weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

one dimensional topological dynamical systems of discrete group actions in this survey it is given a brief account of recent research achievements in the study of 1 dimensional topological dynamical systems of discrete group actions the content contains expansiveness ping pong game and geometric entropy invariant sets and invariant measures topological k transitivity sensitivity and devaney s chaos bounded euler classes and classifications some open questions are reported

global poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation both local and global a r weighted poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation in john domains are established as applications of the poincare inequalities for green s operator the a r weighted sobolev imbedding inequalities for green s operator in john domains are also proved

solvable groups acting on the line actions of discrete groups on the real line are considered when the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line in the process of obtaining these results sufficient conditions are also determined for the existence of invariant quasi invariant measures for abelian solvable groups acting on the line it turns out for example that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi invariant measure and that any abelian group of c diffeomorphisms has an invariant measure an example is given to show that some restrictions are necessary in order to obtain such conclusions some applications to the study of codimension one foliations are indicated introduction if g is a group of homeomorphisms of the line it is reasonable to ask for a dynamic description of how g acts when reasonable restrictions are placed on g for example in 9 it is shown that if g is finitely generated and nilpotent then there is a g invariant borel measure on r which is finite on compact sets this already says much about g the proof is quite different from classical arguments which guarantee the existence of a finite invariant measure when an amenable group acts on a compact hausdorff space 6 in that one uses pseudogroup properties of g rather than group properties specifically nonexponential growth of orbits rather than nilpotence of g it is reasonable to determine what can be said when g is solvable but as pointed out in 9 solvable groups need not have invariant measures e g subgroups of the affine group which contain nontrivial translations and nontrivial dilations such examples suggest consideration of a more general invariance property a measure is called quasi invariant for g if each element of g multiplies the measure by a nonzero constant which depends on the particular element of g usual lebesgue measure is quasi invariant for the affine group so it is reasonable to ask if every finitely generated solvable g has a quasi invariant measure it turns out that a large class of solvable groups including polycyclic groups acting on the line must have quasi invariant measures but there are solvable groups which do not have such measures an important step in the investigation of solvable groups is the consideration of infinitely generated abelian groups if differentiability is assumed stronger received by the editors october 30 1980 and in revised form august 3 1982 1980 mathematics subject classification primary 57e25 secondary 58f11 58f15 58f18

on muger s centralizer in braided equivariantized fusion categories abstract we provide a general formula for muger s centralizer of any fusion subcategory of a braided fusion category containing a tannakian subcategory this entails a description for muger s centralizer of all fusion subcategories of a group theoretical braided fusion category

group actions on one manifolds ii extensions of holder s theorem this self contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of lie groups and their discrete subgroups in this paper we consider groups which act on r with restrictions on the fixed point set of each element one result is a topological characterization of affine groups in diff 2 r as those groups whose elements have at most one fixed point

regulating flows topology of foliations and rigidity a flow transverse to a foliation is regulating if in the universal cover an arbitrary orbit of the flow intersects every leaf of the lifted foliation this implies that the foliation is r covered that is its leaf space in the universal cover is homeomorphic to the reals we analyse the converse of this implication to study the topology of the leaf space of certain foliations we prove that if a pseudo anosov flow is transverse to an r covered foliation and the flow is not an r covered anosov flow then the flow is regulating for the foliation using this we show that several interesting classes of foliations are not r covered finally we show a rigidity result if an r covered anosov flow is transverse to a foliation but is not regulating then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow

weighted integral inequalities for solutions of the a harmonic equation we prove the parametric versions of ar weighted integral inequalities for differential forms satisfying the a harmonic equation these results can be considered as extensions of the classical inequalities for sobolev functions

arλ ω weighted imbedding inequalities for a harmonic tensors abstract we prove new versions of a r weighted imbedding inequalities for a harmonic tensors locally and globally the results are used to estimate the integrals of a homotopy operator t from the banach space l s l to the sobolev space w 1 s l 1 l 1 2 n and sobolev poincare imbedding inequality with a r weight

rigidity for c 1 actions on the interval arising from hyperbolicity i solvable groups we consider abelian by cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the abelian subgroup we show that if such a group acts faithfully by c 1 diffeomorphisms of the closed interval with no global fixed point at the interior then the action is topologically conjugate to that of an affine group moreover in case of non abelian image we show a rigidity result concerning the multipliers of the homotheties despite the fact that the conjugacy is not necessarily smooth some consequences for non solvable groups are proposed in particular we give new proofs examples yielding the existence of finitely generated locally indicable groups with no faithful action by c 1 diffeomorphisms of the interval

a symmetry problem related to wirtinger s and poincare s inequality j u 1 1 u p u q among periodic functions in w 1 1 with mean value zero and for p q 1 the minimizers are scaled in such a way that their maximum is 1 and their minimum is m 0 1 they raise the question if m 1 since this implies symmetries for the minimizers their answer is positive for q 2p and negative for q 1 practically the same question is raised in 2 and 3 if m 1 minimizers must be odd or symmetric in the sense that u x u x it was shown in 3 that for p q 2 the minimizer is symmetric and that for p 2 q it is not symmetric moreover in 2 one can find that the minimizer is symmetric for q 2p and nonsymmetric for for q 4p 1 the nonsymmetry result was derived by egorov in the following way restricted to the following set a

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

galois theory for braided tensor categories and the modular closure abstract given a braided tensor category c with conjugate dual objects and irreducible unit together with a full symmetric subcategory s we define a crossed product c s this construction yields a tensor category with conjugates and an irreducible unit a category is a category enriched over vect c with positive operation a galois correspondence is established between intermediate categories sitting between c and c s and closed subgroups of the galois group gal c s c aut c c s of c the latter being isomorphic to the compact group associated with s by the duality theorem of doplicher and roberts denoting by d c the full subcategory of degenerate objects i e objects which have trivial monodromy with all objects of c the braiding of c extends to a braiding of c s iff s d under this condition c s has no non trivial degenerate objects iff s d if the original category c is rational i e has only finitely many isomorphism classes of irreducible objects then the same holds for the new one the category c c d is called the modular closure of c since in the rational case it is modular i e gives rise to a unitary representation of the modular group sl 2 z if all simple objects of s have dimension one the structure of the category c s can be clarified quite explicitly in terms of group cohomology

poincare inequalities with the radon measure for differential forms we establish the local and global poincare inequalities with the radon measure for the solutions to the nonlinear elliptic partial differential equation for differential forms

some examples of conjugatep harmonic differential forms abstract there has been new interest developing in the study of the hodge theory and p harmonic differential forms in recent years largely pertaining to applications in quasi conformal analysis electromagnetic theory relativity theory mathematical physics nonlinear potential theory etc in this paper we develop an effective method to find conjugate p harmonic differential forms in r n by applying hodge theory in calculations with the p harmonic equation as applications of the method we obtain some interesting examples of conjugate p harmonic differential forms