on the central configurations of the planar restricted four body problem abstract this article is devoted to answering several questions about the central configurations of the planar 3 1 body problem firstly we study bifurcations of central configurations proving the uniqueness of convex central configurations up to symmetry secondly we settle the finiteness problem in the case of two nonzero equal masses lastly we provide all the possibilities for the number of symmetrical central configurations and discuss their bifurcations and spectral stability our proofs are based on applications of rational parametrizations and computer algebra

general thermostatistical formalisms invariance under uniform spectrum translations and tsallis q additivity the recent proposal by tsallis of a generalized thermostatistics devised to treat systems endowed with long range interactions long range memory eects or a fractal like relevant phase space has raised interesting and profound issues concerning the properties of general thermostatistical formalisms in the present paper we identify families of thermostatistical formalisms that share some fundamental characteristics the canonical ensemble s invariance under uniform translations of the hamiltonian s energy spectrum is shown to be a universal property veried by any thermostatistical formalism based upon linear mean energy constraints we also provide multiparametric families of entropies exhibiting tsallis q additivity law c 1999 elsevier science b v all rights reserved

quantum and classical pseudogroups part ii differential and symplectic pseudogroups the category of sympletic pseudospaces analogical to the category of pseudospaces in the sense of 2 is introduced and used to define symplectic pseudogroups structures analogical to pseudogroups 3 or quantum groups 4 it is shown that symplectic pseudogroups are in one to one correspondence with manin groups also introduced in this paper the set theoretical part of these structures has been described in i

quantum corrections for general partition functions abstract the classical limit for quite general partition functions is obtained using coherent states in this framework a general procedure is presented to obtain all the quantum corrections to the classical limit in particular the first and second order corrections are worked out explicitly and the classical limit for the tsallis thermostatistics is determined the results of this work generalize the ones obtained by wigner phys rev 40 1932 749 for usual statistical mechanics

global hyperbolicity of renormalization for c r unimodal mappings in this paper we extend m lyubichi s recent results on the global hyperbolicity of renormalization of quadratic like germs to the space of cr unimodal maps with quadratic critical point we show that in this space the boundedtype limit sets of the renormalization operator have an invariant hyperbolic structure provided r i 2 a with a close to one as an intermediate step between lyubichi s results and ours we prove that the renormalization operator is hyperbolic in a banach space of real analytic maps we construct the local stable manifolds and prove that they form a continuous lamination whose leaves are c1 codimension one banach submanifolds of the ambient space and whose holonomy is c1 for some 0 we also prove that the global stable sets are c1 immersed codimension one submanifolds as well provided r i 3 a with a close to one as a corollary we deduce that in generic oneparameter families of cr unimodal maps the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a cantor set with hausdorff dimension less than one

variational methods in nonextensive tsallis statistics a comparative study abstract in the context of nonextensive tsallis statistics sensible examples are employed in order to compare the variational method proposed in phys rev lett 80 1998 218 with the one in j phys a 26 1993 l893 these examples are such that they can be solved exactly thus a comparison of the exact result with the approximated ones is possible the analysis of these examples indicates that the calculation related to the first variational method is easier to work with than the other method in particular when one is trying to apply the last variational method the calculation of the approximated case can be more cumbersome than solving the exact one

le langage des espaces et des groupes quantiques we study the foundations of the differential calculus in quantum geometry the notions of differential quantum space and cone are introduced generalizing a construction of manin to a quantum cone we associate the quantum group of its linear automorphisms preserving the differentials and deduce a de rham complex on this group we give examples of differential calculi on quantum hyperplanes and quantum linear groups

nonlinear fokker planck equations and generalized entropies abstract we consider maximum entropy solutions to nonlinear fokker planck equations within general thermostatistical formalisms we show that the family of generalized nonextensive entropies introduced by tsallis is the only one that has an associated family of nonlinear fokker planck equations endowed with time dependent maxent solutions of a generalized gaussian type consequently the natural association that arises between tsallis thermostatistical formalism and special families of nonlinear fokker planck equations is not a universal feature of general thermostatistical formalisms but a particular feature of the tsallis one

on the existence of feigenbaum s fixed point we give a proof of the existence of ac2 even solution of feigenbaum s functional equation g text x lambda 0 1 g text g lambda 0 x g text 0 1 whereg is a map of 1 1 into itself it extends to a real analytic function over

periodic orbits symbolic dynamics and topological entropy for the restricted 3 body problem this paper concerns the restricted 3 body problem by applying topological methods we give a computer assisted proof of the existence of some classes of periodic orbits the existence of symbolic dynamics and we give a rigorous lower estimate for the topological entropy

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

automatic differentiation for fourier series and the radii polynomial approach abstract in this work we develop a computer assisted technique for proving existence of periodic solutions of nonlinear differential equations with non polynomial nonlinearities we exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system we compute a numerical fourier expansion of the periodic orbit for the augmented system and prove the existence of a true solution nearby using an a posteriori validation scheme the radii polynomial approach the problems considered here are given in terms of locally analytic vector fields i e the field is analytic in a neighborhood of the periodic orbit hence the computer assisted proofs are formulated in a banach space of sequences satisfying a geometric decay condition in order to illustrate the use and utility of these ideas we implement a number of computer assisted existence proofs for periodic orbits of the planar circular restricted three body problem pcrtbp

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

a shorter proof of the existence of the feigenbaum fixed point we use the leray schauder fixed point theorem to prove the existence of an analytic fixed point for the period doubling accumulation renormalization operator our argument does not however show that the linearization of the renormalization operator at this fixed point is hyperbolic

perturbation and variational methods in nonextensive tsallis statistics a unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on tsallis entropy is given here in the case of the variational method the bogoliubov inequality is generalized in a very natural way following the feynman proof for the usual statistical mechanics the inequality turns out to be form invariant with respect to the entropic index q the method is illustrated with a simple example in classical mechanics the formalisms developed here are expected to be useful in the discussion of nonextensive systems

variational calculus with constraints on general algebroids variational calculus on a vector bundle e equipped with a structure of a general algebroid is developed together with the corresponding analogs of euler lagrange equations constrained systems are introduced in the variational and geometrical settings the constrained euler lagrange equations are derived for analogs of holonomic vakonomic and nonholonomic constraints this general model covers the majority of first order lagrangian systems which are present in the literature and reduces to the standard variational calculus and the euler lagrange equations in classical mechanics for e tm

formal symplectic groupoid of a deformation quantization we give a self contained algebraic description of a formal symplectic groupoid over a poisson manifold m to each natural star product on m we then associate a canonical formal symplectic groupoid over m finally we construct a unique formal symplectic groupoid with separation of variables over an arbitrary kahler poisson manifold

a q generalization of laplace transforms the laplace transform is generalized by using the q exponential function eqx 1 1 q x 1 1 q that emerges from tsallis non extensive statistical mechanics and some of its properties are obtained the usual transform is recovered as a limiting case q1 the use of the q laplace transform is illustrated by establishing a relation between the classical canonical q partition function and the density of states

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

generalized statistics variational perturbation approximation using q deformed calculus abstract a principled framework to generalize variational perturbation approximations vpas formulated within the ambit of the nonadditive statistics of tsallis statistics is introduced this is accomplished by operating on the terms constituting the perturbation expansion of the generalized free energy gfe with a variational procedure formulated using q deformed calculus a candidate q deformed generalized vpa gvpa is derived with the aid of the hellmann feynman theorem the generalized bogoliubov inequality for the approximate gfe are derived for the case of canonical probability densities that maximize the tsallis entropy numerical examples demonstrating the application of the q deformed gvpa are presented the qualitative distinctions between the q deformed gvpa model vis a vis prior gvpa models are highlighted

tsallis nonextensive statistics with normalized q expectation values thermodynamical stability and simple illustrations abstract we prove that the specific heat c q q is nonnegative for q 0 1 in the tsallis generalized canonical ensemble with normalized q expectation values in this way we show the thermodynamic stability for this ensemble we also analyze the behavior of some thermodynamic functions such as internal energy and specific heat in the classical and quantum cases considering a spectrum of energy e n e n r e r 0

a nonextensive maximum entropy approach to a family of nonlinear reaction diffusion equations we consider a monoparametric family of reaction diffusion equations endowed with both a nonlinear diffusion term and a nonlinear reaction one that possess exact time dependent particular solutions of the tsallis maximum entropy maxent form the evolution of these solutions is governed by a system of three coupled nonlinear ordinary differential equations that are integrated numerically a simple population dynamics interpretation provides a qualitative understanding of the behaviour of the q maxent solutions when the reaction term vanishes the time dependent distributions studied here reduce to the previously known tsallis maxent solutions for the nonlinear diffusion equation

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

chaotic motions in the restricted four body problem via devaney s saddle focus homoclinic tangle theorem abstract we prove the existence of chaotic motions in an equilateral planar circular restricted four body problem crfbp establishing that the system is not integrable the proof works by verifying the hypotheses of a topological forcing theorem for hamiltonian vector fields on r 4 which hypothesizes the existence of a transverse homoclinic orbit in the energy manifold of a saddle focus equilibrium we develop mathematically rigorous computer assisted arguments for verifying these hypotheses and provide an implementation for crfbp due to the hamiltonian structure this also establishes the existence of a blue sky catastrophe and hence an analytic family of periodic orbits of arbitrarily long period at nearby energy levels our method works far from any perturbative regime and requires no mass symmetry additionally the method is constructive and yields additional byproducts such as the locations of transverse connecting orbits quantitative information about the invariant manifolds and bounds on transport times

a variational perturbation approximation method in tsallis non extensive statistical physics abstract for the generalized statistical mechanics based on the tsallis entropy a variational perturbation approximation method with the principle of minimal sensitivity is developed by calculating the generalized free energy up to the third order in variational perturbation expansion the approximation up to the first order amounts to a variational approach which covers the variational method developed by e k lenzi l c malacarne r s mendes phys rev lett 80 1998 218 and the approximations up to higher orders can systematically improve variational results as an illustrated example the generalized free energy for a classical harmonic oscillator considered in the lenzi s joint work are calculated up to the third order and the resultant approximations up to the first second and third orders are numerically compared with the exact result

formulation of the hellmann feynman theorem for the second choice version of tsallis thermostatistics abstract an approach to formulating the hellmann feynman theorem within the second choice formalism of non extensive statistical mechanics is considered for the state of thermal equilibrium we derive a relation of hellmann feynman type between the derivative of the non extensive free energy with respect to the external parameter and the quantum statistical q average of the derivative of the hamilton operator we also give a proper extension for an arbitrary observable commuting with the hamiltonian some reasons for the usefulness of new formulas are discussed

two long standing problems in tsallis statistics abstract on the basis of tsallis entropy and the joint probability factorization condition two controversial problems existing in tsallis statistics are investigated where one is whether energy is extensive or not and the other is whether it is necessary to introduce the so called generalized zeroth law of thermodynamics or not the results obtained show clearly that like entropy energy is also nonextensive in tsallis statistics and that the zeroth law of thermodynamics has been implicitly used in tsallis statistics since 1988 moreover it is expounded that the standard energy additivity rule adopted by a great number of researchers is not suitable in tsallis statistics because its corollary is in contradiction with the zeroth law of thermodynamics

quantum and classical pseudogroups part i union pseudogroups and their quantization union pseudogroups structures analogical to pseudogroups in the sense of 1 are defined using the category dual to the category of groupoids instead of the category of pseudospaces in the sense of 2 it is shown that these structures are equivalent to double groups in the sense of 3 moreover it is shown that a quantization procedure associates with each finite union pseudogroup a quantum pseudogroup therefore for each finite double group there is a finite pseudogroup

period doubling forc2 mappings the rigorous treatment of period doubling cascades developed by lanford for analytic families is extended to coverc2 families for any 0 this requires spectral analysis of the linearised doubling operator on spaces of differentiable mappings and a version of stable manifold theory which takes account of the non differentiability of the doubling operator on these spaces

information gain within nonextensive thermostatistics we discuss the information theoretical foundations of the kullback information gain recently generalized within a nonextensive thermostatistical formalism general properties are studied and in particular a consistent test for measuring the degree of correlation between random variables is proposed in addition minimum entropy distributions are discussed and the h theorem is proved within the generalized context

a new branch of mountain pass solutions for the choreographical 3 body problem we prove the existence of a new branch of solutions of mountain pass type for the periodic 3 body problem with choreographical constraint at first we describe the variational structure of the action functional associated to the choreographical three body problem in mathbb r 3 in the second part using a bisection algorithm we provide a numerical non rigorous solution of mountain pass type for this problem in a rotating frame with angular velocity 1 5 the last step consists in the rigorous computer assisted proof of the existence of a full branch of solutions for the problem starting from the mountain pass solution detected numerically

a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps rigorous results abstract in this paper we prove rigorous results on persistence of invariant tori and their whiskers the proofs are based on the parameterization method of x cabre e fontich r de la llave the parameterization method for invariant manifolds i manifolds associated to non resonant subspaces indiana univ math j 52 2 2003 283 328 x cabre e fontich r de la llave the parameterization method for invariant manifolds ii regularity with respect to parameters indiana univ math j 52 2 2003 329 360 the invariant manifolds results proved here include as particular cases of the usual strong stable and strong unstable manifolds but also include other non resonant manifolds the method lends itself to numerical implementations whose analysis and implementation is studied in a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical algorithms preprint 2005 a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical implementation and examples preprint 2005 the results are stated as a posteriori results namely that if one has an approximate solution which is not degenerate then one has a true solution not too far from the approximate one this can be used to validate the results of numerical computations

continuation of homoclinic orbits in the suspension bridge equation a computer assisted proof abstract in this paper we prove existence of symmetric homoclinic orbits for the suspension bridge equation u u e u 1 0 for all parameter values 0 5 1 9 for each a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using chebyshev series the proof is computer assisted and combines the uniform contraction theorem and the radii polynomial approach which provides an efficient means of determining a set centered at a numerical approximation of a solution on which a newton like operator is a contraction

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators