the minimal period problem for nonconvex even second order hamiltonian systems abstract in this paper we study the minimal period problem for even autonomous second order hamiltonian systems defined on r nwithout any convexity assumption by using the variational methods we obtain estimates on the minimal period of the corresponding nonconstant periodic solution of the superquadratic and asymptotically linear hamiltonian systems

bogomolov multipliers and retract rationality for semidirect products abstract let g be a finite group the bogomolov multiplier b 0 g is constructed as an obstruction to the rationality of c v g where g g l v is a faithful representation over c we prove that for any finite groups g 1 and g 2 b 0 g 1 g 2 b 0 g 1 b 0 g 2 under the restriction map if g n g 0 with gcd n g 0 1 then b 0 g b 0 n g 0 b 0 g 0 under the restriction map for any integer n we show that there are non direct product p groups g 1 and g 2 such that b 0 g 1 and b 0 g 2 contain subgroups isomorphic to z p z n and z p n z respectively on the other hand if k is an infinite field and g n g 0 where n is an abelian normal subgroup of exponent e satisfying that e k we will prove that if k g 0 is retract k rational then k g is also retract k rational provided that certain local conditions are satisfied this result generalizes previous results of saltman and jambor 18

boundedness in asymmetric oscillations abstract in this paper we are concerned with the boundedness of all the solutions of the equation x ax bx f t wheref t is a smooth 2 periodic function aandbare positive constants a b

statistical properties of skew endomorphisms with bernoulli base abstract we analyse numerically the statistical properties of a class of mappings of the 2 torus t 2 onto itself whose investigation is suggested by some models of modulated diffusion these transformations can be written as a skew product of the endomorphism bp x px mod 1 p z 1 0 1 on the 1 torus t 1 0 1 and a translation on t 1 under suitable assumptions the skew product can be proven to be mixing w r t the lebesgue measure central limit cl and functional central limit fcl properties are numerically checked for analytic observables the result is remarkable because the mappings show no hyperbolic or quasi hyperbolic structure crucial for the proof of central limit theorem and donsker s invariance principle in all of the dynamical systems where these properties have been established up to now moreover cl and fcl behaviours seem to hold also in the case of purely ergodic endomorphisms and even for observables whose correlations do not decay at infinity

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

on bogomolov multiplier and ш rigidity of groups abstract we show the existence of groups with trivial bogomolov multiplier such that the bogomolov multiplier of a central product of these groups is non trivial we then prove that finite quasisimple groups and freest special p groups are rigid in the sense that all their class preserving automorphisms are inner as an application we show the existence of non nilpotent rigid groups and rigid p groups of odd order having non trivial bogomolov multiplier

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

some applications of symbolic dynamics techniques to toral skew endomorphisms a class of dynamical systems of the 2 torus is considered these systems have the form of a skew product between the bernoulli endomorphism defined on the 1 torus and a translation on itself symbolic dynamics techniques allow one to single out wide classes of observables which show an exponential decay of correlations for some observables the rate of correlation decay can be explicitly estimated

coleman automorphisms of finite groups an automorphism sigma of a finite group g whose restriction to any sylow subgroup equals the restriction of some inner automorphism of g shall be called coleman automorphism named for d b coleman who s important observation from 2 especially shows that such automorphisms occur naturally in the study of the normalizer mathcal n of g in the units mathcal u of the integral group mathbb z g let out col g be the image of these automorphisms in out g we prove that out col g is always an abelian group based on previous work of e c dade who showed that out col g is always nilpotent we prove that if no composition factor of g has order p a fixed prime then out col g is a p prime group if o p g 1 it suffices to assume that no chief factor of g has order p if g is solvable and no chief factor of g rm o 2 g has order 2 then mathcal n g mathcal z where mathcal z is the center of mathcal u this improves an earlier result of s jackowski and z marciniak

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

fractal properties of scattering singularities we discuss properties of the set of scattering singularities in regions of irregular scattering we show how a symbolic organisation of the set can be used to determine the fractal dimension and the scaling function this yields information on the distribution of lyapunov exponents of bounded orbits the specific model studied is the motion of a particle in a plane elastically reflected by three circular discs centred on the corners of an equilateral triangle

roots of unity and unbounded motions of an asymmetric oscillator r 2 z the general questions that we have in mind are under what conditions ona b and f t can we say that 1 has unbounded solutions are there equationsin the class 1 where unbounded and periodic solutions can coexist equation 1 was first considered by dancer in 2 3 and 4 and byfucik in 7 they studied the periodic and dirichlet boundary valueproblems for 1 and looked at this equation as a model of the so called equations with jumping nonlinearities we sum up some of the results in 2 3 consider the family of curves in the a b parameter space 1 a 1 b 2p p 1 2 2 when a b is not in any of these curves then 1 has at least one 2 periodic solution however if a b lies in one of these curves there existfunctions f t for which 1 has no periodic solution more results on theperiodic problem for 1 have been obtained by lazer and mckenna 8 fabry 5 and fabry and fonda 6 the known results on the existenceof periodic solutions already shed some light on the boundedness problem actually the second theorem of massera implies that if 1 has no periodic

unramified brauer groups of finite and infinite groups the bogomolov multiplier is a group theoretical invariant isomorphic to the unramified brauer group of a given quotient space we derive a homological version of the bogomolov multi plier prove a hopf type formula find a five term exact sequence corresponding to this invariant and describe the role of the bogomolov multiplier in the theory of central extensions a new description of the bogomolov multiplier of a nilpotent group of class two is obtained we define the bogomolov multiplier within k theory and show that proving its triviality is equivalent to solving a long standing problem posed by bass an algorithm for computing the bogomolov multiplier is developed

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

some results on solutions of minimal period to superquadratic hamiltonian systems then it is well known that rabinowitz s theorem see 8 guarantees the existence for every t 0 of at least one nontrivial solution to l but he left open the question to find a solution having minimal period t when h is convex some results have been recently obtained about such a problem via duality methods the problem has been completely solved by ambrosetti mancini see 2 7 with a suitable hypothesis on the second derivative of h and by clarke ekeland see 4 replacing hj ha and h5 with a subquadratic hypothesis in another framework rabinowitz determines in 9 a family of solutions to l having minimal periods which diverge in the superquadratic case see theorem 2 34 under the

periodic solutions of some resonant hamiltonian systems abstract based on the galerkin approximation and the morse inequality an existence theorem of periodic solutions of the hamiltonian system j z h t z is given where h t z is asymptotically linear and resonant at infinity existence of periodic solutions of the resonant asymmetric p laplacian equation is also discussed

on a theorem by ekeland and lasry concerning the number of periodic hamiltonian trajectories this note deals with the existence of iz distinct modes of vibration for hamihonian systems with n degrees of freedom on a prescribed energy surface this problem has been studied in a recent remarkable paper by ekeland and lasry 5 who have given an important contribution to the complete extension in the large of the local results of liapunov ri weinstein 111 and moser s the purpose of the present paper is to give a new proof of the result of 5 jointly with a slight improvement

applications of clark duality to periodic solutions with minimal period for discrete hamiltonian systems by making use of clark duality perturbation technique and dual least action principle some results on the existence of subharmonic solutions with minimal period to second order subquadratic discrete hamiltonian systems are obtained

existence of multiple periodic orbits of hamiltonian systems on positive type hypersurfaces in r2n abstract this paper deals with the fixed energy problem of hamiltonian systems in r 2n the main result is an existence theorem of multiple periodic orbits for hamiltonian vector fields on a class of contact type but not necessarily star shaped hypersurfaces this is the first multiplicity result for hypersurfaces that are more general than star shaped ones the main result is achieved by using critical point theory for a special hamiltonian function and central gravity of the proof relies on a delicate construction of such hamiltonian function

a model of modulated diffusion ii numerical results on statistical properties we investigate numerically the statistical properties of a model of modulated diffusion for which we have already computed analytically the diffusion coefficientd our model is constructed by adding a deterministic or random noise to the frequency of an integrable isochronous system we consider in particular the central limit theorem and the invariance principle and we show that they follow wheneverd is positive and for any magnitude of the noise we also investigate the asymptotic distribution in a case whend 0

double resonance with landesman lazer conditions for planar systems of ordinary differential equations abstract we prove the existence of periodic solutions for first order planar systems at resonance the nonlinearity is indeed allowed to interact with two positively homogeneous hamiltonians both at resonance and some kind of landesman lazer conditions are assumed at both sides we are thus able to obtain as particular cases the existence results proposed in the pioneering papers by lazer and leach 1969 27 and by frederickson and lazer 1969 18 our theorem also applies in the case of asymptotically piecewise linear systems and in particular generalizes fabry s results in fabry 1995 10 for scalar equations with double resonance with respect to the dancer fucik spectrum

periodic solutions for a class of non autonomous hamiltonian systems abstract we consider the existence of nontrivial periodic solutions for a superlinear hamiltonian system h j u a t u h t u 0 u r 2 n t r we prove an abstract result on the existence of a critical point for a real valued functional on a hilbert space via a new deformation theorem different from the works in the literature the new deformation theorem is constructed under the cerami type condition instead of palais smale type condition in addition the main assumption here is weaker than the usual ambrosetti rabinowitz type condition 0 h t u u h t u 2 u r 0 this result extends theorems given by li and willem j math anal appl 189 1995 6 32 and li and szulkin j differential equations 112 1994 226 238

solutions of minimal period for a class of nonconvex hamiltonian systems and applications to the fixed energy problem on considere le systeme hamiltonien jz h z ou j x y y x x y r n r n h est non convexe a un bon comportement superquadratique et satisfait une condition simple reliant ses coefficients de superquadraticite et l exposant dans ce cas on demontre que pour t 0 l existence de solutions pour le systeme hamiltonien de periode minimale t

on ш rigidity of groups of order p6 abstract let g be a group and out c g be the group of its class preserving outer automorphisms we compute out c g for all the groups g of order p 6 where p is an odd prime as an application we observe that for most of the rigid groups g of order p 6 its bogomolov multiplier b 0 g is zero

a new fixed point theorem and periodic solutions of nonconservative weakly coupled systems abstract in this paper we prove a new fixed point theorem for the coupling of twist conditions and poincare bohl type conditions as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems

analytical proof of the random phase approximation for a model of modulated diffusion the random phase approximation is rigorously proved for a particular model of modulated diffusion consisting in a 1d integrable hamiltonian map whose phase is modulated by an external markov noise continuity of the diffusion coefficient in the limit of vanishing noise amplitude is also investigated

nonlinear resonance in asymmetric oscillators we study the periodic solutions of equations with asymmetric nonlinearities at resonance with the fucik spectrum we compute the associated topological degree and prove existence multiplicity and stability of large amplitude oscillations for equations with a small friction term such equations can be viewed e g as simple models for investigating vertical oscillations of long span suspension bridges the results are typically of a nonlinear nature as some of the situations observed cannot occur with a linear equation c 1998 academic press

periodic and unbounded motions in asymmetric oscillators at resonance abstract we consider the existence of periodic and unbounded solutions for the asymmetric oscillator x a x b x g x p t where x max x 0 x max x 0 a and b are two positive constants p t is a 2 periodic smooth function and g x satisfies lim x x 1 g x 0 we have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well defined functions p and h in this paper we consider some critical cases and obtain some new sufficient conditions for the existence of 2 periodic and unbounded solutions in particular the obtained periodic solutions are new and cannot be deduced from landesman lazer s and fabry fonda s existence conditions in contrast with many existing results the function g x may be unbounded or oscillatory without asymptotic limits

chaotic scattering modelled by an inclined billiard abstract chaotic scattering as observed in satellite encounters and in other problems is modelled by a simple billiard a point particle bounces on two disks and in addition is subjected to a constant acceleration in an appropriate limit this problem reduces to a piecewise linear mapping we consider a one parameter family of orbits obtained by starting the particle at rest from a constant height we find that this family exhibits chaotic scattering it has an infinite number of discontinuities forming a self similar cantor set with the help of symbolic dynamics the structure of the family can be completely elucidated

class preserving automorphisms of finite groups abstract we construct a family f of frobenius groups having abelian sylow subgroups and non inner class preserving automorphisms we show that any a group that is a finite solvable group with abelian sylow subgroups has a sub quotient belonging to f provided it has a non inner class preserving automorphism as a consequence we obtain that for metabelian a groups or a groups with elementary abelian sylow subgroups class preserving automorphisms are necessarily inner automorphisms the same is true for a finite group whose sylow subgroups of odd order are all cyclic and whose sylow 2 subgroups are either cyclic dihedral or generalized quaternion some applications are given

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

on the number of periodic orbits of hamiltonian systems on positive type hypersurfaces in r2n abstract in this paper we prove some existence theorems of multiple periodic orbits of hamiltonian systems on a class of hypersurfaces in r 2n that are more general than star shaped ones our results extend some earlier works on this subject

the normalizer of a group in the unit group of its group ring let g be a group and r a commutative ring one of the main objects investigated in the theory of group rings is the unit group u s u r g of the group ring rg despite many efforts the structure of this group is very poorly understood especially when the group is infinite the aim of this work is to study the normalizer n s n r g of g in u this group acts by conjugation on g and the group of automorphisms of g obtained in this way is denoted by aut g therefore aut g is isomorphic to nrc r r where c s c r g is the center of u since g n the group inn g of inner automorphisms of g is contained in aut g and by out g we r r denote the group aut grinn g f nrgc out interest in the group n is r motivated by the following properties

unramified brauer groups of finite simple groups of lie type a l we study the subgroup b 0 g of h 2 g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i consisting of all elements which have trivial restrictions to every abelian subgroup of g the group b 0 g serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties v g where v is a faithful complex linear representation of the group g we prove that b 0 g is trivial for finite simple groups of lie type a l

positively homogeneous hamiltonian systems in the plane i try to give a general description of the dynamics of the solutions for a planar hamiltonian system with positively homogeneous hamiltonian function and periodic forcing term most of the results obtained are already known in the special case of a scalar second order differential equation with asymmetric nonlinearity

noether s problem for groups of order 32 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k theorem let g be a finite group of order 32 with exponent e if char k 2 or k is any field containing a primitive eth root of unity then k g is rational over k

a note on the minimal periodic solutions of nonconvex superlinear hamiltonian system in this paper we consider the multiplicity result of minimal periodic solutions for the first order hamiltonian system email protected t email protected h x t we point out a mistake in the literatures m girardi m matzeu solutions of minimal period for a class of nonconvex hamiltonian systems and applications to the fixed energy problem nonlinear anal t m a 10 4 1986 371 382 and tianqing an on the minimal periodic solutions of nonconvex superlinear hamiltonian systems j math anal appl 329 2007 1273 1284 when the potential is even under some stronger restricted conditions we give the correct proof and obtain more distinct nonconstant periodic solutions with minimal period t

periodic solutions of superlinear autonomous hamiltonian systems with prescribed period abstract in this paper we prove an existence theorem of nonconstant periodic solution of superlinear autonomous hamiltonian system x t j h x t with prescribed period under an assumption weaker than ambrosetti rabinowitz type condition 0 h x h x x 2 x r 0 our result extends the pioneering work of rabinowitz of 1978

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

computing the nonabelian tensor squares of polycyclic groups abstract in this paper we develop a theory for computing the nonabelian tensor square and related computations for finitely presented groups and specialize it to polycyclic groups this theory provides a framework for making nonabelian tensor square computations for polycyclic groups and is the basis of an algorithm for computing the nonabelian tensor square for any polycyclic group

universal commutator relations bogomolov multipliers and commuting probability abstract let g be a finite p group we prove that whenever the commuting probability of g is greater than 2 p 2 p 2 p 5 the unramified brauer group of the field of g invariant functions is trivial equivalently all relations between commutators in g are consequences of some universal ones the bound is best possible and gives a global lower bound of 1 4 for all finite groups the result is attained by describing the structure of groups whose bogomolov multipliers are nontrivial and bogomolov multipliers of all of their proper subgroups and quotients are trivial applications include a classification of p groups of minimal order that have nontrivial bogomolov multipliers and are of nilpotency class 2 a nonprobabilistic criterion for the vanishing of the bogomolov multiplier and establishing a sequence of bogomolov s absolute minimal factors which are 2 groups of arbitrarily large nilpotency class thus providing counterexamples to some of bogomolov s claims in relation to this we fill a gap in the proof of triviality of bogomolov multipliers of finite almost simple groups

on the minimal periodic solutions of nonconvex superlinear hamiltonian systems abstract this paper proves a multiplicity result for the minimal periodic solutions of hamiltonian system x t j h x t under a superquadratic hypothesis on h weaker than the ambrosetti rabinowitz type condition we also define a class of homogeneous functions that are more general than the classical ones and prove the rabinowitz s conjecture in the case of h satisfying such new homogeneity

existence uniqueness and stability of oscillations in differential equations with asymmetric nonlinearities we give conditions for the existence uniqueness and asymptotic stability of periodic solutions of a second order differential equation with piecewise linear restoring and 27r periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer with suitable restrictions on the restoring and forcing in the undamped case we give a necessary and sufficient condition

groups of order p 5 and their unramified brauer groups we prove that if p is a prime p 3 and g is a group of order p5 not belonging to the 10 th isoclinism family then the unramified brauer group of g is trivial

periodic solutions of perturbed isochronous hamiltonian systems at resonance we look for periodic solutions of planar systems obtained by adding an asymptotically positively homogeneous nonlinear term to an isochronous hamiltonian system precise computations of the topological degree are obtained by elementary phase plane analysis c 2005 elsevier inc all rights reserved