hyperasymptotics for nonlinear odes i a riccati equation we illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations the example is a riccati equation the main tools that we need are transseries expansions and the riemann sheet structure of the borel transform of the divergent asymptotic expansions hyperasymptotic expansions determine the solutions uniquely a numerical illustration is included

unique normal forms the nilpotent hamiltonian case the problem we consider in this paper namely the characterization of unique normal forms derives from bifurcation theory in a typical situation one has an unperturbed vector field near a degenerate equilibrium in order to study the effect of generic perturbations suitable coordinates are introduced and the simplified system is said to be in normal form a well known case is the hopf bifurcation where a pair of conjugate eigenvalues cross the imaginary axis using averaging one can in this case compute the periodic orbit that splits off from the equilib um a second familiar example is the takens bogdanov bifurcation where two eigenvalues are zero in all these situations it is necessary to make certain choices both in the definition of what constitutes a normal form and in the computation of the transformation the idea here is to use this second kind of choice to sharpen the very definition of normal form while in standard treatments only the linear part of the field plays a role we show that one can effectively use its non linear part to remove additional terms and thus simplify the problem further the underlying philosophy in this type of game is that

coupled cell networks semigroups lie algebras and normal forms we introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a lie algebra this implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself networks without the semigroup property will support normal forms with a more general network architecture but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network we explain how to compute lie brackets and normal forms of coupled cell networks and we characterize the sn decomposition that determines the normal form symmetry this paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid

normal forms for hopf zero singularities with nonconservative nonlinear part abstract in this paper we are concerned with the simplest normal form computation of the systems 0 1 x 2 x f x y 2 z 2 y z y f x y 2 z 2 z y z f x y 2 z 2 where f is a formal function with real coefficients and without any constant term these are the classical normal forms of a larger family of systems with hopf zero singularity indeed these are defined such that this family would be a lie subalgebra for the space of all classical normal form vector fields with hopf zero singularity the simplest normal forms and simplest orbital normal forms of this family with nonzero quadratic part are computed we also obtain the simplest parametric normal form of any non degenerate perturbation of this family within the lie subalgebra the symmetry group of the simplest normal forms is also discussed this is a part of our results in decomposing the normal forms of hopf zero singular systems into systems with a first integral and nonconservative systems

an improved theory of asymptotic unfoldings abstract an asymptotic unfolding of a dynamical system near a rest point is a system with additional parameters such that every one parameter deformation of the original system can be embedded in the unfolding preserving all properties that can be detected by asymptotic methods asymptotic unfoldings are computed using normal and hypernormal form methods we present a simplified and improved method of computing such unfoldings that can be used in any normal form style

quaternionic like bifurcation in the absence of symmetry the authors give some results concerning the existence of bifurcating periodic solutions on nonlinear time evolution equations with the property that criticality is produced by a couple of imaginary eigenvalues with double multiplicity this situation is also compared with the classical hopf case and with the quaternionic bifurcation occurring in the presence of a su 2 symmetry

poincar normal forms for partial differential equations an extension of poincare normal form theory for systems of partial differential equations which have linear ordinary equations as a main part is obtained poincare normal forms for the shallow water equation on a beta plane and in large scale with respect to rossby radius for the variable coriolis parameter are found the normal forms of the shallow water equations contain the charney equation and higher approximation equations for rossby waves and describe the nonlinear interaction of rossby and inertia gravity waves

quantum galois theory for compact lie groups abstract we establish a quantum galois correspondence for compact lie groups of automorphisms acting on a simple vertex operator algebra

multi resurgence and connection to stokes formulae for some linear meromorphic differential systems abstract in this article we consider a linear meromorphic differential system with several levels r 1 r p for any k we prove that the borel transforms of its r k reduced formal solutions are resurgent and we give the general form of all their singularities next under some convenient hypotheses on the geometric configuration of singular points we display exact formulae to express some stokes multipliers of level r k of initial system in terms of connection constants in the borel plane generalizing thus formulae already obtained by m loday richaud and the author for systems with a single level as an illustration we develop one numerical example

applying the method of normal forms to second order nonlinear vibration problems vibration problems are naturally formulated with second order equations of motion when the vibration problem is nonlinear in nature using normal form analysis currently requires that the second order equations of motion be put into first order form in this paper we demonstrate that normal form analysis can be carried out on the second order equations of motion in addition for forced damped nonlinear vibration problems we show that the invariance properties of the first and second order transforms differ as a result using the second order approach leads to a simplified formulation for forced damped nonlinear vibration problems

resurgence stokes phenomenon and alien derivatives for level one linear differential systems abstract a precise description of the singularities of the borel transform of solutions of a level one linear differential system is deduced from a proof of the summable resurgence of the solutions by the perturbative method of j ecalle then we compare the meromorphic classification stokes phenomenon from the viewpoint of the stokes cocycle and the viewpoint of alien derivatives we make explicit the stokes ramis matrices as functions of the connection constants in the borel plane and we develop two examples no assumption of genericity is made

monstrous moonshine from orbifolds we consider the monster module of frenkel lepowsky and meurman as az2 orbifold of a bosonic string compactified by the leech lattice we show that the main conway and norton monstrous moonshine properties stating that the thompson series for each monster group conjugacy class has a modular invariance group of genus zero follow from an orbifold construction based on an orbifold group composed of monster group elements it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each thompson series it is also shown that the power map formula of conway and norton follows from the same vacuum structure finally we demonstrate the validity of the vacuum conjectures for sectors twisted by leech lattice automorphisms in many cases

normal forms symmetry and linearization of dynamical systems we discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system or a parameter dependent family via the poincare normal form approach we discuss this formally at first and later pay attention to the convergence of the linearizing procedure we also discuss some generalizations of our main result

unique normal forms for planar vector fields soit x un germe d un champ vectoriel analytique ou formel en equilibre a o r 2 avec des valeurs propres purement imaginaires a 0 i a 0 0 alors x est formellement equivalent a precisement un des champs vectoriels polynomiaux suivant ou x 1 2 cos y 1 2 sin pour des coordonnees appropriees x y 7b a 0 a 0 7b a k a 0 k 7b b j a 0 a 1 a j j j 1 b j j 0

poincare normal forms and lie point symmetries we study poincare normal forms of vector fields in the presence of symmetry under general i e not necessarily linear diffeomorphisms we show that it is possible to reduce both the vector field and the symmetry diffeomorphism to normal form by means of an algorithmic procedure similar to the usual one for poincare normal forms without symmetry this joint normal form can be given a simple geometric characterization

construction of darboux coordinates and poincare birkhoff normal forms in noncanonical hamiltonian systems abstract we demonstrate a general method to construct darboux coordinates via normal form expansions in noncanonical hamiltonian system obtained from e g a variational approach to quantum systems the procedure serves as a tool to naturally extract canonical coordinates out of the variational parameters and at the same time to transform the energy functional into its poincare birkhoff normal form the method is general in the sense that it is applicable for arbitrary degrees of freedom in arbitrary orders of the local expansion and it is independent of the precise form of the hamilton operator the method presented allows for the general and systematic investigation of quantum systems in the vicinity of fixed points which e g correspond to ground excited or transition states moreover it directly allows to calculate classical and quantum reaction rates by applying transition state theory

on convergent normal form transformations in presence of symmetries abstract we prove a convergence criterion for transformations to poincare dulac normal form that involves the centralizer of the given vector field

modular categories and orbifold models in this paper we try to answer the following question given a modular tensor category with an action of a compact group g is it possible to describe in a suitable sense the quotient category g we give a full answer in the case when is the category of vector spaces in this case g turns out to be the category of representation of drinfeld s double d g this should be considered as the category theory analog of the topological identity pt g bg this implies a conjecture of dijkgraaf vafa e verlinde and h verlinde regarding so called orbifold conformal field theories if is a vertex operator algebra which has a unique irreducible module itself and g is a compact group of automorphisms of and some not too restrictive technical conditions are satisfied then g is finite and the category of representations of the algebra of invariants g is equivalent as a tensor category to the category of representations of drinfeld s double d g we also get some partial results in the non holomorphic case i e when has more than one simple module

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

second order linear differential equations with two irregular singular points of rank three the characteristic exponent abstract for a second order linear differential equation with two irregular singular points of rank three multiple laplace type contour integral solutions are considered an explicit formula in terms of the stokes multipliers is derived for the characteristic exponent of the multiplicative solutions the stokes multipliers are represented by converging series with terms for which limit formulas as well as more detailed asymptotic expansions are available here certain new recursively known coefficients enter which are closely related to but different from the coefficients of the formal solutions at one of the irregular singular points of the differential equation the coefficients of the formal solutions then appear as finite sums over subsets of the new coefficients as a by product the leading exponential terms of the asymptotic behaviour of the late coefficients of the formal solutions are given and this is a concrete example of the structural results obtained by immink in a more general setting the formulas displayed in this paper are not of merely theoretical interest but they also are complete in the sense that they could be and have been implemented for computing accurate numerical values of the characteristic exponent although the computational load is not small and increases with the rank of the singular point under consideration

on transformations into normal form abstract we discuss a certain class of transformations of an ordinary differential equation into normal form near a stationary point these transformations preserve certain special structures and one can derive from them a universally applicable algorithm to determine a normal form up to any desired degree the algorithm can be programmed with the help of any computer algebra system in the same manner transformations into normal form on an invariant manifold are discussed

non perturbative linearization of dynamical systems we consider the classical problem of linearizing a vector field x around a fixed point we adopt a non perturbative point of view based on the symmetry properties of linear vector fields

symbolic computation of normal forms for semi simple cases this paper presents a method and computer programs for computing the normal forms of ordinary differential equations whose jacobian matrix evaluated at an equilibrium involves semi simple eigenvalues the method can be used to deal with systems which are not necessarily described on a center manifold an iterative procedure is developed for finding the closed form expressions of the normal forms and associated nonlinear transformations computer programs using a symbolic computer language maple are developed to facilitate the application of the method the programs can be conveniently executed on a main frame a workstation or a pc machine without any interaction a number of examples are presented to demonstrate the applicability of the method and the computation efficiency of the maple programs

rapidly convergent iteration method for simultaneous normal forms of commuting maps we propose a new rapidly convergent iteration scheme under weak simultaneous arithmetic conditions for overdetermined systems related to simultaneous conjugation of commuting maps the first application is for commuting holomorphic maps in mathcal c n fixing the origin not necessarily diffeomorphisms if n geq 2 the second one is for orientation preserving gevrey circle mappings by means of nonlinear superposition estimates

analysis of critical and post critical behaviour of non linear dynamical systems by the normal form method part ii divergence and flutter abstract the normal form method nfm presented in part i 1 is applied to the analysis of systems near two basic bifurcation conditions divergence and flutter the reduced systems and the normalizing transformation obtained by means of a computer routine implementing the normalization formulae of part i are utilized to determine a the stability of the bifurcating singular point b the local equilibrium paths associated with divergence bifurcation and c the amplitude and frequency or more generally the waveform of post flutter oscillations

versal normal form for nonsemisimple singularities abstract the theory of versal normal form has been playing a role in normal form since the introduction of the concept by v i arnol d in 1 2 but there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields that is the linear part should be done completely first before one computes the nonlinear terms in this paper we address this issue by giving a complete description of a first order calculation in the case of the two and three dimensional irreducible nilpotent cases which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics l 4 problem

beauty and the beast superconformal symmetry in a monster module frenkel lepowsky and meurman have constructed a representation of the largest sporadic simple finite group the fischer griess monster as the automorphism group of the operator product algebra of a conformal field theory with central chargec 24 in string terminology their construction corresponds to compactification on az2 asymmetric orbifold constructed from the torusr24 where is the leech lattice in this note we point out that their construction naturally embodies as well a larger algebraic structure namely a super virasoro algebra with central charge 16 with the supersymmetry generator constructed in terms of bosonic twist fields

lie point symmetries and poincare normal forms for dynamical systems the problem of finding the extended lie point time independent symmetries of autonomous systems of ordinary differential equations is compared with the poincare procedure of reducing the system to linear or normal form showing a close relationship between the two problems some special situations including the classical hopf bifurcation problem are also examined from this point of view

computation of normal forms via a perturbation technique considering qualitative behavior of a non linear dynamical system often leads to first simplifying the differential equations or finding their normal forms a perturbation technique for computing normal forms is presented this technique associated with the method of multiple scales can be used to systematically find a unique form for a given set of differential equations the technique is discussed in detail through the analysis of hopf bifurcation it is shown that for hopf bifurcation the method only requires solving two dimensional matrix systems for any higher order normal forms of a generaln dimensional system with the aid of the symbolic language maple this approach is straightforward and is computationally efficient and fast furthermore a simple verification scheme is given for verifying the normal forms and associated non linear transformations obtained using any methodology examples are presented to demonstrate the applicability of the perturbation technique and the computation efficiency of the maple program

lie point symmetries for autonomous systems and resonance the authors give some results on the problem of finding the lie point symmetries of autonomous systems of differential equations in particular they consider the case in which the nonlinear terms are resonant in the sense of the poincare procedure for reducing the system to normal form and they show that lie symmetries can be characterized in a useful form

on the multisummability of divergent solutions of linear partial differential equations with constant coefficients abstract we consider the cauchy problem for general linear partial differential equations in two complex variables with constant coefficients we obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the cauchy data

lie algebras with finite dimensional polynomial centralizer we give criteria for finite or infinite dimensionality of the polynomial centralizer of the lie algebra of a linear lie group in terms of invariants and relative invariants of the group in the finite dimensional scenario some applications to normal forms and to certain equations with fundamental solutions are presented

symbolic computation of normal forms for resonant double hopf bifurcations using a perturbation technique abstract this paper presents a perturbation method for computing the normal forms of resonant double hopf bifurcations with the aid of a computer algebraic system this technique based on the method of multiple time scales can be used to deal with general n dimensional systems without the application of center manifold theory explicit iterative formulas have been derived for uniquely determining the coefficients of normal forms and associated non linear transformations up to an arbitrary order user friendly symbolic computer programs written in maple have been developed which can be easily applied for computing the normal forms of a given system examples are presented to show the applicability of the methodology and the convenience of using the computer software

two constructions of affine lie algebra representations and boson fermion correspondence in quantum field theory we establish an isomorphism between the vertex and spinor representations of affine lie algebras for types dl 1 and dl 1 2 we also study decomposition of spinor representations using the infinite family of casimir operators and prove that they are either irreducible or have two irreducible components we show that the vertex and spinor constructions of the representations can be reformulated in the language of two dimensional quantum field theory in this physical context the two constructions yield the generalized sine gordon and thirring models respectively already in renormalized form the isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson fermion correspondence

further reductions of normal forms for dynamical systems abstract we propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields with concrete and interesting applications we use lower order nonlinear terms in the normal form for the simplifications of higher order terms our approach is applicable for both the non nilpotent and the nilpotent cases for dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal or unique with respect to equivalence by formal near identity transformations we can compute at the same time a formal diffeormorphism that realizes the normalization comparisons with other methods are given for several examples

spectral sequences and parametric normal forms abstract this paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases in this paper the method is first applied to two simple examples parametric single zero and parametric resonant saddle singularities and then to parametric generalized hopf singularity we provide the simplest parametric normal form for these cases

on the convergence of normalizing transformations in the presence of symmetries it is shown that under suitable conditions involving in particular the existence of analytic constants of motion the presence of lie point symmetries can ensure the convergence of the transformation of a vector field or dynamical system into normal form

normal forms and nonlinear symmetries we give some general theorems and extensions of previous results concerning the problem of transforming an algebra of vector fields into poincare normal form by means of a unifying algebraic language we show the possibility of obtaining either a parallel or joint normal form of the vector fields in a definite way which simplifies the construction of normal forms providing a precise restriction on their structure the application to finite dimensional dynamical systems and their lie point symmetries is also discussed

convergent normal forms of symmetric dynamical systems it is shown that the presence of lie point symmetries of non hamiltonian dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical system or vector field into poincare dulac normal form

vertex operator algebras generalized doubles and dual pairs let v be a simple vertex operator algebra and g a finite automorphism group then there is a natural right g action on the set of all inequivalent irreducible v modules let cal s be a finite set of inequivalent irreducible v modules which is closed under the action of g there is a finite dimensional semisimple associative algebra a alpha g cal s for a suitable 2 cocycle alpha naturally determined by the g action on cal s such that a alpha g cal s and the vertex operator algebra v g form a dual pair on the sum of v modules in cal s in the sense of howe in particular every irreducible v module is completely reducible v g module

on k summability of formal solutions for a class of partial differential operators with time dependent coefficients we study the k summability of divergent formal solutions for the cauchy problem of a certain class of linear partial differential operators with time dependent coefficients the problem is reduced to a k summability property of formal solutions for a linear similar ordinary differential equation associated with the cauchy problem

hypernormal form theory foundations and algorithms abstract hypernormal forms unique normal forms simplest normal forms are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra the baider theory of the campbell hausdorff group is refined by a study of its subgroups to determine the smallest substages into which the hypernormalization process can be divided this leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work a concrete interpretation of jan sanders spectral sequence for hypernormal forms is presented examples are given and a proof is given for a little known theorem of belitskii expressing the hypernormal form space in the inner product style as the kernel of a higher order differential operator

linearizability and integrability of vector fields via commutation abstract in this paper we consider complex smooth and analytic vector fields x in a neighborhood of a nondegenerate singular point it is proved the equivalence between linearizability and commutation i e the existence of a commuting vector field y such that the lie brackets x y 0 for complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point it is also proved the equivalence between integrability and the existence of a smooth vector field y such that y is a normalizer of x i e x y x

g elliptic systems and the genus zero problem for m_ 24 toward the end of the 1970s j h conway and s p norton wrote a quite unique and influential paper 3 in which they produced incontrovertible evidence based partly on ideas of j mckay and j g thompson 13 that the fischer griess monster simple group m was intimately related to the modular function j x familiar from the theory of modular forms thus was so called moonshine brought into the world among other things it was conjectured that there is a natural infinite dimensional complex vector space v which carries a z grading into finite dimensional subspaces f v x e vx 0 v2 0 such that each vi is a cmmodule moreover the graded character chvt q j2n dimf tf of f should be the fourier expansion of j r namely q 196884 21 493 760 if the indeterminate q is interpreted as e for r in the upper half plane h there should in addition be an analogous interpretation for

orbital linearization around equilibria of vector fields in rn and lie symmetries abstract in this paper we study some relationships between analytic vector fields x in r n which are analytically orbitally linearizable in a neighborhood of an equilibrium point and the structure of the set n x of its locally analytic normalizers analytic infinitesimal generators of lie point orbitally symmetries in that neighborhood characterization and necessary and sufficient conditions for the orbital linearization of x via the existence of special elements in n x are given

calculus of twisted vertex operators starting from an arbitrary isometry of an arbitrary even lattice twisted and shifted vertex operators are introduced under commutators these operators provide realizations of twisted affine lie algebras this construction generalizing a number of known ones is based on a self contained calculus

on quantum galois theory the goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the operator content of orbifold models to explain what this might mean and to clarify the title of the paper we will assume that the reader is familiar with the algebraic formulation of 2 dimensional cft in the guise of vertex operator algebras voa see b flm and dm for more information on this point in the paper dvvv several ideas are proposed concerning the structure of a holomorphic orbifold in other words if v is a holomorphic voa and if g is a finite group of automorphisms of v then the sub voa v g of g invariants is itself a voa and the subject of dvvv is very much concerned with speculation on the nature of the v modules it turns out to be more useful at least for purpose of inductive proofs to take v to be a simple voa we will then see that v g is also simple whenever g is a finite group of automorphisms of v one consequence of our main results is the following

integration of perturbed initial value problems through reduction theory we propose a method to integrate an initial value problem formed by the differential equations of a perturbed dynamical system plus the initial condition the approach consists of several steps first of all if the original system does not enjoy a continuous symmetry it is transformed up to a certain order of approximation m into an equivalent but equivariant one after truncation at order m second this new symmetry allows us to define a reduction map which produces in its whole phase space a splitting of the transformed equations specifically the latter system is decoupled into two subsystems also initial value problems one of them defined in the orbit space associated with the reduction and the other linear in its coordinates as a third step the reduced subsystems are analytically resolved or numerically integrated using standard methods finally the solution of the original system is recovered by inverting back the reduction map and thereafter the analytic transformations we apply our procedure to three examples

on hopf bifurcation and control for a delay systems abstract hopf bifurcation of a delay differential equation modeling two enterprise interaction mechanism is investigated for the demands of computer simulation applications the discrete time two enterprise interaction model is proposed by using the forward euler scheme by discussing the characteristic equation of linearized part the local stability criterion of the positive fixed point is presented time lag is selected as the bifurcation parameter then it is demonstrated that variation of time lag may result in hopf bifurcation furthermore derivations for the directions and stability of hopf bifurcation are also given for the purpose of controlling hopf bifurcation a time delayed feedback control scheme is proposed finally we simulate the main results

analysis of critical and post critical behaviour of non linear dynamical systems by the normal form method part i normalization formulae abstract a method based on normal form theory is presented to study the dynamical behaviour of a system in the neighbourhood of a nearly critical equilibrium state associated with a bifurcation condition explicit formulae for the normalization procedure are derived these formulae can be numerically programmed avoiding usual complicated algebraic calculations and making the method effectively applicable for n dimensional systems rather general bifurcations can be included e g non linear flutter hopf bifurcation divergence and internal resonance

filtration preserving mappings and centralizers within graded lie algebras we use spectral sequence techniques to compute centralizers of elements within graded lie algebras and the methods are then applied to the calculation of unique normal forms of elements within one parameter matrix lie algebras a finiteness criterion for unique normal forms is presented

symmetries of dynamical systems and convergent normal forms it is shown that under suitable conditions involving in particular the existence of analytic constants of motion the presence of lie point symmetries can ensure the convergence of the transformation taking a vector field or dynamical system into normal form

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

normal form theory and spectral sequences abstract the concept of unique normal form is formulated in terms of a spectral sequence as an illustration of this technique some results of baider and churchill concerning the normal form of the anharmonic oscillator are reproduced the aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

on normal form calculations in impact oscillators normal form calculations are useful for analysing the dynamics close to bifurcations however the application to non smooth systems is a topic for current research here we consider a class of imp

multisummability of formal solutions of some linear partial differential equations abstract let u t x n 1 un x tn be a formal power series in one variable t satisfying formally p t x t x u t x f t x t x c c d the purpose of the present paper is to introduce a class of linear partial differential operators and show that u t x is multisummable provided p t x t x belongs to this class

nonabelian orbifolds and the boson fermion correspondence for a positive integerl divisible by 8 there is a bosonic holomorphic vertex operator algebra voa v gamma l associated to the spin lattice l for a broad class of finite groupsg of automorphisms of v gamma l we prove the existence and uniqueness of irreducibleg twisted v gamma l modules and establish the modular invariance of the partition functionsz g h for commuting elements ing in particular for any finite group there are infinitely many holomorphic voas admittingg for which these properties hold the proof is facilitated by a boson fermion correspondence which gives a voa isomorphism between v gamma l and a certain fermionic construction and which extends work of frenkel and others

an orbifold theory of genus zero associated to the sporadic groupm24 let v gamma l be the self dual or holomorphic bosonic conformal field theory associated with the spin lattice l of rankl divisible by 24 in earlier work of the authors we showed how it is possible to establish the existence and uniqueness of irreducibleg twisted sectors for v gamma l for certain automorphismsg of v gamma l and to establish the modular invariance of the space of partition functionsz g h corresponding to commuting pairsg h of elements in certain groupsg of automorphisms of v gamma l in the present work we show that if we takel 24 andg the sporadic simple groupm24 then the corresponding orbifold has thegenus zero property that is eachz g h is either identically zero or ahauptmodul i e it generates the field of functions on the subgroup ofsl2 which fixesz g h which then necessarily has genus zero

on the asymptotic and numerical solution of linear ordinary differential equations an investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank we introduce a classification of the solutions into two types explicit and implicit for the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached no such sector exists for implicit solutions in consequence the two types of solution have different uniqueness properties another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions we also investigate the computation of the solutions by numerical integration of the differential equation it is shown that it is the exception rather than the rule for the integration process to be stable particularly so for implicit solutions to overcome these instabilities we develop boundary value methods complete with error analysis numerical examples illustrate the computation of both explicit and implicit solutions and also the associated stokes multipliers

on the representations and formula presented equivariant normal form for solenoidal hopf zero singularities abstract in this paper we deal with the solenoidal conservative lie algebra associated to the classical normal form of hopf zero singular system we concentrate on the study of some representations and z 2 equivariant normal form for such singular differential equations first we list some of the representations that this lie algebra admits the vector fields from this lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one degree of freedom hamiltonian system and the third one depends upon the first two variables this representation is governed by the associated poisson algebra to one sub family of this lie algebra euler s form vector potential and clebsch representation are other representations of this lie algebra that we list here we also study the non potential property of vector fields with hopf zero singularity from this lie algebra finally we examine the unique normal form with non zero cubic terms of this family in the presence of the symmetry group z 2 the theoretical results of normal form theory are illustrated with the modified chua s oscillator

brief paper linearization through state immersion of nonlinear systems admitting lie symmetries in this paper it is shown that if a nonlinear system admits a lie symmetry that can be transformed into its poincare dulac normal form by a state diffeomorphism then under some technical conditions such a nonlinear system can be immersed into a linear one this allows us to compute in closed form the flow all algebraic invariant curves through semi invariants of the nonlinear system and lyapunov functions to study stability properties

shifted vertex operator algebras and g elliptic systems shifted vertex operator algebras and superalgebras are defined both in bosonic and in fermionic pictures their isomorphism boson fermion correspondence is shown partition functions of bosonic shifted vertex operator algebras are realized analytically and are interpreted as string path integrals over elliptic curves their modular properties follow and elliptic systems are constructed based on fermionic shifted vertex operator algebras

multisummability of formal solutions to the cauchy problem for some linear partial differential equations abstract the paper considers the cauchy problem for linear partial differential equations of non kowalevskian type in the complex domain it is shown that if the cauchy data are entire functions of a suitable order the problem has a formal solution which is multisummable the precise bound of the admissible order of entire functions is described in terms of the newton polygon of the equation

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

embedding and splitting ordinary differential equations in normal form abstract we introduce an embedding of real or complex n dimensional space k n as an algebraic variety v which is determined by the action of a linear one parameter group every analytic vector field on k n corresponds to some embedded vector field on v for a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non autonomous linear systems examples and applications are mostly concerned with poincare dulac normal forms embeddings provide a natural setting for perturbations of symmetric systems in particular of systems in normal form up to some degree