infinitely many solutions of p sublinear p laplacian equations abstract existence of infinitely many solutions of the p laplacian equation p u a x u q 2 u f x u u w 0 1 p is proved where 1 q p r n is a bounded domain with smooth boundary a c is allowed to change sign and f c for some 0 is a perturbation term which may not be odd in u

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

the structure of f pure rings for a reduced f finite ring r of characteristic p 0 and q p e one can write open image in new window where m q has no free direct summands over r we investigate the structure of f finite f pure rings r by studying how the numbers a q grow with respect to q this growth is quantified by the splitting dimension and the splitting ratios of r which we study in detail we also prove the existence of a special prime ideal open image in new window r of r called the splitting prime that has the property that r open image in new window r is strongly f regular we show that this ideal captures significant information with regard to the f purity of r

strong and weak f regularity are equivalent for graded rings it is shown that the tight closure of a submodule in a artinian module is the same as its finitistic tight closure when the modules are graded over a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i graded ring over a perfect field as a corollary it is deduced that for such a graded ring strong and weak f regularity are equivalent as another application the following conjecture of hochster and huneke is proved let r m be a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i graded ring over a field with unique homogeneous maximal ideal m then r is weakly f regular if and only if r m is weakly f regular

on the co existence of maximal and whiskered tori in the planetary three body problem in this paper we discuss about the possibility of the coexistence of stable and unstable quasi periodic kolmogorov arnold moser kam tori in a region of the phase space of the three body problem the argument of proof goes along kam theory and especially the production of two non smoothly related systems of canonical coordinates in the same region of the phase space the possibility of which is foreseen for properly degenerate systems by a theorem of nekhoroshev and miscenko and fomenko the two coordinate systems are alternative to the classical reduction of the nodes by jacobi described e g in arnold small denominators and problems of stability of motion in classical and celestial mechanics 18 85 191 1963

elliptic two dimensional invariant tori for the planetary three body problem the spatial planetary three body problem i e one star and two planets modelled by three massive points interacting through gravity in a three dimensional space is considered it is proved that near the limiting stable solutions given by the two planets revolving around the star on keplerian ellipses with small eccentricity and small non zero mutual inclination the system affords two dimensional elliptic quasi periodic solutions provided the masses of the planets are small enough compared to the mass of the star and provided the osculating keplerian major semi axes belong to a two dimensional set of density close to one

cotangent bundle reduction and poincare birkhoff normal forms in this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free lie group action the canonical coordinates enable us to compute poincare birkhoff normal forms of relative equilibria using standard algorithms the case of simple mechanical systems with symmetries is studied in detail as examples we compute poincare birkhoff normal forms for a lagrangian equilateral triangle configuration of a three body system with a morse type potential and the stretched out configuration of a double spherical pendulum

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

the planetary n body problem symplectic foliation reductions and invariant tori the 6n dimensional phase space of the planetary 1 n body problem after the classical reduction of the total linear momentum is shown to be foliated by symplectic leaves of dimension 6n 2 invariant for the planetary hamiltonian mathcal h such foliation is described by means of a new global set of darboux coordinates related to a symplectic partial reduction of rotations on each symplectic leaf mathcal h has the same form and it is shown to preserve classical symmetries further sets of darboux coordinates may be introduced on the symplectic leaves so as to achieve a complete total reduction of rotations next by explicit computations it is shown that in the reduced settings certain degeneracies are removed in particular full torsion is checked both in the partially and totally reduced settings as a consequence a new direct proof of arnold s theorem arnold in russ math surv 18 6 85 191 1963 on the stability of planetary system both in the partially and in the totally reduced setting is easily deduced producing diophantine lagrangian invariant tori of dimension 3n 1 and 3n 2 finally elliptic lower dimensional tori bifurcating from the secular equilibrium are easily obtained

big cohen macaulay algebras and seeds in this article we delve into the properties possessed by algebras which we have termed seeds that map to big cohen macaulay algebras we will show that over a complete local domain of positive characteristic any two big cohen macaulay algebras map to a common big cohen macaulay algebra we will also strengthen hochster and huneke s weakly functorial existence result for big cohen macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic we also show that every seed over a positive characteristic ring maps to a balanced big cohen macaulay algebra that is an absolutely integrally closed adically separated quasilocal domain

quasilength latent regular sequences and content of local cohomology abstract the quasilength of a finitely generated module that is killed by a power of a finitely generated ideal i is introduced it is the length of a shortest filtration of the module with factors that are cyclic modules killed by i this notion is then used to define a notion of content for the dth local cohomology module of a ring or module with support in an ideal generated by d elements in the case of a ring which is central this content is a real number between 0 and 1 it is not known whether it is independent of the choice of the d generators nor whether it can change if the generators are replaced by powers in positive prime characteristic it is shown that the content is always 0 or 1 this is an open question in equal characteristic 0 and in mixed characteristic it is conjectured that if the elements form a system of parameters in a local ring of dimension d then the content is 1 this is proved if the ring contains a field in mixed characteristic it is an open question in dimension 3 or more and implies the direct summand conjecture the relationship between the notions of quasilength and content and the property that a sequence of elements can become a regular sequence on a ring or module after base change is explored

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

existence of almost cohen macaulay algebras implies the existence of big cohen macaulay algebras abstract in 1 the dagger closure is extended over finitely generated modules over noetherian local domain r m and it is proved to be a dietz closure in this short note we show that it also satisfies the algebra axiom of 9 and this leads to the following result of this paper for a complete noetherian local domain if it is contained in an almost cohen macaulay domain then there exists a balanced big cohen macaulay algebra over it

a posteriori kam theory with optimal estimates for partially integrable systems abstract in this paper we present a posteriori kam results for existence of d dimensional isotropic invariant tori for n dof hamiltonian systems with additional n d independent first integrals in involution we carry out a covariant formulation that does not require the use of action angle variables nor symplectic reduction techniques the main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates which may cause difficulties and underperformance when quantifying the hypotheses of the kam theorem in such reduced coordinates the results include ordinary and generalized iso energetic kam theorems the approach is suitable to perform numerical computations and computer assisted proofs

sharp nekhoroshev estimates for the three body problem around periodic orbits abstract we construct a local nekhoroshev like result of stability with sharp constants for the planar three body problem both in the planetary and in the restricted circular case by using the periodic averaging technique our constructions can be generalized to any near integrable hamiltonian system whose unperturbed hamiltonian is quasi convex the dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account this allows for a deep analytical understanding of the limits of such techniques in insuring nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases finally two examples with concrete values are considered one for the planetary case and one for the restricted case

pure subrings of regular rings are pseudo rational we prove a generalization conjectured by aschenbrenner and schoutens 2003 of the hochster roberts boutot kawamata theorem let r s be a pure homomorphism of equicharacteristic zero noetherian local rings if s is regular then r is pseudo rational and if r is moreover q gorenstein then it is pseudo log terminal

dagger closure in regular rings containing a field abstract we prove that dagger closure is trivial in regular domains containing a field and that graded dagger closure is trivial in polynomial rings over a field we also prove that heitmann s full rank one closure coincides with tight closure in positive characteristic under some mild finiteness conditions furthermore we prove that dagger closure is always contained in solid closure and that the forcing algebra for an element contained in dagger closure is parasolid

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

dimension and singularity theory for local rings of finite embedding dimension abstract in this paper we develop an algebraic theory for local rings of finite embedding dimension several extensions of krull dimension are proposed which are then used to generalize singularity notions from commutative algebra finally variants of the homological theorems are shown to hold in equal characteristic this theory is then applied to noetherian local rings in order to get i over a cohen macaulay local ring uniform bounds on the betti numbers of a cohen macaulay module in terms of dimension and multiplicity and similar bounds for the bass numbers of a finitely generated module ii a characterization for being respectively analytically unramified analytically irreducible unmixed quasi unmixed normal cohen macaulay pseudo rational or weakly f regular in terms of certain uniform arithmetic behavior iii in mixed characteristic the improved new intersection theorem when the residual characteristic or ramification index is large with respect to dimension and some other numerical invariants

existence of infinitely many solutions for sublinear elliptic problems abstract let n 2 and r n be a bounded domain in the present paper we show the existence of infinity many solutions of nonlinear dirichlet boundary value problem u g x u u h 0 1 where g r r is a continuous function satisfying lim t 0g x t t for all x

closure operations that induce big cohen macaulay modules and classification of singularities abstract geoffrey dietz introduced a set of axioms for a closure operation on a complete local domain r so that the existence of such a closure operation is equivalent to the existence of a big cohen macaulay module these closure operations are called dietz closures in complete rings of characteristic p 0 tight closure and plus closure satisfy the axioms we define module closures and discuss their properties for many of these properties there is a smallest closure operation satisfying the property in particular we discuss properties of big cohen macaulay module closures and prove that every dietz closure is contained in a big cohen macaulay module closure using this result we show that under mild conditions a ring r is regular if and only if all dietz closures on r are trivial finally we show that solid closure in equal characteristic 0 integral closure and regular closure are not dietz closures and that all dietz closures are contained in liftable integral closure

on the formulation of new explicit conditions for steepness from a former result of n n nekhoroshev the application of the nekhoroshev theorem to many problems arising in different fields of physics and astronomy depends on a non degeneracy property called steepness that a suitable hamiltonian approximation must satisfy since steepness is implicitly defined we have the problem of recognizing whether a given function is steep or not for this purpose we here consider some sufficient conditions for steepness provided by nekhoroshev in 1979 based on the solvability of a collection of systems depending on the number n of degrees of freedom the derivatives of the function up to a certain order r and some auxiliary parameters these conditions are really explicit only for r 2 corresponding to quasi convexity and for r 3 instead for r 4 the conditions are implicit since they require an elaborate computation of the closure of a certain set in this paper we first revisit nekhoroshev s result and we show that the number of parameters in the collections of systems can be suitably reduced th

the dual variational principle and elliptic problems with discontinuous nonlinearities on etudie des problemes aux valeurs limites elliptiques du type u f u p x x u 0 x ou r n est un domaine borne et f peut avoir des discontinuites on cherche des solutions en utilisant le principe d action duale de clarke

infinitely many solutions for sublinear indefinite nonlocal elliptic equations perturbed from symmetry abstract this paper is concerned the existence of infinitely many solutions for perturbed sublinear indefinite elliptic equations involving the nonlocal operator by using a variant of rabinowitz s perturbation method we study the effect of high order non odd perturbations which break the symmetry of the associated energy functional and show how a sequence of small negative energy solutions persists under broken symmetry situations

ideals in a perfect closure linear growth of primary decompositions and tight closure this paper is concerned with tight closure in a commutative noetherian ring r of prime characteristic p and is motivated by an argument of k e smith and i swanson that shows that if the sequence of frobenius powers of a proper ideal a of r has linear growth of primary decompositions then tight closure of a commutes with localization at the powers of a single element it is shown in this paper that provided r has a weak test element linear growth of primary decompositions for other sequences of ideals of r that approximate in a certain sense the sequence of frobenius powers of a would not only be just as good in this context but in the presence of a certain additional finiteness property would actually imply that tight closure of a commutes with localization at an arbitrary multiplicatively closed subset of r work of m katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the frobenius powers of a has only finitely many maximal members this paper develops through a careful analysis of the ideal theory of the perfect closure of r strategies for showing that tight closure of a specified ideal a of r commutes with localization at an arbitrary multiplicatively closed subset of r and for showing that the union of the associated primes of the tight closures of the frobenius powers of a is actually a finite set several applications of the strategies are presented in most of them it was already known that tight closure commutes with localization but the resulting affirmative answers to katzman s question in the various situations considered are believed to be new

uniform annihilators of local cohomology abstract in this paper we study the properties of noetherian rings containing uniform local cohomological annihilators it turns out that all such rings should be universally catenary and locally equidimensional we will prove a necessary and sufficient condition for such rings which enables us to show that if a locally equidimensional ring r is the image of a cohen macaulay ring then r has a uniform local cohomological annihilator moreover we will give a positive answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13 about excellent rings with dimension no more than 5

integral perfectoid big cohen macaulay algebras via andre s theorem the main result of this article is to prove that any noetherian local domain of mixed characteristic maps to an integral perfectoid big cohen macaulay algebra the proof of this result is based on the construction of almost cohen macaulay algebras in mixed characteristic due to yves andre moreover we prove that the absolute integral closure of a complete noetherian local domain of mixed characteristic maps to an integral perfectoid big cohen macaulay algebra

tight closure of finite length modules in graded rings in this article we look at how the equivalence of tight closure and plus closure or frobenius closure in the homogeneous m coprimary case implies the same closure equivalence in the nonhomogeneous m coprimary case in standard graded rings although our result does not depend upon dimension the primary application is based on results known in dimension 2 due to the recent work of h brenner we also demonstrate a connection between tight closure and the m adic closure of modules extended to r or r we finally show that unlike the noetherian case the injective hull of the residue field over r or r contains elements that are not killed by any power of the maximal ideal of r this fact presents an obstruction to one possible method of extending our main result to all modules

flow reconstruction and invariant tori in the spatial three body problem abstract we deal with the spatial three body problem in the various regimes where the hamiltonian is split as the sum of two keplerian systems plus a small perturbation by averaging over the mean anomalies truncating higher order terms and using singular reduction theory we get a one degree of freedom hamiltonian system departing from the analysis performed in 39 concerning the relative equilibria of this reduced system we carry out the reconstruction of the kam tori surrounding the motions associated to each elliptic equilibrium the existence of five dimensional kam tori for the spatial three body problem is established these tori surround various types of motions from circular to near rectilinear passing through coplanar or perpendicular