local spectral gap in simple lie groups and applications we introduce a novel notion of local spectral gap for general possibly infinite measure preserving actions we establish local spectral gap for the left translation action gamma curvearrowright g whenever gamma is a dense subgroup generated by algebraic elements of an arbitrary connected simple lie group g this extends to the non compact setting works of bourgain and gamburd invent math 171 83 121 2008 j eur math soc jems 14 1455 1511 2012 and benoist and de saxce invent math 205 337 361 2016 we present several applications to the banach ruziewicz problem orbit equivalence rigidity continuous and monotone expanders and bounded random walks on g in particular we prove that up to a multiplicative constant the haar measure is the unique gamma invariant finitely additive measure defined on all bounded measurable subsets of g

on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

arithmeticity of holonomy groups of lie foliations the point of this paper is to prove arithmeticity of a class of finitely generated groups that arise naturally in a purely geometric context namely we prove arithmeticity of the holonomy group of a lie foliation of a compact manifold assuming the existence of a metric for which the leaves are symmetric spaces of nonpositive curvature and rank at least 2 this provides a partial answer to the question raised a number of years ago by a haefliger 13 of determining the possible holonomy groups of lie foliations to indicate the nature of the questions we are considering let us first recall that if n is a riemannian manifold a fundamental question is to understand the relationship of the topology of n and the geometry of n or for many purposes the geometry of n there has been a wide variety of work on this type of question from a variety of viewpoints in particular we can ask how the geometry of n or n controls the nature of rc1 n one can ask a more general question namely to understand to what extent the geometry of n controls the actions of rc1 n on a manifold t a standard construction associates to every such action a foliation of a manifold m which has all leaves locally isometric to n in which t appears naturally as a submanifold of m transverse to the foliation and the 7 n action on t controls the way in which the leaves of this foliation are tied together we can thus reinterpret our last question and broaden its context as well by formulating the following question determine the extent to which the geometry of a simply connected riemannian manifold x controls the way in which the leaves of a foliation are tied together if we assume the leaves are all locally isometric to x in general it is of course difficult to understand actions of a group on a manifold and correspondingly difficult to understand foliations with leaves locally isometric to a given x it is natural to begin an investigation of such actions with the case in which the action on t preserves some geometric structure e g a riemannian metric in terms of the associated foliation this becomes a special case of the

ergodic theory group representations and rigidity in these lectures we discuss some topics concerning the relationship of ergodic theory representation theory and the structure of lie groups and their discrete subgroups

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

invariant functionals and the uniqueness of invariant norms let be a representation of a compact group g on a banach space x the question we address is whether x carries a unique invariant norm in the sense that is the unique norm on x for which is a representation we characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when x is a dual banach space and is a x x continuous representation of g on x such that g consists of x x continuous operators we illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces lp where is a locally compact hausdorff space equipped with a positive radon measure and g acts on as a group of continuous invertible measure preserving transformations

boundedness and unboundedness of solutions for reversible oscillators at resonance abstract in this paper we study the problem of boundedness of solutions for the perturbed asymmetric oscillator with damping x f x a x b x g x p t where a b lies on one of the fucik spectrum curves since the system in general is non hamiltonian we have to introduce reversibility assumptions to apply the twist theorem for reversible maps we then obtain a sharp result concerning the boundedness of all solutions

real analytic actions of lattices this paper introduces a new approach to the study of lattice actions on compact manifolds the group sl n z acts by real analytic di eomorphisms on the sphere sny1 via the action on the space of in nite rays based at 0 2 r we believe this action is minimal in the following sense conjecture i any smooth action of a nite index subgroup of sl n z n 2 on a compact r manifold factors through a nite group action if r ny 1

on the mautner phenomenon and groups with property t this paper is inspired by moore s work on the mautner phenomenon 18 in 18 the best possible result of this kind has been established for a one parameter subgroup of a lie group our purpose here is to extend moore s result to a cyclic subgroup of a lie group or of an algebraic group over a local field we make various applications of this result to study groups with property t first we show in theorem 1 2 that moore s result can be extended to a cyclic subgroup of a lie group as applications we have the following theorems

boundedness in nonlinear oscillations at resonance abstract in this paper we are concerned with the boundedness of all the solutions for the semilinear duffing equation x n 2 x x p t with the lazer leach condition where p t is a smooth 2 periodic function and the perturbation x is bounded

groups of measure preserving transformations ii on etudie les proprietes d invariance d independance et de disjonction des groupes de transformation

estimating mean concentrations under transformation for environmental data with detection limits the reporting procedures for potentially toxic pollutants are complicated by the fact that concentrations are measured using small samples that include a number of observations lying below some detection limit furthermore there is often a small number of high concentrations observed in combination with a substantial number of low concentrations this results in small nonnormally distributed censored samples this article presents maximum likelihood estimators for the mean of a population based on censored samples that can be transformed to normality the method estimates the optimal power transformation in the box cox family by searching the censored data likelihood maximum likelihood estimators for the mean in the transformed scale are calculated via the expectation maximization algorithm estimates for the mean in the original scale are functions of the estimated mean and variance in the transformed population confidence intervals are computed using the delta method and the nonparametric percentil

gaussian approximations in high dimensional estimation abstract several estimation techniques assume validity of gaussian approximations for estimation purposes interestingly these ensemble methods have proven to work very well for high dimensional data even when the distributions involved are not necessarily gaussian we attempt to bridge the gap between this oft used computational assumption and the theoretical understanding of why this works by employing some recent results on random projections on low dimensional subspaces and concentration inequalities

boundedness of solutions for semilinear duffing equations in this paper we are concerned with the boundedness of all the solutions and the existence of quasi periodic solutions for some semilinear duffing equationx g x e t wheree t is a smooth 2 periodic function and0 g x x k

extension and separation properties of positive definite functions on locally compact groups continuing earlier work we investigate two related aspects of the set p g of continuous positive definite functions on a locally compact group g the first one is the problem of when for a closed subgroup h of g every function in p h extends to some function in p g the second one is the question whether elements in g h can be separated from h by functions in p g which are identically one on h

non inner amenability of the thompson groups t and v abstract in this paper we prove that the thompson groups t and v are not inner amenable in particular their group von neumann algebras do not have property moreover we prove that if the reduced group c algebra of t is simple then the thompson group f is non amenable furthermore we give a few new equivalent characterizations of amenability of f

semilinear duffing equations crossing resonance points abstract in this paper using a generalized form of the poincare birkhoff theorem and a fixed point theorem we prove under weaker conditions two theorems for the equation x g x p t p t p t 2 of which one shows the existence of a harmonic solution the other that the equation may have an infinite number of harmonic solutions in the resonance case this is an enhancement of the results already obtained

systemes dynamiques non commutatifs et moyennabilite on introduit la notion de fonction de type positif relativement a un systeme dynamique c dont les proprietes sont analogues a celles des fonctions de type positif classiques on montre que dans le cas d un groupe discret g agissant sur une algebre de von neumann m la moyennabilite de l action est equivalente a l existence d une suite bornee de fonctions de type positif sur g a support fini a valeurs dans le centre z m de m qui converge vers 1 pour la topologie de la convergence simple ultrafaible

on the peroidic bvp for the forced duffing equation with jumping nonlinearity on etudie le probleme suivant u cu f t u e t p p sur 0 t u t u 0 0 u t u 0 0 c est une fonction arbitraire dans l 1 0 t r c r et l application non lineaire f est a saut

property t and rigidity for actions on banach spaces we study property t and the fixed point property for actions on lp and other banach spaces we show that property t holds when l2 is replaced by lp and even a subspace quotient of lp and that in fact it is independent of 1 p we show that the fixed point property for lp follows from property t when 1 p 2 e for simple lie groups and their lattices we prove that the fixed point property for lp holds for any 1 p if and only if the rank is at least two finally we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive banach spaces

multiplicity of periodic solutions of semilinear duffing s equation at resonance in this paper we prove an infinity of periodic solutions to the periodically forced nonlinear duffing equation under new conditions

isotropy of semisimple group actions on manifolds with geometric structure we prove results concerning isotropy of noncompact semisimple lie group actions that preserve pseudo riemannian or affine structures on compact or finite volume manifolds for example if a lie group g acts locally faithfully on a connected compact complete affine manifold m and preserves the affine structure then the connected component of the identity of each isotropy subgroup is a compact extension of a solvable group this is a consequence of the result that nontrivial affine actions of connected groups locally isomorphic to sl 2 inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i on such manifolds cannot have fixed points we also prove that if g is connected semisimple without compact factors and acts isometrically and topologically transitively on a connected finite volume pseudo riemannian m then the action must be everywhere locally free this answers affirmatively a special case of a conjecture by g d ambra and m gromov with only the assumption that no factor of g acts trivially local freeness is also assured when m is compact and complete

nonparametric maximum likelihood estimation from samples with irrelevant data and verification bias abstract suppose that some measurements come from a distribution f that is of interest and others come from another irrelevant distribution g some measurements from f are verified and known to be from f the other unverified measurements may be from f or from g not all measurements from f are equally likely to be verified and no measurement from g is ever verified this model applies to measurements of low concentrations obtained using gas chromatography mass spectroscopy for example as is shown in this article how well a feature t f of f can be estimated when there are unverified data depends on what can be assumed about f g and the conditional probability v x of verifying a measurement of x from f if f g and v are unrestricted then more than one choice of f g v gives the same distribution p of the observable x and thus t f cannot be uniquely estimated from data but if the set of values of t f that correspond to a distribution p of x is small enough then it is reasonable to try t

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

invariant measures for algebraic actions zariski dense subgroups and kazhdan s property t let k be any locally compact non discrete field we show that finite invariant measures for k algebraic actions are obtained only via actions of compact groups this extends both borel s density and fixed point theorems over local fields for semisimple solvable groups resp we then prove that for k algebraic actions finitely additive finite invariant measures are obtained only via actions of amenable groups this gives a new criterion for zariski density of subgroups and is shown to have representation theoretic applications the main one is to kazhdan s property t for algebraic groups which we investigate and strengthen

the action of a solvable group on an infinite set never has a unique invariant mean theorem 1 of the paper proves a conjecture of j rosenblatt on nonuniqueness of invariant means for the action of a solvable group g on an infinite set x the same methods used in this proof yield even a more general result nonuniqueness still holds if g is an amenable group containing a solvable subgroup h such that card g h caxd h

product groups acting on manifolds we analyse volume preserving actions of product groups on riemannian manifolds under a natural spectral irreducibility assumption we prove the following dichotomy either the action is measurably isometric in which case there are at most two factors or the action is infinitesimally linear which means that the derivative cocycle arises from unbounded linear representations of all factors as a first application this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group another application is a strong restriction for actions of non linear groups we prove our results by means of a new cocycle superrigidity theorem of independent interest in analogy to zimmer s programme

uniqueness of invariant means for measure preserving transformations for some compact abelian groups x e g t n 2 and hii 1 z2 the group g of topological automorphisms of x has the haar integral as the unique g invariant mean on l x ax this gives a new characterization of lebesgue measure on the bounded lebesgue measurable subsets 8 of rn n 3 it is the unique normalized positive finitely additive measure on 8 which is invariant under isometries and the transformation of rn xi xn xi x2 x2 i x other examples of as well as necessary and sufficient conditions for the uniqueness of a mean on l q x 3 p which is invariant by some group of measure preserving transformations of the probability space x 3 p are described 0 introduction let 8 be the ring of bounded lebesgue measurable sets in rn or in sn the n dimensional unit sphere in rn and let xn be the lebesgue measure on 8 normalized by xn j 1 where j 0 1 or by xn sn 1 respectively the classical characterization by lebesgue of is that it is the unique positive real valued functionf on 8 which satisfies these three conditions a f jn 1 orf sn 1 respectively b f is invariant under isometries c f is countably additive in 1923 banach 1 studied the question of ruziewicz as to whetherf is still unique when c is replaced by c0 f is finitely additive banach gave a negative answer to this question for r 1 r2 and s1 but for r n 3 or sn n 2 this question is still unanswered in this paper we will prove a theorem which comes close to a solution of this problem for rn n 3 theorem 3 10 if f satisfies a b and c0 and if f is invariant under the transformation of r n n 3 given by xi x2 x f xi x2 x2 9 xn then f xn our method of proof will give several other theorems similar to this one in theorem 3 10 we do not need the full strength of b we need only know that f is absolutely continuous with respect to x i e for n e 3 f n 0 whenever n 0 the fact that for rn n 3 or sn n 2 a b and c0 imply this absolute continuity of f is an observation of tarski which is proved like this received by the editors march 3 1980 presented to the society january 10 1981 ams mos subject classifications 1970 primary 28a70 43a07 43a40 1981 american mathematical society 0002 9947 81 0000 0268 04 50

uniqueness of norm on lp g and c g when g is a compact group let g be a compact group if the trivial representation of g is not weakly contained in the left regular representation of g on l 2 g and x is either l p g for 1oppn or c g then we show that every complete norm jj on x that makes translations from x jj into itself continuous is equivalent to jjjj p or jjjj n respectively if 1oppn and every left invariant linear functional on l p g is a constant multiple of the haar integral then we show that every complete norm jj on l p g that makes translations from l p g jj into itself continuous and that makes the map t lt from g into l l p g jj bounded is equivalent to jjjj p

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

random generation in semisimple algebraic groups over local fields abstract let g be a semisimple algebraic group over a local field k of characteristic p if the universal covering map g sc g is inseparable then no open subgroup of g k is finitely generated otherwise for any compact open subgroup g with probability 1 two randomly chosen elements generate an open subgroup of g k our main tool is a recent theorem by r pink characterizing compact groups linear over a local field

action of amenable groups and uniqueness of invariant means abstract it is proved in this paper that assuming the continuum hypothesis there exists an amenable group g acting on an infinite set x such that there is only one g invariant mean on l x

a topological tits alternative let k be a local eld and gln k a linear group over k we prove that either contains a relatively open solvable subgroup or it contains a relatively dense free subgroup this result has applications in dynamics riemannian foliations and pronite groups

stabilizers for ergodic actions of higher rank semisimple groups let g be a connected semisimple lie group with finite center and r rank 2 suppose that each simple factor of g either has r rank 2 or is locally isomorphic to sp l n or f4 20 we prove that any faithful irreducible properly ergodic finite measure preserving action of g is essentially free we extend the result to reducible actions and actions of lattices

discrete groups and non riemannian homogeneous spaces a basic question in geometry is to understand compact locally homogeneous manifolds i e those compact manifolds that can be locally modelled on a homogeneous space j h of a finite dimensional lie group h this means that there is an atlas on a manifold m consisting of local diffeomorphisms with open sets in j h where the transition functions between these open sets are given by translations by elements of h a basic example is the case of m j h f where f c h is a discrete subgroup acting freely and properly discontinuously on j h with a compact quotient these examples can be abstractly characterized as the complete case see g we then call f a cocompact lattice on j h when j is compact j h has an h invariant riemannian metric so that m itself becomes a riemannian manifold in a natural way thus the class of locally homogeneous riemannian manifolds includes but is not exhausted by the locally symmetric spaces for h aff rn and j gl n r the spaces j h f are the complete affinely flat manifolds a non riemannian example which has been extensively studied when h is simple and j is not compact the situation is far from being well understood some special series of such homogeneous spaces admitting a cocompact lattice have been constructed by kulkarni ku and by t kobayashi k for example kulkarni shows the existence of such a discrete group f for the homogeneous spaces so 1 2n so 2 2n and kobayshi does the same for u 1 n so 2 2n on the other hand the constructions they give for these and other similar series are quite special and there has been the general suspicion that most homogeneous spaces j h with h simple and j not compact do not admit a cocompact lattice we recall that when j is compact i e the riemannian case such f always exist by a result of borel b one general situation in which it is known that no such f exists is the case in which j h is noncompact and of reductive type and r rank h r rank j this situation was studied by calabi markus cm wolf w and kobayashi k and the nonexistence of f in this situation is a special case of a more general result known as the calabi markus phenomenon kobayashi and ono ko use the hirzebruch proportionality principle to show that some other special series do

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

the product replacement algorithm and kazhdan s property t a problem of great importance in computational group theory is to generate nearly uniformly distributed random elements in a finite groupg a good example of such an algorithm should start at any given set of generators use no prior knowledge of the structure of g and in a polynomial number of group operations return the desired random group elements see bb2 then these random elements can be used further to determine the structure of g in a pioneer paper bb1 babai found such an algorithm which provably generates nearly uniformly distributed random elements in o log g group multiplications too many for practical applications a different heuristic the product replacement algorithm was designed by leedham green and soicher lg and studied by celler et al in clmno in spite of the fact that very little theoretical justification was known practical experiments showed excellent performance so it quickly became the most popular practical algorithm to generate random group elements and was included in the two most frequently used group algebra packages gap sc and magma bcp a systematic and quantitative approach was carried out by diaconis and saloffcoste ds1 ds2 see also bb2 cg but their results did not reveal the mystery of the truly outstanding performance of the algorithm the aim of this paper is to propose a conceptual explanation based on kazhdan s property t from representation theory of lie groups and to improve some of the previous estimates on the running time the product replacement algorithm works as follows clmno given a finite group g let k g be the set of k tuples g g1 gk of elements of g such that g1 gk g we call elements of k g the generating k tuples given a generating k tuple g1 gk define a move on it in the following way choose uniformly a pair i j such that 1 i 6 j k then apply one of the following four operations with equal probability

regression by local fitting methods properties and computational algorithms abstract local regression is a procedure for estimating regression surfaces by the local fitting of linear or quadratic functions of the independent variables in a moving fashion that is analogous to how a moving average is computed for a time series the advantage of the methodology over the global fitting of parametric functions of the independent variables by least squares the current paradigm in regression studies is that a much wider class of regression functions can be estimated without distortion in this paper we discuss the methods their statistical properties and computational algorithms

decision and detection limits for linear homoscedastic assays this paper focuses on the effect of estimating the calibration parameters on the decision and detection limits for linear homoscedastic assays assay decision and detection limits are defined for linear homoscedastic assays expressions for these limits are derived when the calibration parameters are known and unknown these expressions together with others previously reported in the literature are used to calculate limits for a particular example of a cyclic amp assay the various methods of computing decision and detection limits are compared in a simulation study the results demonstrate the importance of taking into consideration the variation in the estimators of the calibration parameters

almost periodic operators in let g be a locally compact group a g the fourier algebra of g b g the fourier stieltjes algebra of g and vn g the von neumann algebra generated by the left regular representation a of g then a g is the predual of vn g vn g is a b g module and a g is a closed ideal of b g let ap g t e vn g u 4 ut is a compact operator from a g into vn g the space of almost periodic operators in vn g let c g be the c algebra generated by i x x e g then c g c ap g for a compact g let e be the rank one operator on l2 g that sends h e l2 g to the constant function f h x dx we have the following results 1 there exists a compact group g such that e e ap g c g 2 for a compact lie group g e e ap g x e e c g xt l g has a unique left invariant mean x g is semisimple 3 if g is an extension of a locally compact abelian group by an amenable discrete group then ap g cq5 g 4 let g fr the free group with r generators 1 r oo if t e vn g and u u t is a compact operator from b g into vn g then t e c g

applications of kazhdan s property t to automatic continuity let g be a compact group which has the strong kazhdan s property t and let be an isometric representation of g on l p for some measure space and 1 p we characterize the continuity of all the linear operators from l p which commute with translations and the property of p being the unique norm up to equivalence on l p with respect to which all translations are continuous in terms of the finite dimensionality of the subspace of all invariant func

cocycles of ergodic group actions and vanishing of first cohomology for s arithmetic groups introduction in this paper we study cocycles of ergodic actions of s arithmetic subgroups of semisimple lie groups of higher rank we show that under suitable hypotheses the algebraic hull of the cocycle is reductive in the special case of the trivial action on a point this gives a new proof of the vanishing of first cohomology for s arithmetic nonarithmetic groups

rigidity of warped cones and coarse geometry of expanders abstract we study the geometry of warped cones over free minimal isometric group actions and related constructions of expander graphs we prove a rigidity theorem for the coarse geometry of such warped cones namely if a group has no abelian factors then two such warped cones are quasi isometric if and only if the actions are finite covers of conjugate actions as a consequence we produce continuous families of non quasi isometric expanders and superexpanders the proof relies on the use of coarse topology for warped cones such as a computation of their coarse fundamental groups

nondetects detection limits and the probability of detection abstract when chemists cannot quantify the concentration in a field sample they report nondetect instead of a numerical measurement a data analyst faced with environmental data containing nondetects might assume that all nondetects are zeros all nondetects are smaller than the smallest detect numerical measurement or if a detection limit is reported that all nondetects are below the detection limit this article shows that these assumptions can be incorrect and suggests better alternatives the measurements that are likely to be reported as nondetect can be described by a new concept the probability of acceptance where acceptance means that a measurement passes the requirements for being reported as a detect the 90th percentile of the probability of acceptance curve is a reasonable upper bound or censoring limit for a measurement reported as nondetect the probability of acceptance also suggests the complexity of the data analysis task for example assuming that all nondetects are smaller than

oscillatory escape in a duffing equation with a polynomial potential t x 0 v t 1 x v t x 1 a large class of such systems turns out to be near integrable at infinity asa consequence most large amplitude solutions are quasiperiodic withtwo frequencies and furthermore all solutions are bounded for all time the main distinguishing feature of this class of systems is the super quadratic behavior of the potential v for large x the near integrabilityat infinity is due to the difference in time scales between the time dependence of the potential on the one hand and the rapid oscillations ofhigh amplitude solutions on the other there has been much work on the

boundedness of solutions for equations with p laplacian and an asymmetric nonlinear term abstract for p 1 and p s s p 2 s we are concerned with the boundedness of solutions for the equation p x p x p x f t x where x max x 0 x max x 0 and f t x is 2 periodic in t when p 1 p p 1 p 2 n the resonant situation and f has limits f t as x there is a function z plays a central role for the boundedness of solutions more precisely if z is of constant sign then all solutions are bounded moreover such condition also guarantees the boundedness when near a fucik curve

on dense free subgroups of lie groups abstract we give a method for constructing dense and free subgroups in real lie groups in particular we show that any dense subgroup of a connected semisimple real lie group g contains a free group on two generators which is still dense in g and that any finitely generated dense subgroup in a connected non solvable lie group h contains a dense free subgroup of rank 2 dim h this answers a question of carriere and ghys and it gives an elementary proof to a conjecture of connes and sullivan on amenable actions which was first proved by zimmer

rigidity of group actions on homogeneous spaces iii consider homogeneous g h and g f for an s algebraic group g a lattice acts on the left strictly conservatively the following rigidity results are obtained morphisms factors and joinings defined apriori only in the measurable category are in fact algebraically constrained arguing in an elementary fashion we manage to classify all the measurable commuting with the action assuming ergodicity we find they are algebraically defined

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

a small twist theorem and boundedness of solutions for semilinear duffing equations at resonance abstract we obtain a new variant of moser s small twist theorem and apply this new version to investigate the boundedness of solutions for the following semilinear duffing equation x n 2 x g x p t where p is a 2 periodic smooth function and lim x x 1 g x 0 we obtain some sharp sufficient conditions for the boundedness of all solutions to the above equation at resonance unlike many existing results in the literature where the function g is required to be a bounded function with asymptotic limits our main results here allow g be unbounded or oscillatory without asymptotic limits

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

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov

quasi periodic solutions of a forced asymmetric oscillator at resonance abstract it is proved the existence of aubry mather sets and infinitely many subharmonic solutions to an equation of the form u au bu u p t where u max u 0 u max u 0 r r is continuous p 0 2 r is continuous and 2 periodic we deal with the situation when a b are two positive constants satisfying 1 a 1 b 2 n n n