keeping track of limit cycles this article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields it introduces the notion of limit periodic sets and shows that the hilbert s 16th problem can be reduced to the existence of a uniform bound on its cyclicity it also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the noetherian property of the bautin ideal

stein compacts in levi flat hypersurfaces we explore connections between geometric properties of the levi foliation of a levi flat hypersurface m and holomorphic convexity of compact sets in m or bounded in part by m applications include extendability of cauchy riemann functions solvability of the b equation approximation of cauchy riemann and holomorphic functions and global regularity of the neumann operator

on the local solvability for top degree forms in hypo analytic structures l is the differential operator in the differential complex defined by the hypo analytic structure for most i differential forms of top degree m n the solution u should be an equivalence class of currents of bidegree m n 1 in the same paper it is also shown that the nonexistence of hypoanalytic functions h whose real parts have local extrema ensures the local solvability of 1 at least in the cases where either the tangent or the cotangent structure bundles have rank exactly equal to one and are real analytic vector bundles in the present paper we show that this remains true in the general case when the rank of the cotangent structure bundle is equal to 1 this is established by modifying the method in c h in this case the existence of hypo analytic functions whose real parts have local extrema is seen to be equivalent to the nonvanishing of the n 1 dimensional homology of the fibres of the structure see end of section 2 and cf t2 when it is the rank of the tangent structure bundle which is equal to 1 the result is derived in section 3 by reinterpreting moyer s proof of the necessity of condition nv i wish to thank paulo cordaro and jorge hounie for looking at the manuscript and pointing out errors and misprints needless to say if there are any left it is entirely my fault

characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method abstract in this paper a new approximation method is introduced to characterize a so called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with f convex functions of order 2 in this method an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point in order to prove the equivalence between the original multiobjective programming problem and its associated f approximated vector optimization problem the suitable f convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem

necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields a necessary and sufficient condition is proved for the validity of the poincare lemma in degreeq 1 in the differential complex attached to a locally integrable structure of codimension one in spaces of hyperfunctions the base manifold is only assumed to be smooth the hyperfunctions are defined in the hypo analytic structure associated to a smooth first integral z the condition is that the singular homology of the fibres of the map z be trivial in dimensionq 1 by the approximation formula of bt the germ of this fibration at a point is independent of the choice of the first integral z

strongly pseudoconvex cr structures over small balls part i an a priori estimate the purpose of the present series of papers is to develop the theory of harmonic integrals on strongly pseudoconvex cr structures over small balls along the line developed by d c spencer c b morrey j j kohn and l nirenberg 5 6 7 8 for strongly pseudoconvex complex manifolds we consider a strongly pseudoconvex cr structure on a manifold of real dimension 2n 1 in part i we establish the a priori estimate for the neumann boundary value problem on the complex associated with the structure in the case the structure is induced by an embedding in c and restricted to a small ball of special type provided i c q c n 3 where q is the degree of differential forms we consider in part ii we derive the regularity theorem of solutions of the neumann boundary value problem based on the a priori estimate of part i as an application we prove in part il that when n 5 the structure is realized on a neighborhood of a reference point by an embedding in cn we note here that l nirenberg gave an example of a strongly pseudoconvex cr structure which cannot be induced by such an embedding when n 2 9 l boutet de monvel proved that in the case of such structures over a compact manifold we can find such an embedding when n 3 1 there are two major differences between the cases of strongly pseudoconvex integrable almost complex structures and cr structures firstly the principal parts of the associated laplacians in cr structures are no longer elliptic secondly we have characteristic points on the boundary as a result when we follow over the predecessor s calculation in our case we end up with terms which may not be absorbed in the main non negative terms fortunately these leftover terms depend mostly on the levi form of the cr structure and the hessian form with respect to the cr structure of the equation of the boundary of our

local integrability of mizohata structures in this work we study the local integrability of strongly pseudoconvex mizohata structures of rank n 2 and co rank 1 these structures are locally generated in an appropriate coordinate system t 1 t n x by flat perturbations of mizohata vector fields m j t j it j x j 1 n for this we first prove the global integrability of small perturbations of the structure generated by z 1 z n 1 j z j 2 n defined over a manifold c s where s is simply connected

boundary holder and estimates for local solutions of the tangential cauchy riemann equation nous etudions la resolubilite locale de l operateur de cauchy riemann tangentiel sur un voisinage w d un point z 0 d une sous variete cr generique q concave m de codimension quelconque de c n nous construisons une formule d homotopie pour le b sur lorsque est l intersection de m et d un domaine strictement pseudoconvexe nous obtenons ainsi un operateur de resolution pour toute forme b fermee sur nous en deduisons des estimations l p et des estimations holderiennes jusqu au bord pour la solution de l equation de cauchy riemann tangentielle sur

turbulence amalgamation and generic automorphisms of homogeneous structures we study topological properties of conjugacy classes in polish groups with emphasis on automorphism groups of homogeneous countable structures we first consider the existence of dense conjugacy classes the topological rokhlin property we then characterize when an automorphism group admits a comeager conjugacy class answering a question of truss and apply this to show that the homeomorphism group of the cantor space has a comeager conjugacy class answering a question of akin hurley and kennedy finally we study polish groups that admit comeager conjugacy classes in any dimension in which case the groups are said to admit ample generics we show that polish groups with ample generics have the small index property generalizing results of hodges hodkinson lascar and shelah and arbitrary homomorphisms from such groups into separable groups are automatically continuous moreover in the case of oligomorphic permutation groups they have uncountable cofinality and the bergman property these results in particular apply to automorphism groups of many stable 0 categorical structures and of the random graph in this connection we also show that the infinite symmetric group s has a unique non trivial separable group topology for several interesting groups we also establish serre s properties fh and fa

the range of the tangential cauchy riemann operator over a small ball in this paper we study the closed range property of tangential cauchy riemann equations of an abstract cr structure over a small ball let d be the set x e r 1 1x1 4 kuranishi 141 and akahori 11 have studied the l theory

real analytic solvability for differential complexes associated to locally integrable structures abstract inspired by the work of suzuki 12 on the concept of real analytic solvability for first order analytic linear partial differential operators we extend his results for the differential complexes associated to analytic locally integrable structures of corank one we prove that such notion of solvability is related to the smooth solvability condition introduced by f treves 13 in 1983 in our arguments the natural extension to closed forms of the well known baouendi treves approximation formula the so called approximate poincare lemma cf 1 14 plays a key role

local solvability for real analytic involutive structures of tube type of corank one abstract this work is a continuation of p d cordaro e r da silva local solvability in corank one involutive real analytic structures indiana univ math j 53 2004 1605 1628 we study the problem of local solvability in degree one for the differential complex associated to a real analytic involutive structure of tube type of corank one in the present case we obtain more accurate results

bounded normal generation and invariant automatic continuity abstract we study the question of how quickly products of a fixed conjugacy class in the projective unitary group of a ii1 factor von neumann algebra cover the entire group our result is that the number of factors that are needed is essentially as small as permitted by the 1 norm in analogy to results of liebeck and shalev for non abelian finite simple groups as an application of the techniques we prove that every homomorphism from the projective unitary group of a finite factor to a polish sin group is continuous a result which is even new for pu n moreover we show that the projective unitary group of a ii1 factor carries a unique polish group topology

weakly almost periodic functions model theoretic stability and minimality of topological groups we investigate the automorphism groups of aleph 0 categorical structures and prove that they are exactly the roelcke precompact polish groups we show that the theory of a structure is stable if and only if every roelcke uniformly continuous function on the automorphism group is weakly almost periodic analysing the semigroup structure on the weakly almost periodic compactification we show that continuous surjective homomorphisms from automorphism groups of stable aleph 0 categorical structures to hausdorff topological groups are open we also produce some new wap trivial groups and calculate the wap compactification in a number of examples

hilbert integrals singular integrals and radon transforms ii etude des transformations de radon singulieres et des integrales de hilbert appliquees aux problemes aux valeurs limites probleme de neumann

local solvability for top degree forms in a class of systems of vector fields we study the local exactness at the top degree level in the differential complex defined by a smooth locally integrable structure of rank n in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i if z denotes a local first integral of the structure it is proved that the vanishing of the local cohomology in degree n is implied by the absence of compact connected components of the fibers z const this adds one more result towards the verification of a conjecture due to f treves regarding the vanishing of the local cohomology of such complexes of differential operators

on the local solvability and the local integrability of systems of vector fields on etudie des systemes surdetermines de champs vectoriels dans un sous ensemble ouvert de r n 1 de type l j t j j t x x j 1 m ayant des coefficients analytiques on suppose que les champs de vecteurs l j satisfont la condition de frobenius

local solvability of overdetermined systems defined by commuting first order differential operators on demontre comment des theoremes de duistermaat hormander sur la propagation des singularites peuvent etre utilises pour construire des solutions locales lisses pour une classe de systemes surdetermines lineaires d ordre 1 d equations aux derivees partielles

nonexistence of homotopy formula for 0 1 forms on hypersurfaces in mathbb c 3 soit g une 0 1 forme definie sur une hypersurface strictement convexe dans c 3 et on suppose que b g 0 mais b g est petit on considere la resolution locale approchee de l equation b u g

local existence theorems with estimates for b on weakly pseudo convex cr manifolds in particular we prove the solvability for 0 1 on co for 1 q n 2 when m is a pseudo convex cr hypersurface of finite type in the sense defined in d angelo 11 and c co is the intersection of m with a flat hypersurface m0 see theorem 1 in sect i near a point zo of finite type we also show that there exists a neighborhood basis co of z0 such that 0 1 is solvable in co for any ct satisfying 0 2 see corol lary 1 1 we also obtain the interior regularity of the solutions for 0 1 in the sobolev spaces and show that the range of b is closed in the frechet norm l p q co loc see theorem 2 and corol lary 2 1 in sect 1 since the existence of the solution u of 0 1 was proved on co for any c satisfying 0 2 in co theorems 1 and 2 are semi global results when we assume that mo is only levi flat and with a stein ne ighborhood basis then we prove that for any relatively compact subset co c co there exists a u satisfying 0 2 in co see theorem 3 we note that the local solvability for 0 1 was not known previously even on a small subset in co on general weakly pseudo convex cr manifolds in this paper we shall prove the

continuous first order logic and local stability we develop continuous first order logic a variant of the logic described by chang and keisler 1966 we show that this logic has the same power of expression as the framework of open hausdorff cats and as such extends henson s logic for banach space structures we conclude with the development of local stability for which this logic is particularly well suited

global solvability of real analytic involutive systems on compact manifolds the focus of this work is the smooth global solvability of a linear partial differential operator mathbb l associated to a real analytic closed non exact 1 form b defined on a real analytic closed n manifold that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co rank one we define an appropriate covering projection widetilde m rightarrow m such that the pullback of b has a primitive widetilde b and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of widetilde b are connected as a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator when m is orientable we prove a connection between the global solvability of mathbb l and that of mathbb l n 1 which is the last non trivial operator of the complex in particular we prove that mathbb l is globally solvable if and only if mathbb l n 1 is globally solvable in the smooth category we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a morse 1 form i e when the structure is of mizohata type

local extension of boundary holomorphic forms on real hypersurfaces in cn abstract let m be a real hypersurface in c n with n 3 and the levi form at p m has q 1 positive eigenvalues 1 q n 2 we solve one sided local closed extension problem near p for 0 r forms for 1 r q

necessary and sufficient conditions for isolated local minima of nonsmooth functions we consider the mathematical programming problem find inf f x mid x in c where f is an arbitrary extended real valued function and c a subset of a finite dimensional space we give necessary and sufficient optimality conditions for this problem generalizing previous results of a auslendex this journal 22 1984 pp 239 254