the local structure of a liouville vector field 1 introduction a liouville vector field on a symplectic manifold is a smooth vector field whose flow consists of conformally flat transformations nagano 7 characterized the cotangent bundle of the singular set of a 1 form whose exterior derivative is symplectic by means of a liouville field in this work we investigate the local structure of a liouville vector field of a 2n dimensional symplectic manifold p q which vanishes on a k dimensional isotropic submanifold q of p some of the eigenvalues of its linear part are assumed to be in resonance a vector field with this properties appears in a natural geometric problem it arises as the gradient field with respect to some kaihler metric of the square of the distance function to an isotropic submanifold in nagano s theorem the singular set is a lagrangian submanifold and the eigenvalues of the linear part of the liouville field at the singular points are 0 and 1 here we assume that at every point q of q the vector field vanishes and the eigenvalues of its 1 jet are 1 1 2 and 0 this means precisely that in some coordinates the 1 jet of at q is

estimating isogenies on elliptic curves the object of this paper is to prove a new type of estimate for isogenies between elliptic curves this has several diophantine applications effective versions of serre s galois irreducibility theorem and shafarevich s theorem for example which are presented in another paper mw3 later articles will deal with the corresponding problems for abelian varieties of arbitrary dimension right at the beginning we emphasize that we are identifying elliptic curves e with weierstrass equations

recovering asymptotics of metrics from fixed energy scattering data from fixed energy scattering data is studied it is shown that if two such metrics g1 g2 have scattering data at some fixed energy which are equal up to smoothing then there exists a diffeomorphism fixing infinity such that g1 g2 is rapidly decreasing given the scattering matrix at two energies it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously these results also hold for a wide class of scattering manifolds

prime power terms in elliptic divisibility sequences we study a problem on specializations of multiples of rational points on elliptic curves analogous to the mersenne problem we solve this problem when descent via isogeny is possible by giving explicit bounds on the indices of prime power terms in elliptic divisibility sequences associated to points in the image of a nontrivial isogeny we also discuss the uniformity of these bounds assuming the hall lang conjecture

existence of continuous solutions and stability for a general iterative functional equation in this paper c 0 solutions for a general form of iterative functional equation are considered when the coefficient of the first order iterate can be zero which is called the leading coefficient problem for polynomial like iterative equations moreover the stability of those c 0 solutions is also discussed finally as an application we give an algorithm to compute the piecewise linear solutions of polynomial like iterative equations

a simple proof of the grobman hartman theorem for nonuniformly hyperbolic flows abstract we give a simple and direct proof of the grobman hartman theorem for nonautonomous differential equations obtained from perturbing a nonuniform exponential dichotomy in particular we do not need to pass through discrete time and obtain the result as a consequence of a corresponding result for maps to the best of our knowledge this is the first direct approach for nonuniform exponential dichotomies we also show that the conjugacies are continuous in time and holder continuous in space in addition we describe the dependence of the conjugacies on the perturbation and we obtain a reversibility result for the conjugacies of reversible differential equations we emphasize that the additional work required to consider nonuniform exponential dichotomies is substantial

multiplicity estimates on group varieties tions the field k will be the field q of algebraic numbers we remark at this point that the results remain true if we take instead of the field of complex numbers c its p adic analogue c for some fixed prime p and for k a corresponding subfield kp of cp we restrict ourselves to the complex case in order to avoid some minor complications appearing in the p adic domain these complications always arise when analytic functions come in our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p adic domain these difficulties can be avoided by a purely algebraic approach if we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero but we prefer to avoid such an approach in order to keep the text understandable also for those who are mainly interested in the applications in transcendence

multilevel monte carlo for stochastic differential equations with small noise we consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by brownian motions in the commonly occurring small noise regime we consider i standard monte carlo methods combined with numerical discretization algorithms tailored to the small noise setting and ii a multilevel monte carlo method combined with a standard euler maruyama implementation under the assumptions we make on the underlying model the multilevel method combined with euler maruyama is often found to be the most efficient option moreover under a wide range of scalings the multilevel method is found to give the same asymptotic complexity that would arise in the idealized case where we have access to exact samples of the required distribution at a cost of o 1 per sample a key step in our analysis is to analyze the variance between two coupled paths directly as opposed to their l 2 distance careful simulations are provided to illustrate the asymptotic results

the set of all iterates is nowhere dense in c 0 1 0 1 we prove that if a mixing map f 0 1 0 1 belongs to the c 0 closure of the set of iterates and f 0 0 f 1 1 then f is an iterate itself together with some one dimensional techniques it implies that the set of all iterates is nowhere dense in c 0 1 0 1 giving the final answer to the question of a bruckner p humke and m laczkovich

a melnikov method for homoclinic orbits with many pulses we present an extension of the melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near integrable systems the melnikov function in this situation is the sum of the usual melnikov functions evaluated with some appropriate phase delays we show that a nonfolding condition which involves the logarithmic derivative of the melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist

a grobman hartman theorem for general nonuniform exponential dichotomies for a nonautonomous dynamics with discrete time given by a sequence of linear operators am we establish a version of the grobman hartman theorem in banach spaces for a very general nonuniformly hyperbolic dynamics more precisely we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ec n determined by a sequence n for all sufficiently small lipschitz perturbations am fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators am we also show that all conjugacies are holder continuous we note that the usual exponential behavior is included as a very special case when n n but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when n logn

exponential dichotomies almost periodic structurally stable differential systems and an example abstract in this paper we introduce concepts of almost periodic topological in short apt equivalence and almost periodic structural in short aps stability of differential systems and a necessary and sufficient condition for aps stability of differential systems furthermore we discuss the inheritability of the aps stability and density of aps stable differential systems in a certain space of uniformly almost periodic differential systems

on the field of definition of p torsion points on elliptic curves over the rationals let s mathbb q d be the set of primes p for which there exists a number field k of degree le d and an elliptic curve e mathbb q such that the order of the torsion subgroup of e k is divisible by p in this article we give bounds for the primes in the set s mathbb q d in particular we show that if p ge 11 p ne 13 37 and p in s mathbb q d then p le 2d 1 moreover we determine s mathbb q d for all d le 42 and give a conjectural formula for all d ge 1 if serre s uniformity problem is answered positively then our conjectural formula is valid for all sufficiently large d under further assumptions on the non cuspidal points on modular curves that parametrize those j invariants associated to cartan subgroups the formula is valid for all d ge 1

one parameter families of elliptic curves over ℚ with maximal galois representations let e be an elliptic curve over q and let q e n be its nth division field in 1972 serre showed that if e is without complex multiplication then the galois group of q e n q is as large as possible that is gl2 z nz for all integers n coprime to a constant integer m e q depending at most on e q serre also showed that the best one can hope for is to have gl2 z nz gal q e n q 2 for all positive integers n we study the frequency of this optimal situation in a one parameter family of elliptic curves over q and show that in essence for almost all one parameter families almost all elliptic curves have this optimal behavior

forts of quadratic polynomials under iteration abstract since some dynamical behaviors of a one dimensional mapping are influenced by the number of forts attention is paid to the change of the number under iteration for simple computation we work on quadratic polynomials we use the theory of polynomial complete discrimination system to give a symbolic algorithm for the number of forts of iterated polynomials and apply the algorithm to quadratic functions which proves an alternative result that the number either persists to be 1 or tends to infinity under iteration we further compute the number for iterates of order 2 3 7 in the above infinity case and obtain critical values of the parameter at which the number changes those changes with finitely many data display a conjectured fibonacci rule

normal forms and linearization of resonant vector fields with multiple eigenvalues abstract we discuss the linearization and normal forms of resonant vector fields x x a x a x where a has one double or triple eigenvalue or a pair of double eigenvalues we present a simple way of identifying the resonant monomials that have to appear in its normal form and also effective conditions on the nonlinearity a x for the resonant vector field to be linearizable

embedding smooth diffeomorphisms in flows abstract in this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces first we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial then using this result and the normal form theory we obtain a class of local c k diffeomorphisms for k n which admit embedding vector fields with some smoothness finally we prove that for any k n under the coefficient topology the subset of local c k diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local c k diffeomorphisms

coupling sample paths to the thermodynamic limit in monte carlo estimators with applications to gene expression abstract many biochemical systems appearing in applications have a multiscale structure so that they converge to piecewise deterministic markov processes in a thermodynamic limit the statistics of the piecewise deterministic process can be obtained much more efficiently than those of the exact process we explore the possibility of coupling sample paths of the exact model to the piecewise deterministic process in order to reduce the variance of their difference we then apply this coupling to reduce the computational complexity of a monte carlo estimator motivated by the rigorous results in 1 we show how this method can be applied to realistic biological models with nontrivial scalings

a grobman hartman theorem for nonuniformly hyperbolic dynamics abstract we establish a version of the grobman hartman theorem in banach spaces for nonuniformly hyperbolic dynamics we also consider the case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time more precisely we consider sequences of lipschitz maps a m f m such that the linear parts a m admit a nonuniform exponential dichotomy and we establish the existence of a unique sequence of topological conjugacies between the maps a m f m and a m furthermore we show that the conjugacies are holder continuous with holder exponent determined by the ratios of lyapunov exponents with the same sign to the best of our knowledge this statement appeared nowhere before in the published literature even in the particular case of uniform exponential dichotomies although some experts claim that it is well known in this case we are also interested in the dependence of the conjugacies on the perturbations f m we show that it is holder continuous with the same holder exponent as the one for the conjugacies we emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial in particular we need to consider several additional lyapunov norms

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

sharp regularity of linearization for c 1 1 hyperbolic diffeomorphisms c 1 linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions however c 1 smoothness is not enough to guarantee c 1 linearization for c 1 1 hyperbolic diffeomorphisms on banach spaces c 1 linearization was proved under a gap condition together with a band condition of the spectrum in this paper the result of c 1 linearization in banach spaces is strengthened to c 1 beta linearization with a constant beta 0 under a weaker band condition by a decomposition with invariant foliations the weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum furthermore we give an estimate for the exponent beta and prove that the estimate is sharp in the planar case

local contractions of banach spaces and spectral gap conditions abstract we study the linearization problem for a c k 1 k 1 contraction of a banach space e near a fixed point which satisfies a spectral gap condition and a narrow band condition both of order k we also assume that the part of the spectrum in each band satisfies a finite non resonant condition of order k relative to itself together with the part that lies in the larger bands we show that there is a c k linearization for sufficiently small 0 we give a precise estimate on in terms of the gap and band conditions

surjectivity of galois representations associated with quadratic mathbb q curves we prove in this paper an uniform surjectivity result for galois representations associated with non cm mathbb q curves over imaginary quadratic fields using various tools for the proof such as mazur s method isogeny theorems runge s method and analytic estimates of sums of l functions

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2 abstract it is known that any piecewise monotonic function of nonmonotonicity height not less than 2 has no continuous iterative roots of order greater than the number of forts of the function so the following problem arises naturally does such a function have an iterative root of order n not greater than the number of forts we consider the case that the number of forts is equal to n in which there appear possibly only two types t 1 and t 2 of iterative roots i e the roots strictly increasing on the interval stretched on all forts of the given function and the roots strictly decreasing on such an interval respectively we characterize all type t 1 roots of order n and give a necessary condition for a root of order n to be of type t 2 a full description of type t 2 is still an open question

normal form and limit cycle bifurcation of piecewise smooth differential systems with a center abstract in this paper we prove that any center either nondegenerate or degenerate of a planar piecewise c r smooth vector field z is topologically equivalent to that of z 0 x y 1 2 x for y 0 x y 1 2 x for y 0 and that the homeomorphism between z and z 0 is c r smoothness when restricted to each side of the switching line except at the center p we illustrate by examples that there are degenerate centers whose flows are conjugate to that of z 0 and also there exist nondegenerate centers whose flows cannot be conjugate to that of z 0 finally applying the normal form z 0 together with the piecewise smooth equivalence we study the number of limit cycles which can be bifurcated from the center of z

nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems abstract the aim of this paper is to study the normal forms of nonautonomous differential systems for doing so we first investigate the nonuniform dichotomy spectrum of the linear evolution operators that admit a nonuniform exponential dichotomy where the linear evolution operators are defined by nonautonomous differential equations x a t x in r n using the nonuniform dichotomy spectrum we obtain the normal forms of the nonautonomous linear differential equations finally we establish the finite jet normal forms of the nonlinear differential systems x a t x f t x in r n which is based on the nonuniform dichotomy spectrum and the normal forms of the nonautonomous linear systems

pade approximations and diophantine geometry abstract using methods of pade approximations we prove a converse to eisenstein s theorem on the boundedness of denominators of coefficients in the expansion of an algebraic function for classes of functions parametrized by meromorphic functions this result is applied to the tate conjecture on the effective description of isogenies for elliptic curves

codimension 2 bifurcation of twisted double homoclinic loops a local active coordinates approach is employed to obtain bifurcation equations of twisted double homoclinic loops under the condition of one twisted orbit we obtain the existence and uniqueness and of the 1 1 double homoclinic loop 2 1 double homoclinic loop 2 1 right homoclinic loop 1 1 large homoclinic loop 2 1 large homoclinic loop and 2 1 large period orbit for the case of double twisted orbits we obtain the existence or non existence of 1 1 double homoclinic loop 1 2 double homoclinic loop 2 1 double homoclinic loop 2 2 double homoclinic loop 2 1 large homoclinic loop 1 2 large homoclinic loop 2 2 large homoclinic loop 2 2 right homoclinic loop 2 2 large homoclinic loop 2 2 left homoclinic loop and 2 2 large period orbit moreover the bifurcation surfaces and their existence regions are given besides bifurcation sets are presented on the 2 dimensional subspace spanned by the first two melnikov vectors

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

quadratic chabauty for modular curves and modular forms of rank one in this paper we provide refined sufficient conditions for the quadratic chabauty method on a curve x to produce an effective finite set of points containing the rational points x mathbb q with the condition on the rank of the jacobian of x replaced by condition on the rank of a quotient of the jacobian plus an associated space of chow heegner points we then apply this condition to prove the effective finiteness of x mathbb q for any modular curve x x 0 n or x mathrm ns n of genus at least 2 with n prime the proof relies on the existence of a quotient of their jacobians whose mordell weil rank is equal to its dimension and at least 2 which is proven via analytic estimates for orders of vanishing of l functions of modular forms thanks to a kolyvagin logachev type result

sternberg theorems for random dynamical systems in this paper we prove the smooth conjugacy theorems of sternberg type for random dynamical systems based on their lyapunov exponents we also present a stable and unstable manifold theorem with tempered estimates that are used to construct conjugacy 2005 wiley periodicals inc

on the marked length spectrum of generic strictly convex billiard tables in this paper we show that for a generic strictly convex domain one can recover the eigendata corresponding to aubry mather periodic orbits of the induced billiard map from the maximal marked length spectrum of the domain

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

unlikely intersections with isogeny orbits in a product of elliptic schemes fix an elliptic curve e 0 without cm and a non isotrivial elliptic scheme over a smooth irreducible curve both defined over the algebraic numbers consider the union of all images of a fixed finite rank subgroup of arbitrary rank of e 0 g also defined over the algebraic numbers under all isogenies between e 0 g and some fiber of the g th fibered power mathcal a of the elliptic scheme where g is a fixed natural number as a special case of a slightly more general result we characterize the subvarieties of arbitrary dimension inside mathcal a that have potentially zariski dense intersection with this set in the proof we combine a generalized vojta remond inequality with the pila zannier strategy

how violently do smooth iterative roots oscillate abstract a crisis in the smoothness of iterative roots appears at fixed points the local c r smoothness of iterative roots near a hyperbolic fixed point is well known but its global c r extension to both fixed points is generically impossible because of their violent oscillation near the next hyperbolic fixed point in this study we consider how violently these smooth iterative roots oscillate we prove that their first order derivatives are bounded but their r th r 2 order derivatives are generically unbounded in addition we prove that the global c 1 smoothness is of vital significance for the global higher order derivatives as a corollary we show that iterative roots on the maximal interval oscillate with c 1 boundedness but c r r 2 unboundedness

special points on fibered powers of elliptic surfaces consider a fibered power of an elliptic surface we characterize its subvarieties that contain a zariski dense set of points that are torsion points in fibers with complex multiplication this result can be viewed as a mix of the manin mumford and andre oort conjecture and is related to a conjecture of pink the main technical tool is a new height inequality we also use it to give another proof of a case of gubler s result on the bogomolov conjecture over function fields

on fractional iterates of a brouwer homeomorphism embeddable in a flow we present a method for finding continuous and consequently homeomorphic orientation preserving iterative roots of a brouwer homeomorphism which is embeddable in a flow to obtain the roots we use a countable family of maximal parallelizable regions of the flow which is a cover of the plane the maximal parallelizable regions are unions of equivalence classes of an appropriate equivalence relation we show that if an equivalence class is invariant under the nth iterate of a brouwer homeomorphism g then it is invariant under g we use this fact to prove that each maximal parallelizable region of the flow must be invariant under all homeomorphic orientation preserving iterative roots of the given brouwer homeomorphism

linearization of class c1 for contractions on banach spaces in this work we prove a c 1 linearization result for contraction diffeomorphisms near a fixed point valid in infinite dimensional banach spaces as an intermediate step we prove a specific result of existence of invariant manifolds which can be interesting by itself and that was needed on the proof of our main theorem our results essentially generalize some classical results by p hartman in finite dimensions and a result of x mora j sola morales in the infinite dimensional case it is shown that the result can be applied to some abstract systems of semilinear damped wave equations

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 question on characteristic endpoints for iterative roots of pm functions abstract iterative roots of mappings are of special interests because it defines fractional iteration displays middle procedure of evolution and proposes a weak version of the embedding flow problem for pm functions of height 1 the class of 1 dimensional mappings having the simplest nonmonotonicity the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition and the condition was proved to be necessary for those orders greater than the number of forts plus 1 this suggests an open question about iterative roots without that condition called the question on characteristic endpoints in this paper the question is answered completely in the case that the number of forts is equal to the order although results of nonexistence are also obtained for the case that the number of forts is greater than the order a full description is still open

cohomology equations and commutators of germs of contact diffeomorphisms we study the group of germs of contact diffeomorphisms at a fixed point we prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers the principal ingredient in this proof is a version of the sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure

slow motion for the 1d swift hohenberg equation abstract the goal of this paper is to study the behavior of certain solutions to the swift hohenberg equation on a one dimensional torus t combining results from convergence and ode theory it is shown that solutions corresponding to initial data that is l 1 close to a jump function v remain close to v for large time this can be achieved by regarding the equation as the l 2 gradient flow of a second order energy functional and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v

non monotonic iterative roots extended from characteristic intervals it has been treated as a difficult problem to find iterative roots of non monotonic functions for some pm functions which do not increase the number of forts under iteration a method was given to obtain a non monotonic iterative root by extending a monotone iterative root from the characteristic interval in this paper we prove that every continuous iterative root is an extension from the characteristic interval and give various modes of extension for those iterative roots of pm functions

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

conjugacies for linear and nonlinear perturbations of nonuniform behavior we construct topological conjugacies between linear and nonlinear evolution operators that admit either a nonuniform exponential contraction or a nonuniform exponential dichotomy we consider evolution operators defined by nonautonomous differential equations x a t x f t x in a banach space the conjugacies are obtained by first considering sufficiently small linear and nonlinear perturbations of linear equations x a t x in the case of linear perturbations we construct in a more or less explicit manner topological conjugacies between the two linear flows in the case of nonlinear perturbations we obtain a version of the grobman hartman theorem for nonuniformly hyperbolic dynamics furthermore all the conjugacies that we construct are locally holder continuous provided that the vectors fields are of class c1 as a byproduct of our approach we give conditions for the robustness of strong nonuniform exponential behavior in the sense that under sufficiently small perturbations the structure determined by the stable and unstable bundles persists up to small variations we also show that the constants determining the nonuniform exponential contraction or nonuniform exponential dichotomy vary continuously with the perturbation all the results are obtained in banach spaces

elliptic curves with abelian division fields let e be an elliptic curve over mathbb q and let n ge 1 the central object of study of this article is the division field mathbb q e n that results by adjoining to mathbb q the coordinates of all n torsion points on e overline mathbb q in particular we classify all curves e mathbb q such that mathbb q e n is as small as possible that is when mathbb q e n mathbb q zeta n and we prove that this is only possible for n 2 3 4 or 5 more generally we classify all curves such that mathbb q e n is contained in a cyclotomic extension of mathbb q or equivalently by the kronecker weber theorem when mathbb q e n mathbb q is an abelian extension in particular we prove that this only happens for n 2 3 4 5 6 or 8 and we classify the possible galois groups that occur for each value of n

iterative roots of piecewise monotonic functions with finite nonmonotonicity height abstract it is known that any continuous piecewise monotonic function with nonmonotonicity height not less than 2 has no continuous iterative roots of order n greater than the number of forts of the function in this paper we consider the problem of iterative roots in the case that the order n is less than or equal to the number of forts by investigating the trajectory of possible continuous roots we give a general method to find all iterative roots of those functions with finite nonmonotonicity height

c1 linearization for planar contractions c1 linearization is of special interests because it can distinguish characteristic directions of dynamical systems it is known that planar c1 contractions with a fixed point at the origin o admit c1 linearization with sufficiently small 0 if 1 and admit c1 linearization if log 1 log 2 1 1 where 1 and 2 are eigenvalues of the linear parts of the contractions at o with 0 1 2 1 in this paper we improve the lower bound of to lower the condition of c1 linearization for planar contractions furthermore we prove that the derivatives of transformations in our c1 linearization are holder continuous and give estimates for the holder exponent finally we give a counter example to show that those estimates cannot be improved anymore

normal forms for nonautonomous differential equations abstract we extend henry poincares normal form theory for autonomous differential equations x f x to nonautonomous differential equations x f t x poincares nonresonance condition j n i 1 l i i 0 for eigenvalues is generalized to the new nonresonance condition j n i 1 l i i for spectral intervals

c1 hartman theorem for random dynamical systems abstract the c 1 hartman theorem states that every c 1 1 contraction in r d admits c 1 linearization in this paper we prove c 1 linearizability for c 1 contractive or expansive random diffeomorphisms with appropriate constants 0 1 and 0 1 which implies a random version of the c 1 hartman theorem moreover the conjugacy is proved to be tempered this result strengthens related results even in the deterministic case lowering the c 1 1 condition to c 1 and enhancing the c 1 smoothness of the conjugacy to c 1 in r d in the proof we use a smooth weak stable invariant manifold in the random sense to construct the conjugacy and to overcome difficulties from nonuniformity and measurability

serre s uniformity problem in the split cartan case we prove that there exists an integer p 0 such that x split p q is made of cusps and cm points for any prime p p 0 equivalently for any non cm elliptic curve e over q and any prime p p 0 the image of the galois representation induced by the galois action on the p division points of e is not contained in the normalizer of a split cartan subgroup this gives a partial answer to an old question of serre

a note on iterative roots of pm functions abstract in this paper we study iterative roots of pm functions a special class of non monotone functions problem 2 in w zhang pm functions their characteristic intervals and iterative roots ann polon math lxv 1997 119 128 is solved partly and theorem 4 in that paper is generalized

some cases of serre s uniformity problem we show that if e mathbb q is an elliptic curve without complex multiplication and for which there is a prime q such that the image of bar rho e q is contained in the normaliser of a split cartan subgroup of mathrm gl 2 mathbb f q then bar rho e p surjects onto mathrm gl 2 mathbb f p for every prime p 37 this result complements a previous result by the author we also prove analogue results for certain families of mathbb q curves building on results of ellenberg am j math 126 4 763 787 2004 and le fourn math ann 365 1 2 173 214 2016

linearization of resonant vector fields a method allowing the formal linearization of a large class of vector fields with resonant eigenvalues is presented the admissible nonlinearities being characterized by conditions that are easy to check this method also gives information on the terms that are actually present in a nonlinear normal form of a given resonant vector field

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

complexity of multilevel monte carlo tau leaping tau leaping is a popular discretization method for generating approximate paths of continuous time discrete space markov chains notably for biochemical reaction systems to compute expected values in this context an appropriate multilevel monte carlo form of tau leaping has been shown to improve efficiency dramatically in this work we derive new analytic results concerning the computational complexity of multilevel monte carlo tau leaping that are significantly sharper than previous ones we avoid taking asymptotic limits and focus on a practical setting where the system size is large enough for many events to take place along a path so that exact simulation of paths is expensive making tau leaping an attractive option we use a general scaling of the system components that allows for the reaction rate constants and the abundances of species to vary over several orders of magnitude and we exploit the random time change representation developed by kurtz the key feature of the analysis that allows for the sharper bounds is that when comparing relevant pairs of processes we analyze the variance of their difference directly rather than bounding via the second moment use of the second moment is natural in the setting of a diffusion equation where multilevel monte carlo was first developed and where strong convergence results for numerical methods are readily available but is not optimal for the poisson driven jump systems that we consider here we also present computational results that illustrate the new analysis

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