a criterion of integrability for perturbed nonresonant harmonic oscillators wick ordering of the perturbations in classical mechanics and invariance of the frequency spectrum we introduce an analogue to the renormalization theory of quantum fields into classical mechanics we also find an integrability criterion guaranteeing the convergence of birkhoff s series and an algorithm for modifying the hamiltonian to fix the frequency spectrum of the quasi periodic motions we point out its possible relevance to the transition to chaos

non degenerate liouville tori are kam stable in this short note we prove that a quasi periodic torus with a non resonant frequency that can be diophantine or liouville and which is invariant by a sufficiently regular hamiltonian flow is kam stable provided it is kolmogorov non degenerate when the hamiltonian is smooth respectively gevrey smooth respectively real analytic the invariant tori are smooth respectively gevrey smooth respectively real analytic this answers a question raised in a recent work by eliasson fayad and krikorian 6 we also take the opportunity to ask other questions concerning the stability of non resonant invariant quasi periodic tori in analytic or smooth hamiltonian systems

an alternating explicit implicit domain decomposition method for the parallel solution of parabolic equations explicit implicit domain decomposition eidd is a class of globally non iterative non overlapping domain decomposition methods for the numerical solution of parabolic problems on parallel computers which are highly efficient both computationally and communicationally for each time step in this paper an alternating eidd method is proposed which is algorithmically simple efficient for each time step highly parallel and satisfies a stability condition that imposes no additional restriction to the time step restriction imposed by the consistency condition which guarantees a convergence of order o dth 1n b n o h 2 in an h 1 type norm where n b and n respectively denote the number of gridpoints on the interface boundaries b and the number of gridpoints on the entire discrete domain

on properties not inherited by monoids from their schutzenberger groups we give an example of a monoid with finitely many left and right ideals all of whose schutzenberger groups are presentable by finite complete rewriting systems and so each have finite derivation type but such that the monoid itself does not have finite derivation type and therefore does not admit a presentation by a finite complete rewriting system the example also serves as a counterexample to several other natural questions regarding complete rewriting systems and finite derivation type specifically it allows us to construct two finitely generated monoids m and n with isometric cayley graphs where n has finite derivation type respectively admits a presentation by a finite complete rewriting system but m does not this contrasts with the case of finitely generated groups for which finite derivation type is known to be a quasi isometry invariant the same example is also used to show that neither of these two properties is preserved under finite green index extensions

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

differential systems with fuchsian linear part correction and linearization normal forms and multiple orthogonal polynomials abstract differential systems with a fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations such systems are not necessarily analytically equivalent to their linear part they are not linearizable and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable more generally normal forms are found the corrections and the normal forms are found constructively expansions in multiple orthogonal polynomials and their generalization to matrix valued polynomials are instrumental to these constructions

super exponential stability for generic real analytic elliptic equilibrium points we consider the dynamics in a neighborhood of an elliptic equilibrium point with a diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability generically both in a topological and measure theoretical sense any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point

syntactic and rees indices of subsemigroups abstract we define two different notions of index for subsemigroups of semigroups the right syntactic index and the rees index we investigate the relationships between them and with the group index in particular we show that the syntactic index is a generalisation of both the group index and the rees index we use this fact to prove further similarities between the group index and the rees index it turns out however that very few of the nice properties of the group index are inherited by the syntactic index

convergence versus integrability in birkhoff normal form we show that any analytically integrable hamiltonian system near an equilibrium point admits a convergent birkhoff normalization the proof is based on a new geometric approach to the topic

correction and linearization of resonant vector fields and diffeomorphisms we extend the classical siegel brjuno russmann linearization theorem to the resonant case by showing that under a d brjuno s diophantine condition any resonant local analytic vector field resp diffeomorphism possesses a well defined correction which 1 depends on the chart but in any given chart is unique 2 consists solely of resonant terms and 3 has the property that when substracted from the vector field resp when factored out of the diffeomorphism the vector field or diffeomorphism thus corrected becomes analytically linearizable with a privileged or canonical linearizing map moreover in spite of the small denominators and contrary to a hitherto prevalent opinion the correction s analyticity can be established by pure combinatorics without any analysis

persistence of lagrange invariant tori at tangent degeneracy abstract in the present paper without russmann s nondegenderate condition we show the persistence of kam invariant tori using this theorem we obtain the persistence of invariant tori for degenerate systems rising from restricted three body problems especially for lagrange equilibrium position l2 with certain degeneracy

analytic normalization of analytically integrable differential systems near a periodic orbit abstract for an analytic differential system in r n with a periodic orbit we will prove that if the system is analytically integrable around the periodic orbit i e it has n 1 functionally independent analytic first integrals defined in a neighborhood of the periodic orbit then the system is analytically equivalent to its poincare dulac type normal form this result is an extension of analytically integrable differential systems around a singularity to the ones around a periodic orbit

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

green index and finiteness conditions for semigroups let s be a semigroup and let t be a subsemigroup of s then t acts on s by left and by right multiplication if the complement s t has finitely many strong orbits by both these actions we say that t has finite green index in s this notion of finite index encompasses subgroups of finite index in groups and also subsemigroups of finite rees index complement therefore the question of s and t inheriting various finiteness conditions from each other arises in this paper we consider and resolve this question for the following finiteness conditions finiteness residual finiteness local finiteness periodicity having finitely many right ideals and having finitely many idempotents

an unconditionally stable alternating segment difference scheme of eight points for the dispersive equation a group of new saul yev type asymmetric difference schemes to approach the dispersive equation are given here on the basis of these schemes an alternating difference scheme with intrinsic parallelism for solving the dispersive equation is constructed the scheme is unconditionally stable numerical experiments show that the method has high accuracy copyright 2006 john wiley sons ltd

from dynamical systems to renormalization in this paper we study logarithmic derivatives associated to derivations on completed graded lie algebra as well as the existence of inverses these logarithmic derivatives when invertible generalize the exp log correspondence between a lie algebra and its lie group such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms these concepts stemming from the theory of dynamical systems can be rephrased in the abstract setting of lie algebra and the same difficulties as in perturbative quantum field theory pqft arise here surprisingly one can adopt the same ideas as in pqft with fruitful results such as new constructions of normal forms with the help of the birkhoff decomposition the analogy goes even further locality of counter terms choice of a renormalization scheme and shall lead to more interactions between dynamical systems and quantum field theory

on residual finiteness of monoids their schutzenberger groups and associated actions abstract in this paper we discuss connections between the following properties rfm residual finiteness of a monoid m rfsg residual finiteness of schutzenberger groups of m and rfrl residual finiteness of the natural actions of m on its green s r and l classes the general question is whether rfm implies rfsg and or rfrl and vice versa we consider these questions in all the possible combinations of the following situations m is an arbitrary monoid m is an arbitrary regular monoid every j class of m has finitely many r and l classes m has finitely many left and right ideals in each case we obtain complete answers which are summarised in a table

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

a new adi technique for two dimensional parabolic equation with an integral condition abstract nonclassical parabolic initial boundary value problems arise in the study of several important physical phenomena this paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions a new fourth order finite difference technique based upon the noye and hayman n h alternating direction implicit adi scheme is used as the basis to solve the two dimensional time dependent diffusion equation with an integral condition replacing one boundary condition this scheme uses less central processor time cpu than a second order fully implicit scheme based on the classical backward time centered space btcs method for two dimensional diffusion it also has a larger range of stability than a second order fully explicit scheme based on the classical forward time centered space ftcs method the basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyeet this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods the results of numerical experiments for the new method are presented the central processor times needed are also reported error estimates derived in the maximum norm are tabulated

divergent birkhoff normal forms of real analytic area preserving maps in this paper we provide a real analytic area preserving map that possesses a divergent birkhoff normal form near an elliptic fixed point the method uses the small divisor theory and the work of gong in the study of the hamiltonian system

painleve analysis and normal forms theory nonlinear vector fields have two important types of singularities the fixed points in phase space and the time singularities in the complex plane the first singularities are locally analyzed via normal form theory whereas the second ones are studied by the painleve analysis in this paper normal form theory is used to describe the solutions around their complex time singularities to do so a transformation mapping the local series around the singularities to the local series around a fixed point of a new system is introduced regular normal form theory is then used in this new system it is shown that a vector field has the painleve property only if the associated system is locally linearizable around its fixed points a problem analogous to the classical problem of the center moreover the connection between partial and complete integrability and the structure of local series around both types of singularities are established a new proof of the convergence of the local psi series is given and an explicit method to prove the existence of finite time blow up manifold in phase space is presented

around the stability of kam tori we study the accumulation of an invariant quasiperiodic torus of a hamiltonian flow by other quasi periodic invariant tori we show that an analytic invariant torus t0 with diophantine frequency 0 is never isolated due to the following alternative if the birkhoff normal form of the hamiltonian at t0 satisfies a russmann transversality condition the torus t0 is accumulated by kam tori of positive total measure if the birkhoff normal form is degenerate there exists a subvariety of dimension at least d 1 that is foliated by analytic invariant tori with frequency 0 for frequency vectors 0 having a finite uniform diophantine exponent this includes a residual set of liouville vectors we show that if the hamiltonian h satisfies a kolmogorov non degeneracy condition at t0 then t0 is accumulated by kam tori of positive total measure in 4 degrees of freedom or more we construct for any 0 r c gevrey hamiltonians h with a smooth invariant torus t0 with frequency 0 that is not accumulated by a positive measure of invariant tori

linearization of control systems we address the problem of static state linearization of multi input nonlinear control systems via coordinate transformation necessary and sufficient geometric conditions in terms of certain set of vector fields associated with the system were obtained in the early eighties stating the fact that such set of vector fields should be commutative and of constant rank the state linearization problem i e the finding of linearizing coordinates was thus reduced to solving a set of partial differential equations the objective of this paper is to provide an algorithm allowing to compute explicitly the linearizing state coordinates the algorithm is performed using a maximum of n 1 steps n being the dimension of the system and is made possible by extending the explicit solvability of the flow box theorem to a commutative set of vector fields examples are provided to illustrate the results an extension of the method to dynamic feedback linearization is also outlined

efficient parallel algorithms for parabolic problems domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation implicit schemes are used in order to march with large time steps parallelization is realized by approximating interface values using explicit computation various techniques are examined including a multistep second order explicit scheme and a one step high order scheme we show that the resulting schemes are of second order global accuracy in space and stable in the sense of osher or in l infty they are optimized with respect to the parallel efficiency

unconditional stability of alternating difference schemes with variable time steplengthes for dispersive equation in this paper the difference method with intrinsic parallelism for dispersive equation is studied the general alternating difference schemes with variable time steplengthes are constructed and proved to be unconditionally stable some concrete parallel difference schemes are the special cases of the general schemes such as the alternating group explicit scheme with variable time steplengthes the alternating segment explicit implicit scheme with variable time steplengthes the alternating segment crank nicolson scheme with variable time steplengthes and so on the numerical results are given to show the effectiveness of the present method they show that the variable time steplengthes schemes are more accurate and can be obtained with less computational effort than the equal time steplengthes schemes

isochronous centers of polynomial hamiltonian systems and a conjecture of jarque and villadelprat abstract we study the conjecture of jarque and villadelprat stating that every center of a planar polynomial hamiltonian system of even degree is nonisochronous this conjecture has already been proved for quadratic and quartic systems using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields we are able to describe a very large class of nonisochronous hamiltonian systems of even arbitrarily large degree

unconditional stability of alternating difference schemes with intrinsic parallelism for two dimensional fourth order diffusion equation in this paper the parallel difference schemes for parabolic equation are studied the general alternating difference schemes with intrinsic parallelism for two dimensional fourth order diffusion equation are presented and proved to be unconditionally stable by the energy method two kinds of alternating band explicit abe1 abe2 schemes alternating band explicit implicit abe i scheme and alternating band crank nicolson abc n scheme are the special cases of the general schemes constructed here numerical experiments are performed to examine the accuracy and unconditional stability

total convergence or general divergence in small divisors we study generic holomorphic families of dynamical systems presenting problems of small divisors with fixed arithmetic the characteristic features are delicate problems of convergence of formal power series due to small divisors we prove the following dichotomy we have convergence for all parameter values or divergence everywhere except for an exceptional pluri polar set of parameters we illustrate this general principle in different problems of small divisors as an application we obtain new richer families of non linearizable examples in the siegel problem when the bruno condition is violated generalizing and extending to higher dimension previous results of yoccoz and the author

convergence or generic divergence of the birkhoff normal form we prove that the birkhoff normal form of hamiltorlian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent the same result is proved for the normalization mapping and any formal first integral

unconditional stability of alternating difference schemes with intrinsic parallelism for the fourth order parabolic equation in this paper the difference method with intrinsic parallelism for the fourth order parabolic equation is studied the general alternating difference schemes are constructed and proved to be unconditionally stable the alternating group explicit scheme alternating segment explicit implicit scheme and alternating segment crank nicolson scheme are the special cases of the general schemes constructed here numerical results are presented to examine the convergence stability and accuracy of the general schemes