ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

torsion points on families of squares of elliptic curves in a recent paper we proved that there are at most finitely many complex numbers 0 1 such that the points 2 sqrt 2 2 lambda and 3 sqrt 6 3 lambda are both torsion on the elliptic curve defined by y 2 x x 1 x here we give a generalization to any two points with coordinates algebraic over the field q and even over c this implies a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes

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

periodic solutions of weakly coupled superlinear systems abstract by the use of a higher dimensional version of the poincare birkhoff theorem we are able to generalize a result of jacobowitz 15 and hartman 14 thus proving the existence of infinitely many periodic solutions for a weakly coupled superlinear system

on superquadratic periodic systems with indefinite linear part abstract we consider a second order periodic system with an indefinite linear part and a potential function which is superquadratic but does not satisfy the ar condition using morse critical groups we show that the system has at least one nontrivial solution

outage probability in η μ η μ and κ μ η μ interference limited scenarios in this paper exact closed form expressions are derived for the outage probability op in scenarios where the signal of interest soi experiences or fading the interfering signals are subject to fading and the background noise can be neglected with the only assumption that the parameter is a positive integer number for the interfering signals the derived expressions when the soi is either or distributed are given in elementary terms for maximal ratio combining mrc with independent branches if the pre combining soi power signals are distributed with their corresponding parameters being positive integer or half integer numbers the previous analysis is extended for spatially correlated mrc at the receiver

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

laplace transform of product of generalized marcum q bessel i and power functions with applications integral transforms of special functions often occur in different research and practical areas analyzing the kappa mu fading distribution also known as the generalized rician distribution we find out that the assessment of a few different performance metrics involves the evaluation of an improper integral expressed as the laplace transform of product of marcum q bessel i and power functions we evaluate this integral in a closed form and apply the derived results to assess the probability of detection of unknown signals over kappa mu fading channels as well as to evaluate the outage probability under interference limited kappa mu fading scenarios with kappa mu faded co channel interference

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

periodic solutions of classical hamiltonian systems without palais smale condition abstract in this paper we study the existence of periodic solutions for classical hamiltonian systems without the palais smale condition we prove that the information of the potential function contained in a finite domain is sufficient for the existence of periodic solutions moreover we establish the existence of infinitely many periodic solutions without any symmetric condition on the potential function v

multiple solutions of perturbed superquadratic second order hamiltonian systems abstract in this paper the author proves the existence of infinitely many distinct t periodic solutions of the perturbed second order hamiltonian systems q v q f t under the condition that v is c and superquadratic via minimax methods we also obtain similar results for general nonautonomous second order hamiltonian systems and perturbed lagrangian systems keywords invariance asymmetry a priori growth estimates multiple periodic solutions

periodic solutions of autonomous newtonian systems partially asymptotically linear at infinity abstract the goal of this article is to study the existence continuation and global bifurcation of autonomous newtonian systems partially asymptotically linear at infinity the main idea is to prove a kind of splitting lemma and apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 16

periodic solutions of superquadratic hamiltonian systems with bounded forcing terms on etudie l existence et la multiplicite des solutions periodiques pour le systeme hamiltonien z z t z ou z r r 2n z dz dt j o j j o j etant la matrice identite dans r n

on the classical dynamics of strongly driven anharmonic oscillators abstract we investigate the dynamics of periodically driven anharmonic oscillators in particular we consider values of the coupling strength which are orders of magnitude higher than those required for the overlap of primary resonances we observe a division of phase space into a regular and a stochastic region both regions are separated by a sharp chaos border which sets an upper limit to the stochastic heating of particles its dependence on the coupling strength is studied we construct perpetual adiabatic invariants governing regular motion a bifurcation mechanism leading to the annihilation of resonances is explained

periodic solutions of large norm of hamiltonian systems abstract hamiltonian systems of ordinary differential equations are studied the only assumption made on the hamiltonian is appropriately rapid growth at infinity it is proved that for any given period there is an unbounded sequence of periodic solutions of the system having the given period

periodic solutions for a class of superquadratic hamiltonian systems abstract we prove the existence of nontrivial critical points for a class of superquadratic nonautonomous second order hamiltonian systems by applying condition c to critical point theory and some new solvability conditions of nontrivial periodic solutions are obtained

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

root numbers and ranks in positive characteristic abstract for a global field k and an elliptic curve e over k t silverman s specialization theorem implies rank e k t rank e t k for all but finitely many t p 1 k if this inequality is strict for all but finitely many t the elliptic curve e is said to have elevated rank all known examples of elevated rank for k q rest on the parity conjecture for elliptic curves over q and the examples are all isotrivial some additional standard conjectures over q imply that there does not exist a non isotrivial elliptic curve over q t with elevated rank in positive characteristic an analogue of one of these additional conjectures is false inspired by this for the rational function field k u over any finite field with characteristic 2 we construct an explicit 2 parameter family e c d of non isotrivial elliptic curves over k t depending on arbitrary c d such that under the parity conjecture each e c d has elevated rank

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

periodic solutions of semilinear duffing equations with impulsive effects abstract in this paper we are concerned with the existence of periodic solutions for semilinear duffing equations with impulsive effects firstly for its autonomous equation any motion of the solution is same as the motion of the corresponding equation without impulses until it meets the first impulse time under the influence of impulses these two motions are likely to be quite different we introduce some reasonable assumptions on the impulsive functions to control these differences such that the information valid for the equation without impulses can always be used for the impulsive one basing on poincare birkhoff twist theorem we prove the existence of infinitely many periodic solutions secondly as for the nonautonomous equation where the autonomous case is taken as an auxiliary one we find the relation between the solutions of these two equations and then obtain the existence of infinitely many periodic solutions also by poincare birkhoff twist theorem lastly an example with special impulses satisfying the above assumptions is given

a continuation approach to superlinear periodic boundary value problems this paper deals with the problem of the existence of t periodic solutions for the first order differential system x i t x 1 1 where f 0 t x r r is a caratheodory function in what follows we prove some results for the solvability of the periodic bvp in the case when the dimension of the space is even such a limitation is motivated by our interest in applications to the second order equation d g t u u o 1 2 which takes the form of 1 1 when it is written as the equivalent system u v 0 g t u v 1 3

periodic and subharmonic solutions for a class of superquadratic second order hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic second order hamiltonian systems by the minimax methods in critical point theory especially a local linking theorem due to luan and mao and the generalized mountain pass theorem due to rabinowitz

infinitely many periodic solutions for second order hamiltonian systems in this paper we study the existence of infinitely many periodic solutions for second order hamiltonian systems u uv t u 0 where v t u is either asymptotically quadratic or superquadratic as u

on the bolza problem abstract we consider the following boundary value problem x v x 0 x 0 x 0 x t x 1 where t 0 x 0 x 1 r n are given we prove that this problem has infinitely many solutions provided v is even and superquadratic

a bahri lions theorem revisited abstract in 1988 a bahri and p l lions a bahri p l lions morse index of some min max critical points i application to multiplicity results comm pure appl math 41 1988 1027 1037 studied the following elliptic problem u u p 2 u f x u u h 0 1 where is a bounded smooth domain of r n 2 p 2 n 2 n 2 and f x u is not assumed to be odd in u they proved the existence of infinitely many solutions under an appropriate growth restriction on f in the present paper we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign changing solutions in addition we derive an estimate on the number of their nodal domains we also deal with the corresponding fourth order equation 2 u u p 2 u f x u with both dirichlet and navier boundary conditions as well as with strongly coupled elliptic systems

periodic solutions of duffing s equations with superquadratic potential abstract this paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear duffing s equation x g x p t x x where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g x sign x as x among other results we prove the existence of infinitely many harmonic and subharmonic solutions of any order p p t and if the potential g x of g x satisfies certain conditions of superquadratic growth at the new existence results can be applied to situations in which the more classical superlinear growth condition g x x as x is not satisfied in this manner various preceding theorems are improved and sharpened see the introduction for more details proofs are based on a generalized version of the poincare birkhoff twist theorem due to w ding

special points on fibered powers of elliptic surfaces consider a fibered power of an elliptic surface we characterize its subvarieties that contain a zariski dense set of points that are torsion points in fibers with complex multiplication this result can be viewed as a mix of the manin mumford and andre oort conjecture and is related to a conjecture of pink the main technical tool is a new height inequality we also use it to give another proof of a case of gubler s result on the bogomolov conjecture over function fields

error vector magnitude analysis in generalized fading with co channel interference in this paper we derive the data aided error vector magnitude evm in an interference limited system when both the desired channel and interferers experience independent and nonidentically distributed kappa mu shadowed fading then it is analytically shown that the evm is equal to the square root of number of interferers when the desired channel and interferers do not experience fading furthermore the evm is derived in the presence of interference and noise when the desired channel experiences kappa mu shadowed fading and the interferers experience independent and identical nakagami fading moreover using the properties of the special functions the derived evm expressions are also simplified for various special cases

existence of simple jacobian varieties of genus g with rank at least 4g 5 we prove the existence of infinitely many curves of genus g over the rational number field which have absolutely simple jacobian varieties with mordell weil rank at least 4 g 5

periodic and subharmonic solutions of second order hamiltonian systems abstract some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non autonomous second order hamiltonian systems by the minimax methods in critical point theory

existence and multiplicity of periodic solutions for a class of second order hamiltonian systems we study the existence and multiplicity of periodic solutions of the following second order hamiltonian system email protected t email protected k t u t w t u t 0 the existence of a nontrivial periodic solution is obtained when w is asymptotically linear at infinity and the existence of infinitely many periodic solutions is also obtained when w is superlinear

infinitely many periodic solutions for a second order nonautonomous system in this paper we are interested in multiplicity results for a second order nonautonomous system infinitely many solutions follow from a recent variational principle by b ricceri

morse indices of critical manifolds generated by min max methods with compact lie group actions and applications we derive some upper and lower bounds for morse indices of critical manifolds generated by min max principles for functionals invariant under a general compact lie group or a finite group action the results generalize the similar results in the nonequivariant no group action case in doing so we also generalize the extension theorem of dugundji type in the nonequivariant case to the equivariant group action case as an application we obtain a precise growth estimate for the whole sequence of critical values given by the min max procedure for some superquadratic second order differential equations it is well known that this growth estimate is crucial in showing the existence of multiple solutions of some superquadratic perturbed hamiltonian systems and equations

on the outage probability of optimum combining and maximal ratio combining schemes in an interference limited rice fading channel in this letter we study the performance of optimum combining oc and maximal ratio combining mrc diversity schemes in wireless digital communication systems with rice fading and cochannel interference new simplified closed form expressions are derived for the probability density function pdf as well as the outage probability for oc and mrc when the desired signal has a line of sight and the interferers are assumed to be scattered and further away from the receiver it is assumed that the interferers have equal powers and their number l is larger than the diversity order m i e l m

borsuk ulam theorems for arbitrary s1 actions and applications abstract an s to the first power version of the borsuk ulam theorem is proved for a situation where fix s to the first power may be nontrivial the proof is accomplished with the aid of a new relative index theory applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an s to the first power symmetry minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by hamilton s equations in these arguments one obtains the solutions of the differential equations as critical points of an associated lagrangian by minimaxing the lagrangian over appropriate classes of sets intersection theorems such as are proved in this paper play a crucial role in this process in addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an s to the first power symmetry group

periodic solutions of second order superquadratic hamiltonian systems with potential changing sign ii abstract we consider the periodic solutions of the second order hamiltonian system x x h t v x with v being positive and superquadratic at infinity h being c 1 2 periodic sign changing and all zeros being simple some existence and multiplicity results of periodic solutions are given

outage probability of mrc with arbitrary power cochannel interferers in nakagami fading we propose a new approach to outage probability analysis of predetection maximal ratio combining mrc diversity reception in nakagami m fading channels we generalize prior work in that we consider l independent cochannel interferers with arbitrary powers and fading parameters as well as the effects of additive white gaussian noise awgn our approach results in a general expression for outage probability under very broad assumptions moreover our approach leads to a closed form expression for outage probability in most cases of interest we also provide numerical results that demonstrate the performance improvement obtained through mrc diversity combining in the presence of cochannel interferers

periodic solutions of partially superquadratic second order hamiltonian systems abstract we consider the periodic solutions of the hamiltonian system x v x with v being superquadratic in some directions at infinity some theorems on periodic solutions for superquadratic v are generalized

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

existence of forced oscillations for some nonlinear differential equations abstract this article studies the existence of t periodic solutions for systems of nonlinear second order ordinary differential equations assuming function v to be superquadratic it is shown that this system possesses infinitely many t periodic solutions the proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial some more general results concerning systems of another type are also presented

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

error vector magnitude analysis of fading simo channels relying on mrc reception we analytically characterize the data aided error vector magnitude evm performance of a single input multiple output simo communication system relying on maximal ratio combining mrc having either independent or correlated branches that are nonidentically distributed in particular exact closed form expressions are derived for the evm in eta text mu fading and kappa w mu shadowed fading channels and these expressions are validated by simulations the derived expressions are expressed in terms of lauricella s function of the fourth kind f d n which can be easily computed furthermore we have simplified the derived expressions for various special cases such as independent and identically distributed branches rayleigh fading nakagami m fading and kappa text mu fading additionally a parametric study of the evm performance of the wireless system is presented

a periodic solution for a second order asymptotically linear hamiltonian system abstract we study the existence of a periodic solution of the following second order hamiltonian system u t f t u t 0 t 0 t where f t x k t x w t x assuming that k satisfies the pinching condition and w is asymptotically linear at infinity the existence of a nontrivial periodic solution is obtained via the mountain pass theorem

a note on divisible points of curves let c be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3 we partially answer a question posed by levin on points on c for which a non trivial power lies again on c our results have connections to zilber s conjecture on intersections with tori and yield to methods arising in transcendence theory and the theory of o minimal structures

on a notion of category depending on a functional part i theory and application to critical point theory abstract we introduce the notion of category depending on a fixed functional f defined on a topological space this notion permits us to obtain a better lower bound to the number of critical points of f than the bound obtained with the lusternik schnirelman category we also introduce the notion of truncated category depending on the functional f which permits us to obtain a lower bound to the number of critical points of unbounded functionals finally we extend our result after having introduced the notions of relative category and limit relative category depending on the functional f

periodic solutions for hamiltonian systems without ambrosetti rabinowitz condition and spectrum 0 abstract in this paper we consider the superquadratic second order hamiltonian system u t a t u t h t u t 0 t r our main results here allow the classical ambrosetti rabinowitz superlinear condition to be replaced by a general superquadratic condition and 0 lies in a gap of b where b d 2 d t 2 a t we will study the ground state periodic solutions for this problem the main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by schechter and zou

on forced vibrations of a semilinear wave equation without assumptions of monotonicity abstract we prove the existence of infinitely many periodic solutions of a forced wave equation with dirichlet boundary conditions for a large class of non monotone nonlinear terms

neron models of jacobians over base schemes of dimension greater than 1 we investigate to what extent the theory of neron models of jacobians and of abel jacobi maps extends to relative curves over base schemes of dimension greater than 1 we give a necessary and sufficient criterion for the existence of a neron model we use this to show that in general neron models do not exist even after making a modification or even alteration of the base on the other hand we show that neron models do exist outside some codimension 2 locus

the alpha eta kappa mu fading model new fundamental results a number of new results aiming at facilitating the use of the alpha eta kappa mu fading model are presented these results include 1 fast convergent with no recursions series representations for the envelope probability density function pdf and cumulative distribution function cdf 2 higher order moments of the envelope 3 moment generating function mgf of the envelope 4 asymptotic behavior of the envelope pdf and also of the envelope cdf 5 a procedure for parameter estimation 6 new closed form expressions for particular cases of the envelope distribution 7 new mathematical identity for the generalized laguerre polynomial and the generalized hypergeometric function and 8 approximate but tight series representation for the phase pdf as application examples the following are given 1 outage probability 2 outage capacity 3 amount of fading and 4 bit error rate for which the asymptotic behavior is also presented

asymptotic maximum order statistic for sir in kappa mu shadowed fading using tools from extreme value theory evt it is proved that when the user signal and the interferer signals undergo independent and non identically distributed i n i d kappa mu shadowed fading the limiting distribution of the maximum of l independent and identically distributed i i d signal to interference ratio sir random variables rvs is a frechet distribution it is observed that this limiting distribution is close to the true distribution of maximum for maximum sir evaluated over moderate l furthermore moments of the maximum rv is shown to converge to the moments of the frechet rv in addition the rate of convergence of the actual distribution of the maximum to the frechet distribution is derived and analyzed for different kappa and mu parameters finally results from the stochastic ordering are used to analyze the variation in the limiting distribution with respect to the variation in source fading parameters these results are then used to derive upper bound for the rate in full array selection fas schemes for the antenna selection and the asymptotic outage probability and the ergodic rate in maximum sum capacity msc scheduling systems

bounded height in pencils of finitely generated subgroups in this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric mordell lang theorem by replacing finiteness by emptyness for the intersection of varieties and subgroups all moving in a pencil except for bounded height values of the parameters and excluding identical relations more precisely an instance of the results is as follows consider the torus scheme g r m c over a curve c defined over q and let be a subgroup scheme generated by finitely many sections satisfying some necessary conditions further let v be any subscheme then there is a bound for the height of the points p c q such that for some which does not generically lie in v p lies in the fiber vp we further offer some direct diophantine applications to illustrate once again that the results implicitly contain information absent from the previous bounds in this context

symplectic transformations and periodic solutions of hamiltonian systems abstract symplectic transformations with a kind of homogeneity are introduced which enable us to give a unified approach to some existence problems on the periodic solutions for the first and second order hamiltonian systems

multiplicity of periodic solutions for a class of second order perturbed hamiltonian systems abstract consider the following second order perturbed hamiltonian systems u t f u u g t u t r u 0 u t u 0 u t 0 t 0 where f u k u w u k w c 2 r n r g c 2 r r n r with g t t u g t u k and w is even but g t u maybe has no parity in u assume that w is superquadratic and g is subquadratic at the infinity and k satisfies pinched condition the existence of infinitely many periodic solutions of the above perturbed systems is proved via the bolle s perturbation method which is a generalization of 21 theorem 1 2 and 3 theorem 2 in some sense

critical point theory for indefinite functionals with symmetries letxbe a hilbert space and c1 x r be strongly indefinite assume in addition that a compact lie groupgacts orthogonally onxand that is invariant in order to find critical points of we develop the limit relative category of fournieret al in the equivariant context we use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem in the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension also other conditions of the classical symmetric mountain pass theorem forg z 2 due to ambrosetti and rabinowitz can be weakened considerably for example we are able to deal with infinite dimensional fixed point spaces the proofs consist of a direct reduction to a relative borsuk ulam type theorem this provides a new proof even for the classical mountain pass theorem the abstract critical point theorems are applied to an elliptic system with dirichlet boundary conditions we only need a weak version of the usual superquadraticity condition the linking theorem can be applied to asymptotically linear hamiltonian systems which are symmetric with respect to a generalized symplectic group action

nonlinear oscillations and boundary value problems for hamiltonian systems we prove the existence of solutions of various boundary value problems for nonautonomous hamiltonian systems with forcing terms begin gathered dot x t h p t x t p t g t hfill dot p t h x t x t p t f t hfill end gathered