a summation formula over the zeros of a combination of the associated legendre functions with a physical application by using the generalized abel plana formula we derive a summation formula for the series over the zeros of a combination of the associated legendre functions with respect to the degree the summation formula for the series over the zeros of the combination of the bessel functions previously discussed in the literature is obtained as a limiting case as an application we evaluate the wightman function for a scalar field with a general curvature coupling parameter in the region between concentric spherical shells on a background of constant negative curvature space for the dirichlet boundary conditions the corresponding mode sum contains the series over the zeros of the combination of the associated legendre functions the application of the summation formula allows us to present the wightman function in the form of the sum of two integrals the first one corresponds to the wightman function for the geometry of a single spherical shell and the second one is induced by the presence of the second shell the boundary induced part in the vacuum expectation value of the field squared is investigated for points away from the boundaries the corresponding renormalization procedure is reduced to that for the boundary free part

a note on confined diffusion abstract the random motion of a brownian particle confined in some finite domain is considered quite generally the relevant statistical properties involve infinite series whose coefficients are related to the eigenvalues of the diffusion operator because the latter depend on space dimensionality and on the particular shape of the domain an analytical expression is in most circumstances not available in this article it is shown that the series may in some circumstances sum up exactly explicit calculations are performed for 2d diffusion restricted to a circular domain and 3d diffusion inside a sphere in both cases the short time behaviour of the mean square displacement is obtained

special functions arising in the study of semi linear equations in circular domains rayleigh functions are defined by the formula s l n n 1 1 l n n 2 l where l 1 2 3 l n n 0 are zeros of the bessel function j n x and n 1 2 3 is the number of the zero these functions appear in the classical problems of vibrating circular membranes heat conduction in cylinders and diffraction through circular apertures in the present paper it is shown that a new family of special functions convolutions of rayleigh functions with respect to the bessel index 1 r l m p k p k m q s 1 1 l p q 2 l1 l k s 2 lforl 1 2 m 0 1 2 arises in constructing solutions of semi linear evolution equations in circular domains see also v varlamov convolution of rayleigh functions with respect to the bessel index j math anal appl 306 2005 413 424 as an example of its application a forced cahn hilliard equation is considered in a unit disc with homogeneous boundary and initial conditions construction of its global in time solutions involves the use of r 1 m and r 2 m a general representation of r l m is deduced and on the basis of that a particular result for r 2 m is obtained convenient for computing its asymptotics as m the latter issue is important for establishing a function space to which a solution of the corresponding problem belongs

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

the fundamental group of the quillen complex of the symmetric group abstract in this paper we investigate the fundamental group of the quillen complex of the symmetric group a p s n we show that when p is an odd prime the simplicial complex a p s n is simply connected if and only if 3 p 2 n p 2 or n p 2 p furthermore we determine the fundamental group 1 a p s n in all cases except those where p 5 and n 3 p 3 p 1 and that where p 3 and n 10

topological conjugacy of piecewise monotonic functions of nonmonotonicity height 1 abstract the conjugacy problem is an important topic in the theory of dynamical systems and functional equations in this paper we investigate a class of piecewise monotone and continuous maps with nonmonotonicity height 1 we give a sufficient and necessary condition under which any two of these maps are topologically conjugate and construct a topological conjugacy with an extension method if such a conjugacy exists

a note on the non existence of global solutions of initial boundary value problems for the boussinesq equation utt 3uxxxx uxx 12 u2 xx

simple homotopy types and finite spaces whitehead s theory of simple homotopy types is inspired by tietze s theorem in combinatorial group theory which states that any finite presentation of a group could be deformed into any other by a finite sequence of elementary moves which are now called tietze transformations whitehead translated these algebraic moves into the well known geometric moves of elementary collapses and expansions of finite simplicial complexes

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

singularly perturbed ordinary differential equations with dynamic limits coupled slow and fast motions generated by ordinary differential equations are examined the qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after invariant measures of the parametrised fast flow are employed to describe the limit behaviour rather than algebraic equations which are used in the standard reduced order approach in the case of a unique invariant measure for each parameter the limit of the slow motion is governed by a chattering type equation without the uniqueness the limit of the slow motion solves a differential inclusion the fast flow in turn converges in a statistical sense to the direct integral respectively the set valued direct integral of the invariant measures

global solutions and smoothing effects for semi linear evolution equations in circular domains abstract the paper deals with global in time solutions of semi linear evolution equations in circular domains these solutions are constructed by means of the series of eigenfunctions of the laplace operator in the corresponding region a forced cahn hilliard type equation in a unit disc is considered as an example the current work focuses on revealing the mechanism of nonlinear smoothing i e on tracing the influence of smoothness of the source term on the regularity of solutions of the nonlinear mixed problem to this end convolutions of rayleigh functions with respect to the bessel index are employed the study of this new family of special functions was initiated in v varlamov convolution of rayleigh functions with respect to the bessel index j math anal appl 306 2005 413 424 v varlamov convolutions of rayleigh functions and their application to semi linear equations in circular domains j math anal appl 327 2007 1461 1478 in order to reveal the effect of nonlinear smoothing anisotropic sobolev spaces h s are introduced they are based on the sobolev spaces h s weighted by tangential derivatives so that the index is responsible for the smoothness in and s is the usual sobolev index global in time solutions of the mixed problem in question are constructed and additional smoothness with respect to the angular coordinate is established in the anisotropic sobolev spaces the above mentioned special functions are essentially used for this purpose

blow up and scattering of solution for a generalized boussinesq equation in this paper we consider the cauchy problem for a class of boussinesq equation we obtain the existence and uniqueness of the local solutions for a class of nonlinearity of the perturbation blow up solutions are obtained furthermore the global existence and nonlinear scattering for small amplitude solutions are established

linearization in two dimensions le choix des coordonnees linearisantes peut etre fait pour etre de classe c s ou s depend de la regularite r de la transformation et de la partie lineaire l on determine le supremum des valeurs s s l r comme exposant de holder quand l espace est a 2 dimensions et r un entier superieur a un on determine quand ce supremum est un maximum

a summation formula over the zeros of the associated legendre function with a physical application associated legendre functions arise in many problems of mathematical physics by using the generalized abel plana formula in this paper we derive a summation formula for the series over the zeros of the associated legendre function of the first kind with respect to the degree the summation formula for the series over the zeros of the bessel function previously discussed in the literature is obtained as a limiting case the wightman function for a scalar field with a general curvature coupling parameter is considered inside a spherical boundary on the background of a constant negative curvature space the corresponding mode sum contains the series over the zeros of the associated legendre function the application of the summation formula allows us to present the wightman function in the form of the sum of two integrals the first one corresponds to the wightman function for the bulk geometry without boundaries and the second one is induced by the presence of the spherical shell for points away from the boundary the latter is finite in the coincidence limit in this way the renormalization of the vacuum expectation value of the field squared is reduced to that for the boundary free part

casimir energy of a ball and cylinder in the zeta function technique a simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder when calculating the casimir energy in this approach no exact divergencies appear and no renormalization is needed the starting point of the consideration is the representation of the zeta functions in terms of contour integral further the uniform asymptotic expansion of the bessel function is essentially used after the analytic continuation needed for calculating the casimir energy the zeta functions are presented as infinite series containing the riemann zeta function with rapidly falling down terms the spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in a uniform endless medium under the condition that the velocity of light does not change when crossing the interface as a special case perfectly conducting spherical and cylindrical shells are also consi

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

zeta function of the bessel operator on the negative real axis we investigate the pole structure of the zeta function zeta anu s sigma k lambda k s built out of the eigenvalues lambda k of a bessel operator anu subject to dirichlet boundary conditions at one end of the domain this leads us to the study of zeta anu on the negative real axis where most of the singularities occur

smooth linearization of hyperbolic fixed points without resonance conditions a well known theorem of hartman grobman says that a c1 diffeomorphism f rn rn with a hyperbolic fixed point at 0 can be conjugated to the linear diffeomorphism l df 0 at least in a neighbourhood of 0 in this paper we will show that if is c2 then is differentiably conjugate to l at 0 moreover the conjugacy is holder outside 0 no resonance conditions will be required

the zero dispersion limit of the korteweg de vries equation for initial potentials with non trivial reflection coefficient the inverse scattering method is used to determine the distribution limit as 0 of the solution u x t of the initial value problem ut 6uux 2uxxx 0 u x 0 v x where v x is a positive bump which decays sufficiently fast as x x the case v x 0 has been solved by peter d lax and c david levermore 8 9 10 the computation of the distribution limit of u x t as 0 is reduced to a quadratic maximization problem which is then solved

xi and eta function resummation of finite series general case given a spectrum of positive numbers lambda m from which a xi function z s sigma m lambda sm can be constructed the reorganization of series of the type f s t identical to m lambda smf lambda mt into power series in t is examined in detail using the method of xi function resummation for summand functions f lambda mt having power series expansions in lambda mt with infinite radius of convergence and which satisfy other conditions of a rather general nature the author finds that f s t can be reorganized to f s t n antbn k cktdk in t r s t where r s t vanishes exponentially as t to 0 the numbers an bn cn dn can all be computed in terms of the xi function z s r s t is difficult to evaluate but important general features of this function can be determined the power series expansion of f s t can be regarded as a generalization of the heat kernel expansion for which f lambda mt exp lambda mt and s 0 to nonzero complex variable s which is useful and to many other summand functions f lambda mt remarkably the xi function resummation method can be applied as easily to divergent series f s t as it can to convergent ones the method is therefore both a rearrangement procedure for convergent series and a summation prescription for divergent series

on the quantum zeta function it is remarkable that the quantum zeta function defined as a sum over energy eigenvalues e admits of exact evaluation in some situations for which not a single e be known herein we show how to evaluate instances of z s and of an associated parity zeta function y s for various quantum systems for some systems both z n y n can be evaluated for infinitely many integers n such z y values can be used for example to effect sharp numerical estimates of a system s ground energy the difficult problem of evaluating the analytic continuation z s for arbitrary complex s is discussed within the contexts of perturbation expansions path integration and quantum chaos

convolutions of rayleigh functions and their application to semi linear equations in circular domains abstract rayleigh functions l are defined as series in inverse powers of the bessel function zeros n 0 l n 1 1 n 2 l where l 1 2 is the index of the bessel function j x and n 1 2 is the number of the zeros convolutions of rayleigh functions with respect to the bessel index r l m k l m k l k for l 1 2 m 0 1 2 are needed for constructing global in time solutions of semi linear evolution equations in circular domains v varlamov on the spatially two dimensional boussinesq equation in a circular domain nonlinear anal 46 2001 699 725 v varlamov convolution of rayleigh functions with respect to the bessel index j math anal appl 306 2005 413 424 the study of this new family of special functions was initiated in v varlamov convolution of rayleigh functions with respect to the bessel index j math anal appl 306 2005 413 424 where the properties of r 1 m were investigated in the present work a general representation of r l m in terms of l is deduced on the basis of this a representation for the function r 2 m is obtained in terms of the function an asymptotic expansion is computed for r 2 m as m such asymptotics are needed for establishing function spaces for solutions of semi linear equations in bounded domains with periodicity conditions in one coordinate as an example of application of r l m a forced boussinesq equation u t t 2 b u t 2 u u u 2 f with b const 0 and const r is considered in a unit disc with homogeneous boundary and initial data construction of its global in time solutions involves the use of the functions r 1 m and r 2 m which are responsible for the nonlinear smoothing effect

smooth linearization near a fixed point in this paper we extend a theorem of sternberg and bi leckii we study a vector field or a diffeomorphism in the vicinity of a hyperbolic fixed point we assume that the eigenvalues of the linear part a at the fixed point satisfy qth order algebraic inequalities where q 2 2 then there is ck linearization in the vicinity of the fixed point where k is the q smoothness of a we give an explicit and simple algorithm for com puting k an illustrative example from celestial mechanics is included which shows that our main theorem is an improvement over previously known theories we also show that if the vector field or diffeomorphism depends smoothly on a parameter then the linearizing conjugation de pends smoothly on the parameter

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

convolution of rayleigh functions with respect to the bessel index a convolution of rayleigh functions with respect to the bessel index can be treated as a special function in its own right it appears in constructing global in time solutions for some semilinear evolution equations in circular domains and may control the smoothing effect due to nonlinearity an explicit representation for it is derived which involves the special function x the logarithmic derivative of the function the properties of the convolution in question are established asymptotic expansions for small and large values of the argument are obtained and the graph is presented

a singular perturbation problem for nonlinear damped hyperbolic equations on considere des equations hyperboliques non lineaires amorties u tt au t au a 2 u g u 0 ou a est un operateur positif et g est la derivee de gateaux d une fonctionnelle convexe on etudie le comportement asymptotique des solutions et la convergence des solutions fortes quand o

decay and scattering of solutions for a generalized boussinesq equation abstract in this paper we consider the long time behavior of small solutions of the cauchy problem for a generalized boussinesq equation a scattering operator and the nonlinear scattering for small amplitude solutions of the boussinesq equation are established under certain hypotheses

infinite products partition functions and the meinardus theorem the meinardus theorem provides an asymptotic value for the infinite product generating functions of many different partition functions and thereby asymptotic values for these partition functions themselves under assumptions somewhat stronger than those of meinardus a more specific version of the theorem is derived then this is generalized from the spectrum of integers considered by meinardus to infinite products constructed from an arbitrary spectrum included are infinite products that generate partition functions able to count states in the quantum theory of extended objects there emerges a quite general and explicit method of which only limited details are given here for deriving the asymptotic growth of the state density function of multidimensional quantum objects one finds a universal geometry independent leading behavior with nonleading geometry dependent corrections that the method is capable of providing explicitly

riccati equations and convolution formulae for functions of rayleigh type kishore 1963 proc am math soc 14 527 considered the rayleigh functions n k 1 j k 2n n 1 2 where j k are the non zero zeros of the bessel function j z and provided a convolution type sum formula for finding n in terms of 1 n 1 his main tool was the recurrence relation for bessel functions here we extend this result to a larger class of functions by using riccati differential equations we get new results for the zeros of certain combinations of bessel functions and their first and second derivatives as well as recovering some results of buchholz for zeros of confluent hypergeometric functions

solutions of the boussinesq equation on a periodic domain abstract the existence and the uniqueness of solutions for a linear feedback controlled boussinesq equation on a periodic domain are studied the continuous dependence of the solution on initial data is also proved the proof is based on conservation laws for the boussinesq equation

the structure of the rational solutions to the boussinesq equation rational solutions of the one dimensional boussinesq equation both with zero and nonzero boundary conditions at the infinity are obtained by reducing the known solutions of the kadomtsev petviashvili equation the structure of the found solutions generalizes a family of rational solutions to the korteweg de vries equation to the case of two wave processes

the homotopy types of the posets of p subgroups of a finite group abstract we study the homotopy properties of the posets of p subgroups s p g and a p g of a finite group g viewed as finite topological spaces we answer a question raised by r e stong in 1984 about the relationship between the contractibility of the finite space a p g and that of s p g negatively and describe the contractibility of a p g in terms of algebraic properties of the group g

multidimensional extension of a wentzel kramers brillouin improvement for spherical quantum billiard zeta functions some insight is offered into the dimensional dependence of the wentzel kramers brillouin wkb and improved wkb approximations introduced by steiner in connection with hyperspherical originally circular quantum billiards the accuracy of every new d dimensional picture is estimated by examining the residues of the spectral zeta function poles

a stronger reformulation of webb s conjecture in terms of finite topological spaces abstract we investigate a stronger formulation of webb s conjecture on the contractibility of the orbit space of the p subgroup complexes in terms of finite topological spaces the original conjecture which was first proved by symonds and more recently by bux libman and linckelmann can be restated in terms of the topology of certain finite spaces we propose a stronger conjecture and prove various particular cases by combining fusion theory of finite groups and homotopy theory of finite spaces

sum rules for zeros of bessel functions and an application to spherical aharonov bohm quantum bags the authors study the sum zeta h s sigma jej s over the eigenvalues ej of the schrodinger equation in a spherical domain with dirichlet walls threaded by a line of magnetic flux rather than using green function techniques they tackle the mathematically non trivial problem of finding exact sum rules for the zeros of bessel functions jv which are extremely helpful when seeking numerical approximations to ground state energies these results are particularly valuable if v is neither an integer nor half an odd integer

the zeta function constructed from the zeros of the bessel function this paper studies the function built from the zeros of the bessel function the known first eight terms of the mcmahon expansion with are used to construct an accurate approximation to the quality of this approximation is investigated numerically by comparison with a known but at least numerically little studied integral formula for excellent numerical agreement is found for fixed and variable real s and for fixed s and variable both formulae for therefore seem to work well our approximation also accurately reproduces known special values of important properties of are investigated for the first time including several of its zeros in addition some general theory is presented in two areas i perturbed spectra and ii the interrelationship between functions like representable as infinite products and the functions constructed from their infinite spectrum of zeros

spectral zeta functions for q bessel equations explicit formulae for the zeta functions of the zeros of hahn exton and jackson s q bessel functions are derived they can be regarded as spectral sum rules for some discrete quantum billiards

convolution formulae for functions of rayleigh type n kishore 1963 proc am math soc 14 527 considered the rayleigh functions where the are the non zero zeros of the bessel function and provided a convolution type sum formula for finding in terms of here we investigate corresponding expressions for sums of reciprocal powers of zeros of derivatives and other functions related to bessel functions it turns out that we can get results similar to kishore s expressing in terms of and

sum rules for quantum billiards the authors write explicit integral expressions for sums of inverse powers of the eigenvalues of the laplacian with dirichlet boundary conditions in a simply connected bounded two dimensional domain

zeta functions of nearly circular domains we study the zeta functions of negative integer powers of eigenvalues of the laplacian on two dimensional domains which are close to the unit disc d for closed form expressions are obtained which describe when the technique developed is applied to derive an asymptotic expansion for zeta functions of regular n sided polygons in the limit

the korteweg de vries equation with small dispersion higher order lax levermore theory we utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem begin gathered u t 6u u x in 2 u xxx 0 hfill u left x 0 right v left x right hfill end gathered when 0 we refine the lax levermore theory which gives the weak limit of the solution as 0 by introducing a quantum condition this allows us to calculate the waveform of the local oscillations up to phase shifts when is small

distributional convergence in planar dynamics and singular perturbations abstract motivated by applications to singular perturbations the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane the solutions may converge either to a limit cycle or to a heteroclinic cycle the limit distributions form invariant measures on the limit set the customary gauges of topological distances may not apply to such cases and do not suit the applications the paper employs the prohorov distance between probability measures it is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle the latter may exhibit two paces depending on a relation among the eigenvalues of the hyperbolic equilibria the limit invariant measures are also exhibited the motivation is stemmed from singularly perturbed systems with non stationary fast dynamics and averaging the resulting rates of convergence are displayed for a planar singularly perturbed system and for a general system of a slow flow coupled with a planar fast dynamics

coarsening dynamics of the convective cahn hilliard equation abstract we characterize the coarsening dynamics associated with a convective cahn hilliard equation cch in one space dimension first we derive a sharp interface theory through a matched asymptotic analysis two types of phase boundaries kink and anti kink arise due to the presence of convection and their motions are governed to leading order by a nearest neighbors interaction coarsening dynamical system cds theoretical predictions on cds include the characteristic length l m for coarsening exhibits the temporal power law scaling t1 2 provided l m is appropriately small with respect to the peclet length scale l p binary coalescence of phase boundaries is impossible ternary coalescence only occurs through the kink ternary interaction two kinks meet an anti kink resulting in a kink direct numerical simulations performed on both cds and cch confirm each of these predictions a linear stability analysis of cds identifies a pinching mechanism as the dominant instability which in turn leads to kink ternaries we propose a self similar period doubling pinch ansatz as a model for the coarsening process from which an analytical coarsening law for the characteristic length scale l m emerges it predicts both the scaling constant c of the t1 2 regime i e l m ct 1 2 as well as the crossover to logarithmically slow coarsening as l m crosses l p our analytical coarsening law stands in good qualitative agreement with large scale numerical simulations that have been performed on cch

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

on the dissipative boussinesq equation in a non cylindrical domain abstract we investigate the initial boundary value problem for the one dimensional nonlinear boussinesq equation inside domains with moving ends having both small increasing and decreasing displacements global solvability uniqueness of solutions and the exponential decay to the energy are established provided the initial data are bounded in some sense

the sil nikov problem exponential expansion strong λ lemma c1 linearization and homoclinic bifurcation on introduit une nouvelle theorie appelee le developpement exponentiel et on explore ses interactions avec la theorie de la variete invariante locale les lemmes et la theorie de linerarisation reguliere

sharpness for c 1 linearization of planar hyperbolic diffeomorphisms abstract c 1 linearization preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions planar hyperbolic diffeomorphisms are the most elementary ones of representatively technical difficulties in the study of c 1 linearization in the poincare domain both eigenvalues inside the unit circle s 1 a lower bound 0 was given such that c 1 smoothness with 0 1 admits c 1 linearization our first purpose of this paper is to prove the sharpness of 0 and give a weaker linearization for 0 in the siegel domain one eigenvalue inside s 1 but the other outside s 1 it is known that c 1 smoothness admits c 1 linearization for all 0 1 the second purpose is to prove that the c 1 linearization is actually a c 1 linearization and give sharp estimates for