarily semisimple kirby elements lead to 3 manifold invariants we characterize a class of kirby elements the algebraic kirby elements in terms of the structure maps of a hopf algebra in mathcal c this class is sufficiently large to recover the quantum invariants of 3 manifolds of reshetikhin and turaev of hennings kauffman and radford and of lyubashenko when these are well defined the cases of a semisimple ribbon category and of a category of representations are explored in detail

the group of automorphisms of a semisimple hopf algebra over a field of characteristic 0 is finite introduction suppose that a is a semisimple hopf algebra over a field of characteristic 0 or that a is a semisimple cosemisimple involutory hopf algebra over a field k of characteristic p dim a in this paper we prove that the group authopf a of hopf algebra automorphisms of a is finite we show that the semigroup endhopf a of hopf algebra endomorphisms of a is also finite as a corollary generally the group of hopf algebra automorphisms of a finite dimensional hopf algebra need not be finite for any positive integer n and any field k we construct a family of finite dimensional hopf algebras over k with automorphism group gln k the fact that authopf a is finite is a consequence of a theorem of

non semisimple hopf algebras of dimension p2 abstract let h be a finite dimensional hopf algebra with antipode s of dimension pq over an algebraically closed field of characteristic 0 where p q are odd primes if h is not semisimple then the order of s 4 is p and tr s 2 p is an integer divisible by p 2 in particular if dim h p 2 we prove that h is isomorphic to a taft algebra this completes the classification for the hopf algebras of dimension p 2

semisimple hopf algebras via geometric invariant theory abstract we study hopf algebras via tools from geometric invariant theory we show that all the invariants we get can be constructed using the integrals of the hopf algebra and its dual together with the multiplication and the comultiplication and that these invariants determine the isomorphism class of the hopf algebra we then define certain canonical subspaces i n v i j of tensor powers of h and h and use the invariant theory to prove that these subspaces satisfy a certain non degeneracy condition using this non degeneracy condition together with results on symmetric monoidal categories we prove that the spaces i n v i j can also be described as h i h j a where a is the group of hopf automorphisms of h as a result we prove that the number of possible hopf orders of any semisimple hopf algebra over a given number ring is finite we give some examples of these invariants arising from the theory of frobenius schur indicators and from reshetikhin turaev invariants of three manifolds we give a complete description of the invariants for a group algebra proving that they all encode the number of homomorphisms from some finitely presented group to the group we also show that if all the invariants are algebraic integers then the hopf algebra satisfies kaplansky s sixth conjecture the dimensions of the irreducible representations of h divide the dimension of h

invariants of the euler equations for ideal or barotropic hydrodynamics and superconductivity in d dimensions abstract the hamiltonian formalism for the euler equations of an ideal fluid superconductivity and a barotropic fluid on a d dimensional riemannian manifold is proposed we show that each of these equations has an infinite series of integrals if d is even generalized enstrophies and at least one integral if d is odd generalized helicity we prove that the magnetic hydrodynamics integral f v b is equal to the average linking number of vector fields rot v and b in terms of the ergodic theory all the invariants considered are casimir elements i e invariants of coadjoint action of the corresponding infinite dimensional lie algebras

on the quasitriangular structures of a semisimple hopf algebra in this paper we show that the theorem of 4 on the order of the group of hopf algebra automorphisms aut a of certain semisimple hopf algebras over a field k has an interesting implication for quantum groups if a is a finite dimensional hopf algebra over the field k there may be infinitely many re a a such that a r is a quasitriangular hopf algebra however if a is semisimple and the characteristic of k is 0 or if a is semisimple cosemisimple and involutory and the characteristic of k is p dim a we show in the theorem of this paper that there are at most finitely r ea a such that a r is quasitriangular our proof is based on the fact that the group aut a is finite 4 theorem 33 for such a hopf algebra a we will assume that the reader is familiar with the elementary aspects of the theory of hopf algebras a good general reference is s

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

the jones witten invariant for flows on a 3 dimensional manifold the jones witten functional for knots is generalized for smooth flows on 3 dimensional manifolds explicit computations for the abelian case are given

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

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