on the syntactic transformation semigroup of a language generated by a finite biprefix code abstract let p be a finite biprefix code and let x q s be the syntactic transformation semigroup ts of p we show that if e s is an idempotent then the ts x e qe eqe consists of partial one to one maps we also show that any ts of partial one to one maps divides a ts of partial one to one maps which is the syntactic ts of a finite biprefix code

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

covers for monoids a monoid m is an extension of a submonoid t by a group g if there is a morphism from m onto g such that t is the inverse image of the identity of g our first main theorem gives descriptions of such extensions in terms of groups acting on categories the theory developed is also used to obtain a second main theorem which answers the following question given a monoid m and a submonoid t under what conditions can we find a monoid m and a morphism f from m onto m such that m is an extension of a submonoid t by a group and f maps t isomorphically onto t these results can be viewed as generalisations of two seminal theorems of mcalister in inverse semigroup theory they are also closely related to ash s celebrated solution of the rhodes conjecture in finite semigroup theory mcalister proved that each inverse monoid admits an e unitary inverse cover and gave a structure theorem for e unitary inverse monoids many researchers have extended one or both of these results to wider classes of semigroups almost all these generalisations can be recovered from our two main theorems

on locally reversible languages there exist several works that study the class of reversible languages defined as the union closure of 0 reversible languages their properties and suitable representations in this work we define and study the class of locally reversible languages defined as the union closure of k reversible languages we characterize the class and prove that it is a local positive variety of formal languages we also extend the definition of quasi reversible automata to deal with locally reversible languages and propose a polynomial algorithm to obtain for any given locally k reversible language a quasi k reversible automaton

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

expansions free inverse semigroups and schutzenberger product in this paper we shall present a new construction of the free inverse monoid on a set x contrary to the previous constructions of 9 11 our construction is symmetric and originates from classical ideas of language theory the ingredients of this construction are the free group on x and the relation that associates to a word w of the free monoid on x the set of all pairs u v such that uv w it follows at once from our construction that the free inverse monoid on x can be naturally embedded into the schutzenberger product of two free groups of basis x we shall also give some connections with the theory of expansions as developed by rhodes and birget 2 3

on the varieties of languages associated with some varieties of finite monoids with commuting idempotents abstract eilenberg has shown that there is a one to one correspondence between varieties of finite monoids and varieties of recognizable languages in this paper we give a description of a variety of languages close to the class of piecewise testable languages considered by i simon the corresponding variety of monoids is the variety of j trivial monoids with commuting idempotents this result is then generalized to the case of finite monoids with commuting idempotents whose regular d classes are groups from a given variety of groups

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

well posedness for the navier slip thin film equation in the case of partial wetting we are interested in the thin lm equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid lm driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourthorder parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with p 13 1 4 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l 2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

existence of inverse jacobi multipliers around hopf points in r 3 emphasis on the center problem abstract we consider the center problem at hopf points of analytic systems in r 3 that has a classical solution in the lyapunov center theorem which is given in terms of an analytic first integral here we give a new solution in terms of an analytic inverse jacobi multiplier v the existence of a smooth and non flat inverse jacobi multiplier around a hopf point of saddle focus type is also proved when studying these problems we needed to discuss the relation between inverse jacobi multipliers and center manifolds w c in particular to know under what conditions w c v 1 0 to illustrate our results we solve the center problem for the lu system

well posedness for a class of thin film equations with general mobility in the regime of partial wetting we establish well posedness for the family of thin film equations left begin array ll h t h n h xxx x 0 quad rm in h 0 h 0 h x 1 quad quad rm on partial h 0 end array right 1 with n in 0 frac 14 5 backslash 1 2 the model 1 with n in 0 3 has been used to describe the evolution of a capillary driven thin liquid droplet on a solid substrate in terms of its height profile h geqq 0 the family of thin film equations 1 provides a model problem to investigate contact line propagation in fluid dynamics under relaxed slip conditions the parabolicity of the fourth order parabolic problem degenerates at the free boundary which leads to a loss of regularity at the moving contact point our solutions are regular in terms of the two variables d x and d x 3 n where d x is the distance to the free boundary the main technical difficulty in the analysis of 1 is related to the loss of regularity at the contact points

stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem abstract consider the thin film equation h t h h y y y y 0 with a zero contact angle at the free boundary that is at the triple junction where liquid gas and solid meet previous results on stability and well posedness of this equation have focused on perturbations of equilibrium stationary or self similar profiles the latter eventually wetting the whole surface these solutions have their counterparts for the second order porous medium equation h t h m y y 0 where m 1 is a free parameter both porous medium and thin film equation degenerate as h 0 but the porous medium equation additionally fulfills a comparison principle while the thin film equation does not in this note we consider traveling waves h v 6 x 3 x 2 for x 0 where x y v t and v 0 are free parameters these traveling waves are receding and therefore describe de wetting a phenomenon genuinely linked to the fourth order nature of the thin film equation and not encountered in the porous medium case as it violates the comparison principle the linear stability analysis leads to a linear fourth order degenerate parabolic operator for which we prove maximal regularity estimates to arbitrary orders of the expansion in x in a right neighborhood of the contact line x 0 this leads to a well posedness and stability result for the corresponding nonlinear equation as the linearized evolution has different scaling as x 0 and x the analysis is more intricate than in related previous works we anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated such as droplet rupture

on uniqueness of weak solutions for the thin film equation abstract in any number of space variables we study the cauchy problem related to the fourth order degenerate diffusion equation s h h h 0 this equation derived from a lubrication approximation also models the surface tension dominated flow of a thin viscous film in the hele shaw cell our focus is on uniqueness of weak solutions in the complete wetting regime when a zero contact angle between liquid and solid is imposed in this case we transform the problem by zooming into the free boundary and look at small lipschitz perturbations of a quadratic stationary solution in the perturbational setting the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide optimal conditions on the initial data under which a unique solution exists in fact this solution can be used to define a class of functions in which the original initial value problem has a unique weak solution moreover we show that the interface between empty and wetted regions is an analytic hypersurface in time and space

the navier slip thin film equation for 3d fluid films existence and uniqueness abstract we consider the thin film equation t h h 2 h 0 in physical space dimensions i e one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension which corresponds to the lubrication approximation of the navier stokes equations of a three dimensional viscous thin fluid film with navier slip at the substrate this equation can have a free boundary the contact line moving with finite speed at which we assume a zero contact angle condition complete wetting regime previous results have focused on the 1 1 dimensional version where it has been found that solutions are not smooth as a function of the distance to the free boundary in particular a well posedness and regularity theory is more intricate than for the second order counterpart the porous medium equation or the thin film equation with linear mobility corresponding to darcy dynamics in the hele shaw cell here we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling wave profile this leads to control on the free boundary and in particular its velocity

the lubrication approximation for thin viscous films regularity and long time behavior of weak solutions we consider the fourth order degenerate diffusion equation in one space dimension this equation derived from a lubrication approximation models the surface tension dominated motion of thin viscous films and spreading droplets 15 the equation with f h h also models a thin neck of fluid in the hele shaw cell 10 11 23 in such problems h x t is the local thickness of the the film or neck this paper considers the properties of weak solutions that are more relevant to the droplet problem than to hele shaw for simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets we consider two problems the first has initial data h0 0 and f h h n 0 n 3 we show that there exists a weak nonnegative solution for all time also we show that this solution becomes a strong positive solution after some finite time t and asymptotically approaches its means as t the weak solution is in the classical sense of distributions for 3 8 n 3 and in a weaker sense introduced in 1 for the remaining 0 n 3 8 furthermore the solutions have high enough regularity to just include the unique source type solutions 2 with zero slope at the edge of the support they do not include any of the less regular solutions with positive slope at the edge of the support second we consider strictly positive initial data h0 m 0 and f h h n 0 n for this problem we show that even if a finite time singularity of the form h 0 does occur there exists a weak nonnegative solution for all time t this weak solution becomes strong and positive again after some critical time t as in the first problem we show that the solution approaches its mean as t the main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity we show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result 1996 john wiley sons inc

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

duality preserving discretization of the large time increment methods abstract the large time increment method is a well established iterative computational method for time dependent non linear structural analyses which has the peculiarity to produce at each iteration an approximation of the complete structural response over the whole considered history of loading the method establishes a converging iterative sequence by exploiting the specific structure of the equations governing the continuum problem the relatively large body of literature on the subject has been mostly concerned with assessing conceptual and practical properties of the method by referring to the continuum problem so far computer implementations where based on heuristic considerations and standard finite element technology in the present paper a different approach is followed a space and time discretization which preserves the duality structure of the continuum problem is introduced first then the method is reformulated for the discrete problem it is shown how the so called generalized variable modelling preserves the fundamental duality structure of the continuum problem a proof of convergence of the iterative scheme to the solution of the discrete problem is also outlined

stable manifolds to bounded solutions in possibly ill posed pdes abstract we prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations these equations may be ill posed a single specific solution an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions in the later cases which include several orbits we also establish the invariant manifolds of an orbit depend smoothly on the orbit analytically in the case of quasi periodic orbits and finitely differentiably in the case of more general families we first establish a general abstract theorem which under suitable spectral non degeneracy analyticity assumptions on the linearized equation establishes the existence of the desired manifold related results appear in the literature but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds this makes the results in this paper applicable to some several models of current interest that could not be treated otherwise we discuss in detail the boussinesq equation of water waves similar phenomena happen in other long wave approximations and complex ginzburg landau equation more recently we observed 11 that our results also apply to mean field games since the equations we consider may be ill posed part of the requirements for the stable manifold is that one can define the forward dynamics on them note also that the methods that are based in the existence of dynamics such as graph transform do not apply to ill posed equation we use the methods based on integral equations perron method associated with the partial dynamics but we need to take advantage of smoothing properties of the partial dynamics note that even if the families of solutions we started with are finite dimensional the stable manifolds may be infinite dimensional

well posedness for the navier slip thin film equation in the case of complete wetting we are interested in the thin film equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid film driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourth order parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with 13 14 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

well posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge we investigate regularity and well posedness for a fluid evolution model in the presence of a three phase contact point we consider a fluid evolution governed by darcy s law after linearization we obtain a nonlocal third order operator which contains the dirichlet neumann operator on the wedge with opening angle epsilon 0 we show well posedness and regularity for this linear evolution equation in the limit of vanishing opening angle we show the convergence of solutions to a fourth order degenerate parabolic operator related to the thin film equation in the course of the analysis we introduce and characterize a new type of sum of weighted sobolev spaces which are suitable to capture the singular limit as epsilon to 0 in particular the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem

darcy s flow with prescribed contact angle well posedness and lubrication approximation we consider the spreading of a thin two dimensional droplet on a solid substrate we use a model for viscous fluids where the evolution is governed by darcy s law at the contact point where air and liquid meet the solid substrate a constant non zero contact angle partial wetting is assumed we show local and global well posedness of this free boundary problem in the presence of the moving contact point our estimates are uniform in the contact angle assumed by the liquid at the contact point in the so called lubrication approximation long wave limit we show that the solutions converge to the solution of a one dimensional degenerate parabolic fourth order equation which belongs to a family of thin film equations the main technical difficulty is to describe the evolution of the non smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter varepsilon

biprefix codes inverse semigroups and syntactic monoids of injective automata abstract we prove for injective automata an analogue of pin s result 1978 that any finite monoid divides the syntactic monoid of p for some finite prefix code p pin s result for arbitrary automata is strengthened in that we show divides can be replaced by is a subsemigroup of we give some conditions sufficient to guarantee that if a syntactic monoid is inverse then the automaton is injective a direct proof is given of reutenaur s result 1979 1981 that if the language recognized by an injective automaton is a subsemigroup then it is of the form b for some biprefix code b

thin film equations with partial wetting energy existence of weak solutions abstract the capillarity driven evolution with slip of a thin liquid film over a dry surface is considered in the regime of partial wetting the focus is on the simplest model case of a constant non zero dynamic contact angle in the lubrication approximation for the analytical treatment of the corresponding free boundary problem a new strategy is proposed based on the introduction of an ad hoc class of disjoining pressures which tend to concentrate at triple junctions a first investigation of this approach yields the existence of weak solutions which satisfy the dissipation relation as an inequality and which are different from those with zero contact angle a heuristic argument is also presented in order to clarify the connection between contact angle and dissipation relation it shows that moving droplets which satisfy the dissipation relation as an equality are forced to have the prescribed contact angle

asymptotic equipartition and long time behavior of solutions of a thin film equation we investigate the large time behavior of classical solutions to the thin film type equation ut uuxxx x it was shown in previous work of carrillo and toscani that for non negative initial data u0 that belongs to h 1 r and also has a finite mass and second moment the strong solutions relax in the l 1 r norm at an explicit rate to the unique self similar source type solution with the same mass the equation itself is gradient flow for an energy functional that controls the h 1 r norm and so it is natural to expect that one should also have convergence in this norm carrillo and toscani raised this question but their methods using a different lyapunov functions that arises in the theory of the porous medium equation do not directly address this since their lyapunov functional does not involve derivatives of u here we show that the solutions do indeed converge in the h 1 r norm at an explicit but slow rate the key to establishing this convergence is an asymptotic equipartition of the excess energy roughly speaking the energy functional whose dissipation drives the evolution through gradient flow consists of two parts one involving derivatives of u and one that does not we show that these must decay at related rates due to the asymptotic equipartition and then use the results of carrillo and toscani to control the rate for the part that does not depend on derivatives from this one gets a rate on the dissipation for all of the excess energy

local existence and regularity for a class of degenerate parabolic equations on etudie des equations paraboliques scalaires a une variable d espace de la forme u t f u u x uu xx ou f f u p s est une fonction lisse de u p s r 3 avec f s strictement positive

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

latin a new view and an extension to wave propagation in nonlinear media summary the latin acronym of large time increment method was originally devised as a non incremental procedure for the solution of quasi static problems in continuum mechanics with material nonlinearity in contrast to standard incremental methods like newton and modified newton latin is an iterative procedure applied to the entire loading path in each latin iteration two problems are solved a local problem which is nonlinear but algebraic and miniature and a global problem which involves the entire loading process but is linear the convergence of these iterations which has been shown to occur for a large class of nonlinear problems provides an approximate solution to the original problem in this paper the latin method is presented from a different viewpoint taking advantage of the causality principle in this new view latin is an incremental method and the latin iterations are performed within each load step similarly to the way that newton iterations are performed the advantages of the new approach are discussed in addition latin is extended for the solution of time dependent wave problems as a relatively simple model for illustrating the new formulation lateral wave propagation in a flat membrane made of a nonlinear material is considered numerical examples demonstrate the performance of the scheme in conjunction with finite element discretization in space and the newmark trapezoidal algorithm in time copyright 2017 john wiley sons ltd

generalized midpoint finite element dynamic analysis of elastoplastic systems the application of a four parameter generalized midpoint integration method to small displacement elastoplastic dynamic problems is considered the integration of the governing relations over a time step is formulated as a non linear algebraic problem finite step problem by making use of generalized variables finite element modelling uniqueness and an extremum characterization of the solution of the finite step problem are discussed sufficient conditions for convergence of various iterative algorithms including newton raphson are given the b stability of the proposed procedure is assessed

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

a unified approach for variationally consistent finite elements in elastoplasticity abstract a general variationally consistent approach for elastoplastic finite element analysis is presented a weak formulation of the finite step i e in terms of finite increments elastoplastic boundary value problem is obtained by enforcing the stationarity of a mixed functional of hu washizu type a peculiarity of the present formulation is that the constitutive equations are expressed in weak form also all fields are then independently discretized in terms of so called generalized variables the result is a set of non linear algebraic equations a subset of which can be interpreted as the backward difference time integrated constitutive law of the finite element an iterative procedure where all calculations are at the element and not at the gauss point level resting on newton raphson scheme is proposed for the solution of the governing equations it is shown how the customary displacement approach and also some other mixed formulations in elastoplasticity can be regarded as special cases of the present formulation

non linear b stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity a class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward euler return maps as a particular case more importantly it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary helmholz free energy function b stability since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm the requirement of b stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity the analysis that follows depends critically on the assumption of convexity in particular the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation the proposed algorithms obey the discrete counterpart of this classical principle

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation

a dynamical systems approach for the contact line singularity in thin film flows abstract we are interested in a complete characterization of the contact line singularity of thin film flows for zero and nonzero contact angles by treating the model problem of source type self similar solutions we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system in particular we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points the results are extended to radially symmetric source type solutions in higher dimensions furthermore we give dynamical systems proofs for the existence and uniqueness of self similar droplet solutions in the nonzero dynamic contact angle case

bifurcations under nondegenerated conditions of higher degree and a new simple proof of the hopf neimark sacker bifurcation theorem this paper generalizes the nondegenerated conditions that imply the most common bifurcations in uniparametric families of maps defined on r it also presents a new very simple proof of the hopf bifurcation theorem of maps on r2 see g iooss mathematical studies vol 36 1979 based on one of the results obtained in one dimension and generalizes one of the nondegenerated conditions the hopf condition of the theorem

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given