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

subgroups of polynomial automorphisms with diagonalizable fibers let r be an integral domain over a field k and g a subgroup of the automorphism group of the polynomial ring r x1 xn over r in this paper we discuss when g is diagonalizable under the assumption that g is diagonalizable over the field of fractions of r we are particularly interested in the case where g is a finite abelian group kraft and russell 2014 8 imply that every finite abelian subgroup of autrr x1 x2 is diagonalizable if r is an affine pid over k c one of the main results of this paper says that the same holds for a pid r over any field k containing enough roots of unity

boundedness for the general semilinear duffing equations via the twist theorem abstract in this paper we prove the boundedness of all solutions for the periodic equation x 2 x x g x x t p t where satisfies the diophantine condition x c 19 is bounded p t c 23 and d x i d t j g x t is bounded for 0 i j 21

non linearizable algebraic k actions on affine spaces non linearizable faithful algebraic r r actions of torus on the real affine n space r n are given for each r 1 n 3 when n 4

growth of sobolev norms in time dependent semiclassical anharmonic oscillators abstract we consider the semiclassical schrodinger equation on r d given by i t 2 2 w l x v t x where w l is an anharmonic trapping of the form w l x 1 2 l j 1 d x j 2 l l 2 is an integer and is a semiclassical small parameter we construct a smooth potential v t x bounded in time with its derivatives and an initial datum such that the sobolev norms of the solution grow at a logarithmic speed for all times of order log 1 2 1 the proof relies on two ingredients first we construct an unbounded solution to a forced mechanical anharmonic oscillator then we exploit semiclassical approximation with coherent states to obtain growth of sobolev norms for the quantum system which are valid for semiclassical time scales

simple birational extensions of the polynomial algebra c 3 the abhyankar sathaye problem asks whether any biregular embedding c k c n can be rectified that is whether there exists an automorphism a e aut c n such that a o is a linear embedding here we study this problem for the embeddings c 3 c 4 whose image x c 3 is given in c 4 by an equation p f x y u g x y z 0 where f c x y 0 and g e c x y z under certain additional assumptions we show that indeed the polynomial p is a variable of the polynomial ring c 4 c x y z u i e a coordinate of a polynomial automorphism of c 4 this is an analog of a theorem due to sathaye 1976 which concerns the case of embeddings c 2 c 3 besides we generalize a theorem of miyanishi 1984 giving for a polynomial p as above a criterion for when x p 1 0 c 3

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

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

evolution equations for phase separation and ordering in binary alloys we explore two phenomenological approaches leading to systems of coupled cahn hilliard and cahn allen equations for describing the dynamics of systems which can undergo first order phase separation and order disorder transitions simultaneously starting from the same discrete lattice free energy function in the first approach a quasicontinuum limit is taken for this discrete energy and the evolution of the system is then assumed to be given by gradient flow in the second approach a discrete set of gradient flow evolution equations is derived for the lattice dynamics and a quasicontinuum limit is then taken we demonstrate in the context of bcc fe al binary alloys that it is important that variables be chosen that accommodate the variations in the average concentration as well as the underlying ordered structure of the possible coexistent phases only then will the two approaches lead to roughly the same continuum descriptions we conjecture that in general the number of variables necessary to describe the dynamics of such systems is equal ton 1 n 2 1 wheren 1 is given by the dimension of the span of the bases of the irreducible representations needed to describe the symmetry groups of the possible equilibrium phases andn 2 is the number of chemical components n 1 of these variables are nonconserved and the remaining are conserved and represent the average concentrations for the fe al alloys this implies a description of one conserved order parameter and one nonconserved order parameter the resultant description is given by a cahn hilliard equation coupled to a cahn allen equation via the lower order nonlinear terms the rough equivalence of the two phenomenological methods adds credibility to the validity of the resulting evolution equations a similar description should also be valid for alloy systems in which the structure of the competing phases is more complicated

derivations of r x y z with a slice the question under consideration is whether every locally nilpotent r derivation of r x y z with a slice has kernel a generated by two elements over r where r is a polynomial ring over a field of characteristic zero theorem 1 1 gives a fundamental property of such kernels namely that a is an a2 fibration over r while it is an open question whether every a2 fibration over r is trivial the property of a asserted in the theorem is necessary to the condition that a is a polynomial ring in two variables over r the last section of the paper presents a family of examples n n 1 which are quite simple to define but whose status relative to the kernel question is not known

homogeneous locally nilpotent derivations having slices and embeddings of affine spaces abstract let m and n be positive integers such that n m and let b be a polynomial ring in m n 1 variables over a field k of characteristic 0 we give a bijective correspondence between the equivalence classes of embeddings a m a n and the equivalence classes of sequences of mutually commuting locally nilpotent derivations i 1 i m on b in some form which are homogeneous with respect to a z grading on b and have slices the intersection a of the kernels of i for 1 i m inherits the z grading on b we show that a is a polynomial ring with homogeneous coordinates if and only if the corresponding embedding is rectifiable

the surface induced spatial temporal structures in confined binary alloys abstract this paper examines surface induced ordering in confined binary alloys the hyperbolic initial boundary value problem ibvp is used to describe a scenario of spatiotemporal ordering in a disordered phase for concentration of one component of binary alloy and order parameter with non linear dynamic boundary conditions this hyperbolic model consists of two coupled second order differential equations for order parameter and concentration it also takes into account effects of the memory on the ordering of atoms and their densities in the alloy the boundary conditions characterize surface velocities of order parameter and concentration changing which is due to surface super cooling on walls confining the binary alloy it is shown that for large times there are three classes of dynamic non linear boundary conditions which lead to three different types of attractor s elements for the ibvp namely the elements of attractor are the limit periodic simple shock waves with fronts of discontinuities if is finite then the attractor contains spatiotemporal functions of relaxation type if is infinite and countable then we observe the functions of pre turbulent type if is infinite and uncountable then we obtain the functions of turbulent type

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

fixed point free locally nilpotent derivations of a2 fibrations abstract let r be a regular ring containing q and let a be an a 2 fibration over r we prove in this paper that every fixed point free locally nilpotent r derivation of a has a slice as a consequence an a 2 fibration over a factorial regular ring r is trivial if and only if it has a fixed point free locally nilpotent r derivation our results answer a question raised by freudenburg 2009 11 question 2 which actually was the main motivation of this work

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

quasi periodic solutions of forced isochronous oscillators at resonance abstract we deal with the existence of quasi periodic solutions of forced isochronous oscillators with a repulsive singularity the nonlinearity is a bounded perturbation using a variant of moser s twist theorem of invariant curves due to ortega r ortega boundedness in a piecewise linear oscillator and a variant of the small twist theorem proc london math soc 79 1999 381 413 we show that there are many quasi periodic solutions and the boundedness of all solutions

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

the coexistence of quasi periodic and blow up solutions in a class of hamiltonian systems abstract the coexistence of quasi periodic solutions and blow up phenomena in a class of higher dimensional duffing type equations is proved in this paper moreover we show that the initial point sets for both kinds of solutions are of infinite lebesgue measure in the phase space for the part of quasi periodic solutions the tool we used is the small twist theorem for higher dimensional cases

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

nonlinear resonance in asymmetric oscillations in this paper we study the periodic problem f p x email protected f p x email protected f p x f x t where f p u u p 2u p 1 a 1 email protected 1 p 2n 1 email protected n by computing the associated topological degree we prove existence of periodic solutions and stability of large amplitude oscillations for equations with a small damping term

boundedness in forced isochronous oscillators abstract in this paper we are concerned with the boundedness of all solutions for the forced isochronous oscillator x v x g x f t where v is a so called t isochronous potential the perturbation g is assumed to be bounded and the 2 periodic function f t is smooth using the resonant small twist theorem and averaged small twist theorem established by ortega we will prove the boundedness of all solutions for the above forced isochronous oscillator in the resonant and non resonant cases under some reasonable assumptions respectively

the spatial temporal lamellar structures in the confined ideal polymer blends we consider the formation of self organized spatial temporal oscillating structures in symmetric binary polymer blends confined by two flat walls an influence of these walls on the formation of the oscillating volume structures is studied this phenomenon is simulated by an initial boundary value problem for the conserved order parameter or the concentration of one of the components in a binary mixture under a special choice the dynamical puri binder s boundary conditions these structures look like the lamellar structures the behavior of the order parameter is described by the modified cahn hilliard equation which models so called the non fickian diffusion in the symmetric binary polymer blends the nonlinear dynamical boundary conditions correspond to the process of adsorption desorption on the walls as a result these nonlinear surface processes induce into the volume the spatial temporal asymptotically periodic structures of relaxation pre turbulent or turbulent type with finite countable or non countable points of discontinuities on the period correspondingly the frequency of oscillations on the period follows a power law for the relaxation type and increases exponentially in the other cases

invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term abstract in this paper the boundedness of all solutions of the oscillator x f x x 2 x x p t is discussed where is a positive constant f x y x and p t are smooth functions and p t p t 2

on the cancellation problem for the affine space mathbb a 3 in characteristic p we show that the cancellation conjecture does not hold for the affine space mathbb a 3 k over any field k of positive characteristic we prove that an example of t asanuma provides a three dimensional k algebra a for which a is not isomorphic to k x1 x2 x3 although a t is isomorphic to k x1 x2 x3 x4

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

the structure of a laurent polynomial fibration in n variables abstract bass connell and wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain r is isomorphic to the symmetric algebra of a finitely generated projective r module of rank n in this paper we prove a corresponding structure theorem for a ring a which is a locally laurent polynomial algebra in n variables over an integral domain r viz we show that a is isomorphic to an r algebra of the form sym r q i 1 where q is a direct sum of n finitely generated projective r modules of rank one and i is a suitable invertible ideal of the symmetric algebra sym r q further we show that any faithfully flat algebra over a noetherian normal domain r whose generic and codimension one fibres are laurent polynomial algebras in n variables is a locally laurent polynomial algebra in n variables over r

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

boundedness of semilinear duffing equations at resonance in a critical situation abstract in this paper we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation x n 2 x g x p t with the critical situation that 0 2 p t e i n t d t 2 g g on g and p where n n p t is periodic and g x is bounded

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