on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

spatially periodic instantons nahm transform and moduli this paper establishes that the nahm transform sending spatially periodic instantons instantons on the product of the real line and a three torus to singular monopoles on the dual three torus is indeed a bijection as suggested by the heuristic in the process we show how the nahm transform intertwines to a fourier mukai transform via kobayashi hitchin correspondences we also prove existence and non existence results

existence results for urysohn integral inclusions involving the henstock integral this paper presents two existence results for urysohn integral inclusions in banach spaces the set valued integral involved being the henstock integral in particular the existence of continuous solutions for integral inclusions of volterra and of hammerstein type is obtained our results extend the existence theorems given in literature in the single or set valued case under bochner or pettis integrability hypothesis the compactness of solutions set is also studied

fixed point theorems for set valued maps and existence principles for integral inclusions new fixed point theorems of monch type are presented for set valued maps these theorems are then used to establish general existence principles for hammerstein integral inclusions in banach spaces

solvability of inclusions of hammerstein type abstract we establish a universal rule for solving operator inclusions of hammerstein type in lebesgue bochner spaces with the aid of some recently proven continuation theorem of leray schauder type for the class of so called admissible multimaps examples illustrating the legitimacy of this approach include the initial value problem for perturbation of m accretive multivalued differential equation the nonlocal cauchy problem for semilinear differential inclusion abstract integral inclusion of fredholm and volterra type and the two point boundary value problem for nonlinear evolution inclusion

the stable topology of self dual moduli spaces soit m une 4 variete de riemann compacte connexe orientee et soit g un groupe de lie connexe simple on fixe un entier q m q alors l application inclusion i induit un epimorphisme sur les groupes respectifs d homologie et d homotopie a toutes les dimensions inferieures ou egales a q

algebro geometric analogues of donaldson s polynomials donaldson 5 has introduced polynomials on the second homology of any simply connected closed four manifold m with b odd at least 3 which up to are invariant under orientation preserving diffeomorphisms for every c ibm there is a polynomial 7c m syma h2 s where d c 4c z2 b 1 to define it one considers j c the set which is in a natural way a manifold of gauge equivalence classes of connections on the su 2 bundle on m with c2 c antiself dual with respect to a generic metric on m there is a natural map h2 m h2 j c namely slant product with 88 where p is the universal so 3 principal bundle on m x j c one way of defining 7c m 10 is to construct a compactification of r xc carrying a fundamental class xc one proves that extends to a map h2 m h2 xc then one defines

the atiyah jones conjecture for rational surfaces we show that if the atiyah jones conjecture holds for a surface x then it also holds for the blow up of x at a point since the conjecture is known to hold for and for ruled surfaces it follows that the conjecture is true for all rational surfaces

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity in this article new properties of variable exponent lebesgue and sobolev spaces are examined using these properties we prove the existence of the solution of some parabolic variational inequality

the topology of instanton moduli spaces i the atiyah jones conjecture in this paper we study the global geometry and topology of the moduli spaces of based su 2 instantons over the 4 sphere s4 these instanton moduli spaces have a rich history and have been analyzed from many points of view originally these spaces which we denote by mk were defined as solution spaces modulo gauge equivalence to certain partial differential equations namely the self duality equations associated to the yang mills functional in su 2 gauge theory they have been successfully studied from this point of view by taubes t1 t2 uhlenbeck u and others an important alternative approach was initiated by ward w who related instantons to certain holomorphic bundles on cp3 and was continued by atiyah and ward in aw this allowed the classification of instantons on 4 in terms of quaternionic linear algebra by atiyah drinfeld hitchin and manin adhm this holomorphic approach was further extended by donaldson d who showed that these bundles were determined by their restriction to a cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in c2r2 hurtubise hul then exploited this fact to study the moduli spaces mk as did atiyah a to show that mk arise naturally in the theory of holomorphic maps into loop groups this latter approach was continued by gravesen g atiyah and jones aj obtained the first results and formulated the foundational questions on the global topology of these moduli spaces recall that an element of mk is a based gauge equivalence class of a connection on the principal su 2 bundle over 4 denoted by pk with second chern class k satisfying the self duality equations there is a natural forgetful map to the based equivalence classes of all connections in pk atiyah and jones aj showed that this target space which we denote by bk is homotopy equivalent

family gromov witten invariants for kahler surfaces we use analytic methods to define family gromov witten invariants for kahler surfaces we prove that these are well defined invariants of the deformation class of the kahler structure gromov witten invariants are counts of holomorphic curves in a symplectic manifold x to define them using the analytic approach one chooses an almost complex structure j compatible with the symplectic structure and considers the set of maps f x from riemann surfaces which satisfy the nonlinear elliptic j holomorphic map equation

existence criteria for operator inclusions in abstract spaces fixed point theory for single and multivalued maps is used to obtain general existence principles for nonlinear operator equations and inclusions in abstract spaces

superconformal simple type and witten s conjecture abstract let x be a smooth closed connected orientable four manifold with b 1 x 0 and b x 3 and odd we show that if x has seiberg witten simple type then the so 3 monopole cobordism formula of 14 implies witten s conjecture relating the donaldson and seiberg witten invariants

existence and multiplicity of solutions for a neumann problem involving the p x laplace operator abstract in this paper we study the following nonlinear neumann boundary value problem div u p x 2 u u p x 2 u f x u x t r u v 0 x t r where r n is a bounded domain with smooth boundary u v is the outer unit normal derivative on 0 is a real number p is a continuous function on with inf x p x 1 f r r is a continuous function using the three critical point theorem due to ricceri under the appropriate assumptions on f we establish the existence of at least three solutions of this problem some known results are generalized

on the diffeomorphism types of certain algebraic surfaces i l invariant de donaldson la geometrie de certaines surfaces rationnelles surfaces dolgachev generalites sur des formes de type 1 n la classe canonique d une chambre et un coin le cone de kahler d une bonne surface generique une etude de p cellules les super p cellules

fixed points of discontinuous multivalued increasing operators in banach spaces with applications by using relatively weakly compactness conditions we obtain existence theorems of fixed points for discontinuous multivalued increasing operators in banach spaces as an application we utilize the results obtained in section 2 to lead to the existence principles for integral inclusions of hammerstein type multivalued maps

the nekrasov conjecture for toric surfaces the nekrasov conjecture predicts a relation between the partition function for n 2 supersymmetric yang mills theory and the seiberg witten prepotential for instantons on mathbb r 4 the conjecture was proved independently and using different methods by nekrasov okounkov and nakajima yoshioka we prove a generalized version of the conjecture for instantons on noncompact toric surfaces

chern simons functional and the homology cobordism group for each integral homology sphere y we construct a function y on the set of integers we establish that y depends only on the homology cobordism class of y and that it recovers the froyshov invariant we state a relation between y and fintushel and stern s r invariant we show that the value of y at each integer is related to the critical values of the chern simons functional and we give some topological applications of y in particular we show that if y is trivial then there is no simply connected homology cobordism from y to itself

heegaard floer correction terms and rational genus bounds given an element in the first homology of a rational homology 3 sphere y one can consider the minimal rational genus of all knots in this homology class this defines a function on h1 y z which was introduced by turaev as an analogue of thurston norm we will give a lower bound for this function using the correction terms in heegaard floer homology as a corollary we show that floer simple knots in l spaces are genus minimizers in their homology classes hence answer questions of turaev and rasmussen about genus minimizers in lens spaces

computations of instanton invariants using donaldson floer theory we compute the donaldson su 2 invariants of the double cover of 2 branched over a smooth algebraic curve of degree eight from this we deduce a formula for the relative invariants of the blow up of the gompf nucleusn2 and we show how this gives a blow up formula for a class of 4 manifolds which includes essentially all the simply connected 4 manifolds known to have big diffeomorphism group we apply the result on the nucleus also to prove a formula for the invariants of minimal simply connected elliptic surfaces which reduces the computation to the case of geometric genus one in particular we compute all the donaldson invariants of minimal simply connected elliptic surfaces without multiple fibers our main tool is donaldson floer theory

an obstruction bundle relating gromov witten invariants of curves and kahler surfaces in a previous paper the authors defined symplectic local gromov witten invariants associated to spin curves and showed that the gw invariants of a k ahler surface x with p g 0 are a sum of such local gw invariants this paper describes how the local gw invariants arise from an obstruction bundle in the sense of taubes over the space of stable maps into curves together with the results of our earlier paper this reduces the calculation of the gw invariants of elliptic and general type complex surfaces to computations in the gw theory of curves with additional classes the euler classes of the real obstruction bundles

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

energy gap for yang mills connections i four dimensional closed riemannian manifolds we extend an l2l2 energy gap result due to min oo 28 theorem 2 and parker 30 proposition 2 2 for yang mills connections on principal g bundles p over closed connected four dimensional oriented smooth manifolds x from the case of positive riemannian metrics to the more general case of good riemannian metrics including metrics that are generic and where the topologies of p and x obey certain mild conditions and the compact lie group g is su 2 su 2 or so 3 so 3

rank two bundles on the blow up ofc2 in this paper we study holomorphic rank two vector bundles on the blow up ofc2with vanishing chern class the restriction of such a bundle over the exceptional divisor splits as o j o j for some integerj we denote by mjthe moduli space of holomorphic bundles on the blow up ofc2whose restriction to the exceptional divisor is o j o j we prove that mjis generically a complex projective space of dimension 2j 3

fixed point theory for set valued mappings between topological vector spaces having sufficiently many linear functions abstract we present new fixed point theorems for multivalued maps on hausdorff topological spaces having sufficiently many linear functionals in addition a continuation theory is presented for the maps discussed

study of the solutions to a model porous medium equation with variable exponent of nonlinearity abstract the authors of this paper study the dirichlet problem of the following equation u t div u x t u f u p x t 1 u the existence and uniqueness of weak solutions are proven also the properties of the solutions are studied which include the property of finite speed of propagation of disturbances localization property and the property of vanishing at a finite time etc

uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity abstract in this article new properties of variable exponent lebesque and sobolev spaces were examined using these properties we prove that the solution of some parabolic variational inequality is unique with the given conditions

supersymmetric yang mills theory on a four manifold by exploiting standard facts about n 1 and n 2 supersymmetric yang mills theory the donaldson invariants of four manifolds that admit a kahler metric can be computed the results are in agreement with available mathematical computations and provide a powerful check on the standard claims about supersymmetric yang mills theory

lyapunov opial and beesack inequalities for one dimensional p t laplacian equations we generalize the classical lyapunov opial and beesack inequalities for one dimensional differential equations to nonstandard growth p t laplacian

instantons and geometric invariant theory we show that the yang mills instantons can be described in terms of certain holomorphic bundles on the projective plane the proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries

a nonlinear periodic averaging principle abstract this paper is devoted to a nonlinear averaging principle for periodic solutions of a class of second order inclusions in addition an existence theorem for periodic solutions of such inclusions is established this work which complements the abstract nonlinear averaging principle worked out in couchouron and kamenski nonlin anal 42 2000 1101 makes a synthesis of the methods contained in couchouron and kamenski and couchouron and precup electron j differential equations 4 2002 1 and represents a nonvariational topological approach for boundary values problems

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

monotonic solutions of a quadratic integral equation of volterra type abstract we study a nonlinear quadratic integral equation of volterra type in the banach space of real functions defined and continuous on a bounded and closed interval with the help of a suitable measure of noncompactness we show that the mentioned integral equation has monotonic solutions

multivalued fixed point theorems in terms of weak topology and measure of weak noncompactness abstract we present several fixed point theorems for multimaps in banach spaces which in turn are multivalued versions of the monch daher darbo and sadovskii theorems the main assumptions of our results are formulated in terms of weak topology and an axiomatic definition of measure of weak noncompactness our theorems extend in a broad sense some new and classical results moreover a sadovskii type theorem stated in 2011 is improved

quantum cohomology of the moduli space of stable bundles over a riemann surface we determine the quantum cohomology of the moduli space m of odd degree rank two stable vector bundles over a riemann surface of genus g 1 this work together with 10 complete the proof of the existence of an isomorphism qh m hf s1

the blowup formula for donaldson invariants k k which is calculated in 4 below this formula has been the target of much recent work the abstract fact that there exists such a formula which is independent of x was first proved by c taubes using techniques of t j bryan b and p ozsvath o have independently calculated the coefficients throughb10 x quite recently j morgan and ozsvath have announced a scheme which can recursively compute all of the bk x the special case of the blowup formula for manifolds of simple type see 5 below was first given by p kronheimer and t mrowka however none of the techniques in these cases approach the simplicity of that offered here before presenting the formula we shall first establish notation for the donaldson in variants of a simply connected 4 manifold x with b 1 and odd the hypothesis of simple connectivity is not necessary but makes the exposition easier an orientation of x together with an orientation of h 2 x r is called a homology orientation of x such a homology orientation determines the su 2 donaldson invariant a linear function d dx a x sym h0 x h2 x r

the genus minimizing property of algebraic curves a viable and still unproved conjecture states that if x is a smooth algebraic surface and c is a smooth algebraic curve in x then c realizes the smallest possible genus amongst all smoothly embedded 2 manifolds in its homology class a proof is announced here for this conjecture for a large class of surfaces x under the assumption that the normal bundle of c has positive degree

three applications of instanton numbers we use instanton numbers to i stratify moduli of vector bundles ii calculate relative homology of moduli spaces and iii distinguish curve singularities

a model porous medium equation with variable exponent of nonlinearity existence uniqueness and localization properties of solutions abstract we study the localization properties of weak solutions to the dirichlet problem for the degenerate parabolic equation u t div u x t u f with variable exponent of nonlinearity we prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent x t sufficient for the existence of such properties as finite speed of propagation of disturbances the waiting time effect finite time vanishing of the solution it is shown that the solution may instinct in a finite time even if x 0 in the problem domain but max 0

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

instantons and jumping lines we study the behaviour under deformation of holomorphic bundles of rank 2 over 1 this is then applied to the description of the moduli space tilde m n of framed su 2 instantons of chargen tilde m n is shown to map to n with equidimensional fibers we use this to provide a stratification of tilde m n and compute the strata explicitly to codimension 4 this then yields 1 tilde m n 2 and for the standard moduli spacem n 1 m n 0 forn odd 2 forn even

theory and applications of fuzzy volterra integral equations formulations of fuzzy integral equations in terms of the aumann integral do not reflect the behavior of corresponding crisp models consequently they are ill adapted to describe physical phenomena even when vagueness and uncertainty are present a similar situation for fuzzy odes has been obviated by interpretation in terms of families of differential inclusions the paper extends this formalism to fuzzy integral equations and shows that the resulting solution sets and attainability sets are fuzzy and far better descriptions of uncertain models involving integral equations the investigation is restricted to volterra type equations with mildly restrictive conditions but the methods are capable of extensive generalization to other types and more general assumptions the results are illustrated by integral equations relating to control models with fuzzy uncertainties

higher type adjunction inequalities for donaldson invariants we prove new adjunction inequalities for embedded surfaces in four manifolds with non negative self intersection number using the donaldson invariants these formulas are completely analogous to the ones obtained by ozsvath and szabo using the seiberg witten invariants to prove these relations we give a fairly explicit description of the structure of the fukaya floer homology of a surface times a circle as an aside we also relate the floer homology of a surface times a circle with the cohomology of some symmetric products of the surface

global existence for nonlinear operator inclusions abstract we use fixed point theory for multivalued maps to obtain general existence principles for nonlinear operator inclusions our maps are either of upper semicontinuous or lower semicontinuous type

the so 3 monopole cobordism and superconformal simple type abstract we show that the so 3 monopole cobordism formula from 15 implies that all closed oriented smooth four manifolds with b 1 0 and b 3 and odd with seiberg witten simple type satisfy the superconformal simple type condition defined by marino moore and peradze 28 27 this implies the lower bound conjectured by fintushel and stern 18 on the number of seiberg witten basic classes in terms of topological data

on the homology of su n instantons in this paper we study the homology of the moduli spaces of instantons associated to principal su n bundles over the four sphere this is accomplished by exploiting an iterated loop space structure implicit in the disjoint union of all moduli spaces associated to a fixed su n with arbitrary instanton number and relating these spaces to the known homology structure of the four fold loop space on bsu n moduli spaces of instantons self dual connections with respect to a conformal class of metrics associated to principal g bundles pk g p over s4 have proven to be basic objects in modern geometry here g is any simple compact lie group and k is the integer that classifies the bundle p and is referred to as the instanton number we denote these moduli spaces by at g there is a natural inclusion 0 1 ik g z g qk bg induced by forgetting the self duality condition where we have identified the moduli space of based gauge equivalence classes of all connections on p with the kth component of the four fold loop space q 4bg 5 in a fundamental paper atiyah and jones 5 studied the inclusion 0 1 for g su 2 when s4 has its standard conformally flat metric and posed several fundamental questions in a series of papers taubes cf 24 26 proved several basic existence theorems stability theorems in terms of k and provided a basis framework to describe how the topology of ak changes as k increases taubes work is much more general than we describe here in that he not only studied general lie groups g but also replaced 54 by an arbitrary compact closed riemannian four manifold with arbitrary conformal class of metric in 7 it was observed that over the four sphere but with g sp n the disjoint union of k over all k form a homotopy c4 space and that iterated loop space techniques may be profitably used to study h xk for individual k certain computational results may be obtained immediately from these received by the editors october 17 1989 1980 mathematics subject classification 1985 revision primary 53c55 58e05 secondary 53c57 55p35

solvability of volterra stieltjes operator integral equations and their applications abstract we investigate a class of operator integral equations of volterra stieltjes type and we study the solvability of those equations in the space of continuous functions equations in question create a generalization of numerous integral equations considered in nonlinear analysis the main tool used in our considerations is the technique associated with measures of noncompactness we show the applicability of our existence result in the study of a few integral equations of volterra type

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4