the control of confounding by intermediate variables in epidemiologic studies of the effect of an exposure on disease the crude association of exposure with disease may fail to reflect a causal association due to confounding by one or more covariates most previous discussions of confounding in the epidemiologic literature have considered only point exposure studies that is studies that measure exposure and covariate status only once at start of follow up in this paper we offer definitions of confounding suitable for longitudinal studies that obtain data on exposure covariate and vital status at several points in time an important difference between longitudinal studies and point exposure studies is that in longitudinal studies a time dependent covariate can be simultaneously a confounder and an intermediate variable on the causal pathway from exposure to disease in this paper i propose an estimator the extended standardized risk difference that provides control for confounding by a covariate that is simultaneously a confounder and an intermediate variable

twistor spaces of hypercomplex manifolds are balanced abstract a hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations if in addition there exists a metric on the manifold which is hermitian with respect to the three structures and such that the corresponding hermitian forms are closed the manifold is said to be hyperkahler in the paper non hermitian yang mills connections 13 kaledin and verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric these were first studied in the article on the existence of special metrics in complex geometry 17 by michelsohn in the present article we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced

integrable systems and reductions of the self dual yang mills equations many integrable equations are known to be reductions of the self dual yang mills equations this article discusses some of the well known reductions including the standard soliton equations the classical painleve equations and integrable generalizations of the darboux halphen system and chazy equations the chazy equation first derived in 1909 is shown to correspond to the equations studied independently by ramanujan in 1916

equilibria of set valued maps on nonconvex domains we present new theorems on the existence of equilibria or zeros of convex as well as nonconvex set valued maps defined on compact neighborhood retracts of normed spaces the maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets among other things we show that if k is a compact neighborhood retract with nontrivial euler characteristic in a banach space e and 1 k 2e is an upper hemicontinuous set valued map with nonempty closed convex values satisfying the tangency condition x n tk x 0 for all x c k then there exists xo c k such that 0 c 4 xo here tk x denotes a new concept of retraction tangent cone to k at x suited for compact neighborhood retracts when k is locally convex at x tk x coincides with the usual tangent cone of convex analysis special attention is given to neighborhood retracts having lipschitzian behavior called l retracts below this class of sets is very broad it contains compact homeomorphically convex subsets of banach spaces epi lipschitz subsets of banach spaces as well as proximate retracts our results thus generalize classical theorems for convex domains as well as recent results for nonconvex sets

equilibria of nonconvex valued maps under constraints abstract in the paper the notion of n tangency for set valued maps defined on a subset of a banach space is considered the existence of equilibria of upper semicontinuous map being n tangent to a sleek retract with the nontrivial euler characteristic is established

non local continuity equations and binary darboux transformation of noncommutative anti self dual yang mills equations we present an infinite number of non local continuity equations of noncommutative anti self dual yang mills nc a sdym equations using the induction method of brezin et al 1979 phys lett b 82 442 and relate it to the lax pair and the parametric backlund transformation of the system from the lax pair we derive a binary darboux transformation to generate solutions of the nc a sdym equations

fixed point theorems and morse s lemma for lipschitzian functions abstract we prove a fixed point theorem for set valued mappings defined on a nonempty compact subset x of r n which can be represented by inequality constraints i e x x r n f x 0 f locally lipschitzian and satisfying a nondegeneracy assumption outside of x this class of sets extends significantly the class of convex compact sets with a nonempty interior topological properties of such sets x are proved continuous deformation retract of a ball acyclicity as a consequence of a generalization of morse s lemma for lipschitzian real valued function defined on r n a result also of interest for itself

estimation of controlled direct effects on a dichotomous outcome using logistic structural direct effect models we consider the problem of assessing whether an exposure affects a dichotomous outcome other than by modifying a given mediator the standard approach logistic regression adjusting for both exposure and the mediator is known to be biased in the presence of confounders for the mediator outcome relationship because additional regression adjustment for such confounders is only justified when they are not affected by the exposure inverse probability weighting has been advocated but is not ideally tailored to mediators that are continuous or have strong measured predictors we overcome this limitation by developing inference for a novel class of causal models that are closely related to robins logistic structural direct effect models but do not inherit their difficulties of estimation we study identification and efficient estimation under the assumption that all confounders for the exposure outcome and mediator outcome relationships have been measured and find adequate performance in simulation studies we discuss extensions to case control studies and relevant implications for the generic problem of adjustment for time varying confounding

lectures on classical integrability we review some essential aspects of classically integrable systems the detailed outline of the sections consists of 1 introduction and motivation with historical remarks 2 liouville theorem and action angle variables with examples harmonic oscillator kepler problem 3 algebraic tools lax pairs monodromy and transfer matrices classical r matrices and exchange relations non ultralocal poisson brackets with examples non linear schrodinger model principal chiral field 4 features of classical r matrices belavin drinfeld theorems analyticity properties and lift of the classical structures to quantum groups 5 classical inverse scattering method to solve integrable differential equations soliton solutions spectral properties and the gel fand levitan marchenko equation with examples kdv equation sine gordon model prepared for the durham young researchers integrability school organised by the gatis network this is part of a collection of lecture notes

remarks on the geodesic einstein metrics of a relative ample line bundle in this paper we introduce the associated geodesic einstein flow for a relative ample line bundle l over the total space mathcal x of a holomorphic fibration and obtain a few properties of that flow in particular we prove that the pair mathcal x l is nonlinear semistable if the associated donaldson type functional is bounded from below and the geodesic einstein flow has long time existence property we also define the associated s classes and c classes for mathcal x l and obtain two inequalities between them when l admits a geodesic einstein metric finally in the appendix of this paper we prove that a relative ample line bundle is geodesic einstein if and only if an associated infinite rank bundle is hermitian einstein

estimation of controlled direct effects in time varying treatments using structural nested mean models application to a primary prevention trial for coronary events with pravastatin for the estimation of controlled direct effects i e direct effects controlling intermediates that are set at a fixed level for all members of the population without bias two fundamental assumptions must hold the absence of unmeasured confounding factors for treatment and outcome and for intermediate variables and outcome even if these assumptions hold one would nonetheless fail to estimate direct effects using standard methods for example stratification or regression modeling when the treatment influences confounding factors for such situations the sequential g estimation method for structural nested mean models has been developed for estimating controlled direct effects in point treatment situations in this study we demonstrate that this method can be applied to longitudinal data with time varying treatments and repeatedly measured intermediate variables we sequentially estimate the parameters in two structural nested mean models one for a repeatedly measured intermediate and the other one for direct effects of a time varying treatment the method was applied to data from a large primary prevention trial for coronary events in which pravastatin was used to lower the cholesterol levels in patients with moderate hypercholesterolemia

the 2 1 dimensional nonisospectral relativistic toda hierarchy related to the generalized discrete painleve hierarchy in this paper we will concentrate on the topic of integrable discrete hierarchies in 2 1 dimensions and their connection with discrete painleve hierarchies by considering a 2 1 dimensional nonisospectral discrete linear problem two new 2 1 dimensional nonisospectral integrable lattice hierarchies the 2 1 nonisospectral relativistic toda lattice hierarchy and the 2 1 nonisospectral negative relativistic toda lattice hierarchy are constructed it is shown that the reductions of the two new 2 1 nonisospectral lattice hierarchies lead to the 2 1 dimensional nonisospectral volterra lattice hierarchy and the 2 1 dimensional nonisospectral negative volterra lattice hierarchy we also obtain two new 1 1 dimensional nonisospectral integrable lattice hierarchies and two new ordinary difference hierarchies which are direct reductions of the two 2 1 nonisospectral integrable lattice hierarchies one of the two difference hierarchies yields our previously obtained generalized discrete first painleve dpi hierarchy and another one yields a generalized alternative discrete second painleve alt dpii hierarchy

modular solutions to equations of generalized halphen type solutions to a class of differential systems that generalize the halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group these functions uniformize riemann surfaces of genus zero and have q series with integral coefficients rational maps between these functions are derived implying subgroup relations between their automorphism groups as well as symmetrization maps between the associated differential systems

representations of epi lipschitzian sets abstract a closed subset m of a banach space e is epi lipschitzian i e can be represented locally as the epigraph of a lipschitz function if and only if it is the level set of some locally lipschitz function f e r for which clarke s generalized gradient does not contain 0 at points in the boundary of m i e such that m x f x 0 0 f x if x bd m this extends the characterization previously known in finite dimension and answers to a standing question

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

central extensions of current groups in two dimensions the paper is devoted to generalization of the theory of loop groups to the two dimensional case to every complex riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex lie group g by the jacobian of this surface this extension is topologically nontrivial as in the loop group case orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalg bundles over the surface when the surface is the torus elliptic curve classification of coadjoint orbits is related to linear difference equations in one variable and to classification of conjugacy classes in the loop group we study integral orbits for which the kirillov kostant form is a curvature form for some principal torus bundle the number of such orbits for a given level is finite as in the loop group case conjecturedly they correspond to analogues of integrable modules occurring in conformal field theory

some reductions of the self dual yang mills equations to integrable systems in 2 1 dimensions a reduction of the self dual yang mills sdym equations is studied by imposing two space time symmetries and by requiring that the connection one form belongs to a lie algebra of formal matrix valued differential operators in an auxiliary variable in this article the scalar case and the canonical cases for 2 2 matrices are examined in the scalar case it is shown that the field equations can be reduced to the forced burgers equation in the matrix case several well known 2 1 integrable equations are obtained also examined are certain transformation properties between the solutions of some of these 2 1 equations

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

2 1 dimensional integrable lattice hierarchies related to discrete fourth order nonisospectral problems in this paper we consider the 2 1 dimensional discrete fourth order nonisospectral problem by using the lax technique three new 2 1 dimensional nonisospectral four field integrable lattice hierarchies are constructed their reductions yield three 1 1 dimensional isospectral four field integrable lattice hierarchies due to mlaszak marciniak we make a comparison between the 1 1 dimensional discrete fourth order nonisospectral problem and the third order nonisospectral problem we found that the integrable lattice hierarchies related to the discrete fourth order nonisospectral problem have new characteristics

first integrals of a generalized darboux halphen system a third order system of nonlinear ordinary differential equations depending on three arbitrary parameters is analyzed the system arises in the study of su 2 invariant hypercomplex manifolds and is a dimensional reduction of the self dual yang mills equation the general solution first integrals and the nambu poisson structure of the system are explicitly derived it is shown that the first integrals are multi valued on the phase space even though the general solution of the system is single valued for special choices of parameters

periodic solutions to impulsive differential inclusions with constraints abstract the existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non convex set of state constraints is established by the use a lefschetz type fixed point theorem for set valued maps

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

existence of generalized equilibria we consider the problem of the existence of equilibria generalized equilibria and 8xed points of a correspondence f de8ned on a compact subset m of rn with values in rn when the set m is neither assumed to be convex nor smooth 1 the general framework of the article is the following let f be a correspondence from m to rn that is a map from m to the set of all the subsets of rn the correspondence f is said to be upper semicontinuous u s c if the set x m f x v is open in m for every open set v rn let n be a normal cone correspondence i e at this stage n is a correspondence from m to rn such that for every x m n x

fixed points and equilibria in nonconvex sets tcentre de recherches mathtmatiques universite de montreal montreal quebec h3c 357 canada psteklov institute of mathematics moscow 117966 russia and ildepartment of mathematics and statistics concordia university montreal quebec h4b lr6 canada received 30 september 1993 received forpublication 18 august 1994 key words and phrases multivalued mapping tangent cones zeros fixed points differential inclusion weak invariance nonsmooth analysis 1 introduction

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

canonical structures on anti self dual four manifolds and the diffeomorphism group an unusual and attractive system is studied that arises from the anti self dual asd yang mills equations with maximal translational symmetry and with gauge group the volume preserving diffeomorphisms of an auxiliary four manifold m the resulting equations lead to a system consisting of a volume form together with four independent vector fields on m satisfying three simple lie bracket relations this structure is shown to give rise to a two sphere s worth of closed simple two forms which in turn lead to the standard hyper kahler structure of an anti self dual metric on m the system determines not only the asd metric but also a frame that is proportional to an orthonormal frame it is shown that the freedom in the choice of frame is related to a pair of solutions of a modified laplacian and can always be chosen so that the proportionality factor is unity the plebanski first and second heavenly forms for general asd metrics are written out in terms of these structures the first being the standard description in terms of a complex structure and kahler scalar one of the scalars is interpreted as the generating function for the diffeomorphisms symplectomorphism in line with the origin of the system as the asd yang mills equations with the volume preserving diffeomorphisms as gauge group

description of surfaces associated with grassmannian sigma models on minkowski space we construct and investigate smooth orientable surfaces in su n algebras the structural equations of surfaces associated with grassmannian sigma models on minkowski space are studied using moving frames adapted to the surfaces the first and second fundamental forms of these surfaces as well as the relations between them as expressed in the gauss weingarten and gauss codazzi ricci equations are found the scalar curvature and the mean curvature vector expressed in terms of a solution of grassmanian sigma model are obtained

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

sublevel representations of epi lipschitz sets and other properties epi lipschitz sets in normed spaces are represented as sublevel sets of lipschitz functions satisfying a so called qualification condition canonical representations through the signed distance functions associated with the sets are also obtained new optimality conditions are provided for optimization problems with epi lipschitz set constraints in terms of the signed distance function

the constrained degree and fixed point index theory for set valued maps abstract in the first part of the paper we give a construction of a topological degree theory for set valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds in the second part we propose a fixed point index for inward maps on arbitrary closed convex sets

an algebraic scheme associated with the noncommutative kp hierarchy and some of its extensions a well known ansatz trace method for soliton solutions turns the equations of the non commutative kp hierarchy and those of certain extensions into families of algebraic sum identities we develop an algebraic formalism in particular involving a mixable shuffle product to explore their structure more precisely we show that the equations of the non commutative kp hierarchy and its extension xnckp in the case of a moyal deformed product as derived in previous work correspond to identities in this algebra furthermore the moyal product is replaced by a more general associative product this leads to a new even more general extension of the non commutative kp hierarchy relations with rota baxter algebras are established

generalised monopole equations on kahler surfaces in this article we establish a hitchin kobayashi type correspondence for generalised seiberg witten monopole equations on kahler surfaces we show that the stability criterion we obtain for the existence of solutions coincides with that of the usual seiberg witten monopole equations this enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors in this article we establish a hitchin kobayashi type correspondence for generalised seiberg witten monopole equations on kahler surfaces we show that the stability criterion we obtain for the existence of solutions coincides with that of the usual seiberg witten monopole equations this enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors

integrable systems and modular forms of level 2 a set of nonlinear differential equations associated with the eisenstein series of the congruent subgroup 0 2 of the modular group is constructed these nonlinear equations are analogues of the well known ramanujan equations as well as the chazy and darboux halphen equations associated with the modular group the general solutions of these equations can be realized in terms of the schwarz triangle function s 0 0 1 2 z

estimation of controlled direct effects when regression models adjust for mediators on the causal path from exposure to outcome the regression coefficient of exposure is commonly viewed as a measure of the direct exposure effect this interpretation can be misleading even with a randomly assigned exposure this is because adjustment for post exposure measurements introduces bias whenever their association with the outcome is confounded by more than just the exposure by the same token adjustment for such confounders stays problematic when these are themselves affected by the exposure robins accommodated this by introducing linear structural nested direct effect models with direct effect parameters that can be estimated by using inverse probability weighting by a conditional distribution of the mediator the resulting estimators are consistent but inefficient and can be extremely unstable when the mediator is absolutely continuous we develop direct effect estimators which are not only more efficient but also consistent under a less demanding model for a conditional expectation of the outcome we find that the one estimator which avoids inverse probability weighting altogether performs best this estimator is intuitive computationally straightforward and as demonstrated by simulation competes extremely well with ordinary least squares estimators in settings where standard regression is valid

commuting flows and conservation laws for noncommutative lax hierarchies we discuss commuting flows and conservation laws for lax hierarchies on noncommutative spaces in the framework of the sato theory on commutative spaces the sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the lax hierarchies in terms of pseudodifferential operators noncommutative extension of the sato theory has been already studied by the author and toda and the existence of various noncommutative lax hierarchies are guaranteed in this paper we present conservation laws for the noncommutative lax hierarchies with both space space and space time noncommutativities and prove the existence of infinite number of conserved densities we also give the explicit representations of them in terms of lax operators our results include noncommutative versions of kp kdv boussinesq coupled kdv sawada kotera modified kdv equation and so on

a connection between the einstein and yang mills equations it is our purpose here to show an unusual relationship between the einstein equations and the yang mills equations we give a correspondence between solutions of the self dual einstein vacuum equations and the self dual yang mills equations with a special choice of gauge group the extension of the argument to the full yang mills equations yields einstein s unifield equations we try to incorporate the full einstein vacuum equations but the approach is incomplete we first consider yang mills theory for an arbitrary lie algebra with the condition that the connection 1 form and curvature are constant on minkowski space this leads to a set of algebraic equations on the connection components we then specialize the lie algebra to be the infinite dimensional lie algebra of a group of diffeomorphisms of some manifold the algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act in the self dual case if we choose the connection components from the lie algebra of the volume preserving 4 dimensional diffeomorphism group the resulting equations are the same as those obtained by ashtekar jacobsen and smolin in their remarkable simplification of the self dual einstein vacuum equations an alternative derivation of the same equations begins with the self dual yang mills connection now depending only on the time then choosing the lie algebra as that of the volume preserving 3 dimensional diffeomorphisms when the reduced full yang mills equations are used in the same context we get einstein s equations for his unified theory based on absolute parallelism to incorporate the full einsteinvacuum equations we use as the lie group the semi direct product of the diffeomorphism group of a 4 dimensional manifold with the group of frame rotations of anso 1 3 bundle over the 4 manifold this last approach however yields equations more general than the vacuum equations

structural nested cumulative failure time models to estimate the effects of interventions in the presence of time varying confounders affected by prior treatment standard statistical methods for failure time analysis may be biased methods that correctly adjust for this type of covariate include the parametric g formula inverse probability weighted estimation of marginal structural cox proportional hazards models and g estimation of structural nested accelerated failure time models in this article we propose a novel method to estimate the causal effect of a time dependent treatment on failure in the presence of informative right censoring and time dependent confounders that may be affected by past treatment g estimation of structural nested cumulative failure time models sncftms an sncftm considers the conditional effect of a final treatment at time m on the outcome at each later time k by modeling the ratio of two counterfactual cumulative risks at time k under treatment regimes that differ only at time m inverse probability weights are used to adjust for informative censoring we a

the bolzano mean value theorem and partial differential equations abstract we study the existence of solutions to abstract equations of the form 0 a u f u u k e where a is an abstract differential operator acting in a banach space e k is a closed convex set of constraints being invariant with respect to resolvents of a and a perturbation f satisfies a certain tangency condition such problems are closely related to the so called poincare miranda theorem being the multi dimensional counterpart of the celebrated bolzano intermediate value theorem in fact our main results should be regarded as infinite dimensional variants of bolzano and miranda poincare theorems along with single valued problems we deal with set valued ones yielding the existence of the so called constrained equilibria of set valued maps the abstract results are applied to show existence of strong steady state solutions to some weakly coupled systems of drift reaction diffusion equations or differential inclusions of this type in particular we get the existence of strong solutions to the dirichlet neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type certain aspects of the bernstein theory for bvp for second order ode are studied too no assumptions concerning structural coupling monotonicity cooperativity are undertaken

a self dual yang mills hierarchy and its reductions to integrable systems in 1 1 and 2 1 dimensions the self dual yang mills equations play a central role in the study of integrable systems in this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self dual yang mills flow as the first member we show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1 1 and 2 1 dimensions and can be considered as a universal integrable hierarchy prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators symmetries and conservation laws are discussed

on the solution sets of differential inclusions and the periodic problem in banach spaces abstract in this paper the topological structure of the solution set of a constrained semilinear differential inclusion in a banach space e is studied it is shown that the set of all mild solutions with values in a closed and in general thin subset d e is an r set provided natural boundary conditions and appropriate geometrical assumptions on d which hold e g when d is convex are satisfied applications to the periodic problem and to the existence of equilibria are given

homotopy invariants for tangent vector fields on closed sets abstract in the paper we construct a topological degree theory for single and convex valued tangent vector fields defined on locally compact closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds the degree allows to study vector fields on sets which are neither smooth nor convex and may be applied to study continuation and bifurcation of equilibria in parameterized families of closed sets

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

surgery in cusp neighborhoods and the geography of irreducible 4 manifolds since its inception the theory of smooth 4 manifolds has relied upon complex surface theory to provide its basic examples this is especially true for simply connected 4 manifolds a longstanding conjecture held that every smooth simply connected closed 4 manifold is the connected sum of manifolds each admitting a complex structure with one of its orientations for this conjecture the 4 sphere was considered as the empty connected sum quite recently this was disproved by gompf and mrowka gm1 who produced infinite families of examples of 4 manifolds homeomorphic to the k3 surface and which admit no complex structure with either orientation the discovery of these dramatic counterexamples to the h cobordism theorem in dimension 4 was the most recent success in a long line of applications of donaldson invariants to the study of smooth structures on 4 manifolds beginning with donaldson s d4 and followed by fm1 ko ov1 ov2 which depended on donaldson s f invariant or a suitable generalization and fmm dk e2 eo1 mn2 sa using donaldson s polynomial invariant a common thread running through many of these counterexamples to the h cobordism conjecture is the process of logarithmic transformation which introduces multiple fibres in an elliptic surface in controlled situations this process will change the diffeomorphism type of a given elliptic surface while keeping its homotopy type fixed taking advantage of the complex structure on an elliptic surface this change can often be detected by using the relationship between anti self dual connections over a complex surface and stable holomorphic bundles d2 an alternative and more differential geometric

seiberg witten equations in r4 lie symmetries particular solutions integrability it is shown that the 11 parameter automorphism group of lie algebra of four dimensional euclidean group is a maximal lie symmetries group of the seiberg witten equations in r4 particular explicit solutions which are invariant under so 3 subgroups of the maximal lie symmetries group are constructed it is established that seiberg witten equations do not possess the painleve property nevertheless so 3 invariant solutions obtained are turned to admit a characteristic singularity structure