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

picard groups of singular affine curves over a perfect field this was proved in w1 for the case of an algebraically closed field k in that case the conditions force r to be a dedekind domain for an infinite perfect ground field this is no longer the case for example the conditions are satisfied for the restricted polynomial ring k x k x l where k k is a finite algebraic extension and x is an indeterminate a sharpened form of the main theorem proved in the next section says that a not dedekind domain satisfying the conditions of 0 1 has exactly one singularity and its singularity is analytically isomorphic to that of a restricted polynomial ring we wilt show by example that 0 1 1 no longer implies 0 1 3 if domain is replaced by reduced ring or if infinite is deleted from the hypotheses on k i don t know whether perfection of the ground field k is essential we obtain a partial result for reduced rings if d r is finitely generated then r is seminormal and the intersection graph of the irreducible components of spec r is acyclic

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

cancellation and direct summands in dimension 1 abstract let be a module finite algebra over a commutative noetherian ring of krull dimension 1 we extend roiter s direct summand theorem to arbitrary finitely generated modules obtaining a sharpened form of serre s direct summand theorem in this setting we also extend drozd s cancellation theorem to arbitrary finitely generated modules obtaining a sharpened form of bass s cancellation theorem in this setting a corollary is that over commutative reduced noetherian rings of dimension 1 direct sum cancellation holds in every genus of finitely generated modules this becomes false if the ring has nilpotent elements another corollary is that if direct sum cancellation holds in the genera of modules m and n then it holds in the genus of m n this seems to be new even for the modules that occur in integral representation theory the main thrust of this paper is to close the gap between integral representation theory and the rest of module theory by eliminating hypotheses concerning the existence of maximal orders of finite normalization in the commutative case and allowing our rings to have nilpotent ideals

cancellation over commutative rings of dimension one and two all rings in this paper are commutative and noetherian and all modules are finitely generated we will study the question if a c z b c is a z b here a b and c are modules over a ring r in 1962 chase 4 gave an affirmative answer for r k x y the polynomial ring in two variables over an algebraically closed field k provided a and b are torsionfree and char k trank a in the first section of this paper we extend chase s result to include all two dimensional regular affine k domains most of the paper deals with one dimensional rings in section 2 we prove some general results which we hope will eventally lead to some sort of structure theory for torsionfree modules over one dimensional reduced rings with finite normalization in this context we show that one can cancel from projectives that is the answer is yes when a and b are projective if and only if pit r pit z where g is the normalization when a and b are assumed to be merely torsionfree the problem is more subtle but we have some partial results the last three sections deal with three naturally occurring examples of one dimensional rings coordinate rings of curves quadratic orders and integral group rings of finite abelian groups in each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules many of the ideas in this paper are the direct or indirect results of innumerable conversations with raymond heitmann and lawrence levy in particular levy discovered a serious error in the proof of the main theorem of an earlier version and heitmann showed by example that the statement itself was incorrect i am extremely grateful to both of them for their interest and insight

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

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

longtime closeness estimates for bounded and unbounded solutions of non recurrent duffing equations with polynomial potentials abstract in this paper we consider the littlewood s problem for the second order differential equation x x 2 n 1 q t x l p t l n where the functions p t and q t do not need to be periodic using the theory of non periodic twist mappings which is developed by kunze and ortega in 3 4 5 6 7 we obtained longtime closeness results on the bounded and unbounded solutions of the equation

a proper relaxation of controls with variable shifts abstract in a previous paper written jointly with q j zhu 1992 j math anal appl 169 546 561 we studied optimal control problems defined by functional integral equations and in particular ordinary differential equations with shifts in the controls and with the shifted controls not necessarily separated i e either additively or nonadditively coupled in that paper it was assumed that the domain of the state and control functions is a cartesian product of an interval with a compact metric space and that each shift h j j 1 k has a one dimensional component of the form t 1 d j where d 1 d k are constant possibly noncommensurate delays and advances in the present note we extend those results to the case where each d j is replaced by a function d j that may vary and take on at different times both positive and nonpositive values

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

stable isomorphism of modules over one dimensional rings abstract two r modules m and n are said to be stably isomorphic provided m r n n r n for some n 1 in this paper r is always a commutative reduced one dimensional noetherian ring with finite normalization r and m and n are torsionfree all modules are assumed to be finitely generated there is a natural action of r c the group of units of r c where c is the conductor of r in r on torsionfree r modules and the main theorem of this paper is that the orbits of the induced action of r c are exactly the stable isomorphism classes as applications we show that stably isomorphic modules are actually isomorphic if r is a domain finitely generated as an r algebra with at most one singular real maximal ideal we give several examples of modules that are stably isomorphic but not isomorphic illustrating that these hypotheses cannot be significantly weakened we are grateful to l s levy and the referee for several suggestions that have improved the exposition of this work

problems with relaxed shifted controls and the pointwise maximum principle abstract in previous work j warga a proper relaxation of controls with variable shifts j math anal appl 196 1995 783 793 it has been shown that controls with k variable and unrelated time shifts form a set whose closure in the weak star topology of l 1 t c k 1 is a convex and compact space this provides a basis for establishing the existence of an optimal solution of the shifted control problem by essentially copying the assumptions and arguments used for control problems without shifted controls similarly the necessary optimality conditions for original and relaxed minima that define an extremal previously studied for differential and functional equations apply directly to such problems with shifted controls except that the maximum principle follows directly in integral form only in the present note we first establish the form of dependence of the relaxed shifted controls on the shifts and then use the integral form of the maximum principle to derive a corresponding pointwise maximum principle for optimal ordinary or relaxed shifted controls acting through ordinary differential and certain functional differential equations

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

integral degree of a ring reduction numbers and uniform artin rees numbers abstract the supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree a new invariant of the ring which is finite provided the ring has finite integral closure as a consequence one obtains bounds for the castelnuovo mumford regularity of the rees algebra and for the artin rees numbers

modules over dedekind like rings abstract all finitely generated modules are described over a class of rings that includes the integral group ring zgn of the cyclic group of square free order n some rings of algebraic integers that are not integrally closed in their field of quotients and many subrings of z z z special attention is paid to behavior of these modules in direct sums

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

arithmetic of mori domains and monoids in this paper we introduce weakly c monoids as a new class of v noetherian monoids weakly c monoids generalize c monoids and make it possible to study multiplicative properties of a wide class of mori domains e g rings of generalized power series with coefficients in a field and exponents in a finitely generated monoid the main goal of the paper is to study the question when a weakly c monoid is locally tame after having proved a classification theorem for local tameness we use it to show that every locally tame weakly c monoid whose complete integral closure has finite class group has finite catenary degree and finite set of distances

direct sum cancellation for modules over one dimensional rings let r be a one dimensional noetherian domain with finite normalization r in the eighties the second named author and s wiegand developed a mechanism for studying the cancellation problem for finitely generated torsion free r modules the key idea described in wie84 and ww87 is to represent a given torsion free module m as a pullback m rm m fm rm fm this research was initiated while the author was a visitor at the university of nebraska supported by the fonds zur forderung der wissenschaftlichen forschung partially supported by grants from the national science foundation the national security agency and the mathematical sciences research institute where much of this research was done

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

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

a proper relaxation of shifted and delayed controls abstract we consider optimal control problems defined by functional integral equations with variable shifts in the controls and with the shifted controls nonadditively coupled as a special case we obtain differential equations with delays and or advances in both the state and the control functions we prove that if the original controls are measurable functions u with values in a compact metric space then the k 1 tuples u u u h 1 u h k of shifted controls have relaxed limits that form a compact and convex set this enables us to prove an existence theorem to approximate any relaxed solution with original solutions and to derive necessary conditions generalizeing pontryagin s maximum principle in integral form even for nonsmooth problems we also discuss the dependence on the shifts hj of the set of original controls u of the set of their relaxed limits and of the set of the corresponding solutions of the functional integral equation our results include the proof that the d model of rosenblueth and vinter is proper

relaxation procedures for time delay systems abstract despite the attention that optimal control problems involving time delays have received over the last two decades little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers recently warga has proposed a relaxation procedure for fully nonlinear problems with delays in the state and control variables and showed that the resulting relaxed problem has a solution nevertheless the effect of this relaxation may reduce the infimum cost as we show through an example which belongs to a class of problems with commensurate delays i e the quotient of any two delays is rational for problems of this nature we provide a new relaxation procedure for which the extension is proper i e the infimum costs coincide we also present an abstract relaxation procedure applicable to problems with possibly noncommensurate delays

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

indecomposable modules of large rank over cohen macaulay local rings a commutative noetherian local ring r m k is called dedekind like provided r is one dimensional and reduced the integral closure r is generated by at most 2 elements as an r module and m is the jacobson radical of r if m is an indecomposable finitely generated module over a dedekind like ring r and if p is a minimal prime ideal of r it follows from a classification theorem due to l klingler and l levy that m p must be free of rank 0 1 or 2 now suppose r m k is a one dimensional cohen macaulay local ring that is not dedekind like and let p 1 p t be the minimal prime ideals of r the main theorem in the paper asserts that for each non zero t tuple n 1 n t of non negative integers there is an infinite family of pairwise non isomorphic indecomposable finitely generated r modules m satisfying mp i rp i n i for each i

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

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