an orlicz space approach to superlinear elliptic systems in this paper we study superlinear elliptic systems in hamiltonian form using an orlicz space setting we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

existence and multiplicity of solutions for asymptotically hamiltonian elliptic systems in rn abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain solutions for the above system

solutions of a nonlinear dirac equation with external fields we study the stationary dirac equation ic hbar sum 3 k 1 alpha k partial k u mc 2 beta u m x u r u x u where m x is a matrix potential describing the external field and r x u stands for an asymptotically quadratic nonlinearity modeling various types of interaction without any periodicity assumption for fixed our discussion includes the coulomb potential as a special case and for the semiclassical situation 0 we handle the scalar fields we obtain existence and multiplicity results of stationary solutions via critical point theory

the uniqueness of polynomial crystallographic actions let gamma be a polycyclic by finite group it is proved in 8 that gamma admits a polynomial action of bounded degree on mathbb r n which is properly discontinuous and such that the quotient gamma backslash mathbb r n is compact we prove here that such an action is unique up to conjugation by a polynomial transformation of mathbb r n

on the existence of positive solutions of a perturbed hamiltonian system in rn using the legendre fenchel transformation and the mountain pass theorem due to ambrosetti and rabinowitz we establish an existence result for perturbations of periodic and asymptotically periodic semilinear hamiltonian systems of the type pw u u w2 x v p 1vinrn v v w1 x u q 1uinrn u x v x 0as x u 0 v 0inrn n 2 here the numbers p q 1 are below the critical hyperbola if n 3 that is they satisfy 1 p 1 1 q 1 n 2 n while no additional restrictions on p and q are required if n 2 the functions wi i 1 2 are bounded positive continuous functions

archimedean superrigidity of solvables arithmetic groups abstract let g be a connected solvable linear algebraic group over a number fieldk letsbe a finite set of places ofkthat contains all the infinite places and let o s be the ring ofs integers ofk we define a certain closed subgroup formula of g s v s g kvthat contains g o s and prove that g o s is a superrigid lattice in formula by which we mean that finite dimensional representations g o s gln r more or less extend to representations of formula the subgroup formula may be a proper subgroup of g sfor only two reasons first it is well known that g o s is not a lattice in g sif g has nontrivialk characters so one passes to a certain subgroup g s 1 second g o s may fail to be zariski dense in g s 1 in an appropriate sense in this sense the subgroup formula is the zariski closure of g o s in g s 1 furthermore we note that a superrigidity theorem for many nonsolvables arithmetic groups can be proved by combining our main theorem with the margulis superrigidity theorem

semiclassical states for nonlinear schrodinger equations with sign changing potentials we establish the existence and multiplicity of semiclassical bound states of the following nonlinear schrodinger equation e2 u v x u g x u for x rn u x 0as x where v changes sign and g is superlinear with critical or supercritical growth as u

flat compact hermite lorentz manifolds in dimension 4 we give a classification up to finite cover of flat compact complete hermite lorentz manifolds up to complex dimension 4

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

stochastic machine minimization with constant service times we consider the nonpreemptive scheduling of n 1 jobs on identical parallel machines with job running times all given by some constant the objective is to minimize the expected number of machines needed throughout a schedule subject to the following waiting time constraints at time 0 a timer with random initial value w is started after time w all jobs must either be finished or running on a machine the value of w is not known in advance but its distribution is made available to the scheduler in general if job running times are random there might be situations in which after running jobs on a given number of machines it would be desirable to activate new machines because the remaining times of unfinished jobs are stochastically too large compared to the time remaining on the timer a principal result of this paper is that randomness of job running times is essential to such situations if running times are deterministic then irrespective of the distribution of w an optimal policy initially assigns jobs to some number of machines kn which depends on the distribution of w and maintains that number while there are jobs available until the timer has expired at which point any remaining as yet unscheduled jobs are assigned to new unused machines this paper also analyzes the dependence of kn and the minimal expected cost on the parameters of the timer distribution when w is either exponentially distributed or uniformly distributed on a finite interval the determination of kn turns out to have an intriguing number theoretic flavor

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

on the zariski closure of the linear part of a properly discontinuous group of affine transformations let be a subgroup of the group of affine transformations of the affine space r 2n 1 suppose acts properly discontinuously on r 2n 1 the paper deals with the question which subgroups of gl 2n 1 r occur as zariski closure of the linear part of such a group the two main results of the paper say that so n 1 n does occur as of such a group if n is odd but does not if n is even

homoclinic orbits for a nonperiodic hamiltonian system abstract in this paper we prove the existence and multiplicity of homoclinic orbits for first order hamiltonian systems of the form z j h z t z where h z is asymptotically linear at and is not assumed to be periodic

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

decay symmetry and existence of solutions of semilinear elliptic systems our utmost aim was originally to establish the existence of solutions of system however in this effort we got sidetracked to consider other questions in this way we come to interesting results on asymptotic behavior of solutions on symmetry properties of such solutions and the existence of ground states

the processor minimization problem with independent waiting time constraints abstract this paper studies a dual of classical stochastic scheduling of parallel processor systems the given data are n jobs with running times t 1 t n and waiting time constraints w 1 w n these sequences are independent and each consists of i i d random variables although the t i and w i are not known in advance they are known to be samples from exponential distributions with given parameters scheduling policies are nonpreemptive and have the option for each i of making the i th job wait during 0 w i however by the random deadline w i the i th job must be assigned to a processor if it is still waiting in this paper we find a policy that minimizes the expected number of processors used among all policies of the above type we also estimate the expected number of processors used under this policy there appears to be no simple exact formula for this quantity so we turn to an asymptotic analysis based on a continuous approximation

the radiance obstruction and parallel forms on affine manifolds a manifold m is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine when they are locally linear m is called radiant the obstruction to radiance is a one dimensional class cm with coefficients in the flat tangent bundle of m exterior powers of cm give information on the existence of parallel forms on m especially parallel volume forms as applications various kinds of restrictions are found on the holonomy and topology of compact affine manifolds introduction an affine manifold m is a manifold with a distinguished maximal atlas of charts all of whose coordinate changes are locally affine on such a manifold there is an intrinsic notion of a parallel tensor one whose components in any affine chart are constants more generally there is the notion of a polynomial tensor field of given degree in 1962 l markus conjectured in mk that a compact orientable affine n dimensional manifold has parallel volume form if and only if it is complete meaning that its universal covering is affinely isomorphic to euclidean n space rn the problem of constructing a parallel volume form determines an n dimensional twisted real cohomology class originally studied by j smillie sm2 in this paper we express this class as the nth exterior power of a twisted one dimensional real class which we call the radiance obstruction cm by computing cm in various cohomology theories cech singular de rham and others we are able to exploit cm in several ways to yield more information on the structure of affine manifolds some of these results will appear in a subsequent paper gh3 the basic tool used is a formula proved in 2 6 which expresses the cohomology class of a parallel exterior k form in terms of the k th exterior power of the radiance obstruction akcvf min 2 7 the special case of a parallel volume form on a compact n dimensional manifold m is examined the existence of such a form implies that ancm 0 in 2 8 we show that the affine holonomy group f of such an m cannot preserve a proper affine subspace in gh3 this result will be improved by showing that r preserves no proper semialgebraic subset of rn in 2 10 we show that there tend to be plenty of parallel forms on a compact affine manifold with nilpotent affine holonomy received by the editors may 26 1983 1980 mathematics subject classification primary 57r99 53c05 secondary 53c10 55r25 4c1984 amencan mathematical society 0002 9947 84 1 00 25 per page

on a periodic schrodinger equation with nonlocal superlinear part we consider the choquard pekar equation open image in new window and focus on the case of periodic potential v for a large class of even functions w we show existence and multiplicity of solutions essentially the conditions are that 0 is not in the spectrum of the linear part v and that w does not change sign our results carry over to more general nonlinear terms in arbitrary space dimension n 2

optimal stochastic allocation of machines under waiting time constraints the nonpreemptive scheduling of n geqslant 1 stochastic jobs is considered to minimize the expected number of parallel machines needed to meet given waiting time constraints the number of machines available is unlimited the running times of the jobs are denoted t 1 ldots t n and are taken to be independent samples of an exponentially distributed random variable t with mean 1 job waiting times are to be bounded stochastically by a nonnegative random variable w independent of t 1 ldots t n at time zero a timer is started with an initial value w and job scheduling begins when the timer expires all jobs still waiting for a machine are assigned to available machines only the distributions of w and the job durations t 1 ldots t n are known to the scheduler in advance a scheduling policy is defined which is proven to be optimal when w has an exponential distribution and is asymptotically optimal as n to infty when w is a constant hard deadline in the exponential case

existence and multiplicity of semiclassical solutions for asymptotically hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system e 2 u v x u h u x u v in r n e 2 v v x v h v x u v in r n u x 0 and v x 0 as x where e 0 is a small parameter and the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain the existence of multiple semiclassical solutions for the above system

the existence of solutions for superquadratic hamiltonian elliptic systems on rn abstract this paper is concerned with the following hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where v x c r n r and h x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we obtain infinitely many solutions for the above system

existence and multiplicity of solutions for a non periodic schrodinger equation abstract this paper is concerned with the non periodic schrodinger equation u v x u f x u x r n u x 0 as x under a general spectral assumption the existence and multiplicity of solutions are obtained for asymptotically linear nonlinearity via variational methods

existence of solutions for nonperiodic superquadratic hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u r v x u v in r n v v x v r u x u v in r n u x 0 and v x 0 as x where r x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we prove the existence of solution for the above system

the fundamental group of a compact flat lorentz space form is virtually polycyclic a flat lorentz space form is a geodesically complete lorentzian manifold of zero curvature it is well known see auslander markus 3 that such a space m may be represented as a quotient r where r is an dimensional minkowski space n equals the dimension of m and is a group of lorentz isometries acting properly discontinuously and freely on r in particular the universal covering of m is isometric to r and the fundamental group m is isomorphic to theorem let m be a compact flat lorentz space form then m is virtually poly cyclic recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups this result affirms a conjecture of milnor 13 in a special case for discussion of this conjecture as well as another special case we refer to fried goldman 8 the importance of this result is that it reduces the classification of compact flat lorentz space forms to fairly elementary problems concerning lie algebras and lattices in solvable lie groups we hope to pursue this classification in a future publication for a description of this reduction and the classification in dimension 3 see fried goldman 8 in dimension 4 the classification is worked out in fried 7 one immediate consequence of the structure theory developed in fried goldman 8 1 and kamishima 12 is the following corollary let m be the compact flat lorentz space form then m has a finite covering which is diffeomorphic to a solvmanifold the outline of this paper is as follows in the first section we collect some basic facts about the group e n 1 1 of isometries of dimensional minkowski space in the second section we prove the theorem in the special

on a conjecture by coffman flatto and wright on stochastic machine minimization we investigate a conjecture stated by coffman flatto and wright within the context of a stochastic machine minimization problem with a hard deadline we prove that the conjecture is true

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

strongly indefinite functionals and multiple solutions of elliptic systems we study existence and multiplicity of solutions of the elliptic system left begin array 20 l updelta u h u x u v hfill text in upomega hfill updelta v h v x u v hfill text in upomega quad u x v x 0 quad text on partial upomega hfill end array right where upomega subset mathbb r n n ge 3 is a smooth bounded domain and h in mathcal c 1 overline upomega times mathbb r 2 mathbb r we assume that the nonlinear term h x u v sim left u right p left v right q r x u v text with mathop lim limits left u v right to infty frac r x u v left u right p left v right q 0 where p in 1 2 2 2n n 2 text and q in 1 infty so some supercritical systems are included nontrivial solutions are obtained when h x u v is even in u v we show that the system possesses a sequence of solutions associated with a sequence of positive energies resp negative energies going toward infinity resp zero if p 2 resp p 2 all results are proved using variational methods some new critical point theorems for strongly indefinite functionals are proved