pseudo normal form near saddle center or saddle focus equilibria abstract in this paper we introduce the pseudo normal form which generalizes the notion of normal form around an equilibrium its convergence is proved for a general analytic system in a neighborhood of a saddle center or a saddle focus equilibrium point if the system is hamiltonian or reversible this pseudo normal form coincides with the birkhoff normal form so we present a new proof in these celebrated cases from the convergence of the pseudo normal form for a general analytic system several dynamical consequences are derived like the existence of local invariant objects

high order melnikov method theory and application abstract let d t 0 e be the splitting distance of the stable and unstable manifold of a time periodic second order equation we expand d t 0 e as a formal power series in e as d t 0 e e 0 t 0 e e 1 t 0 e n e n t 0 in this paper we derive an explicit integral formula for e 1 t 0 we also evaluate e 1 t 0 to prove the existence of homoclinic tangles for an equation to which the poincare melnikov method fails to apply

a survey on learning to hash nearest neighbor search is a problem of finding the data points from the database such that the distances from them to the query point are the smallest learning to hash is one of the major solutions to this problem and has been widely studied recently in this paper we present a comprehensive survey of the learning to hash algorithms categorize them according to the manners of preserving the similarities into pairwise similarity preserving multiwise similarity preserving implicit similarity preserving as well as quantization and discuss their relations we separate quantization from pairwise similarity preserving as the objective function is very different though quantization as we show can be derived from preserving the pairwise similarities in addition we present the evaluation protocols and the general performance analysis and point out that the quantization algorithms perform superiorly in terms of search accuracy search time cost and space cost finally we introduce a few emerging topics

exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables abstract we present a numerical study of some billiard tables depending on a perturbative parameter 0 and a hyperbolicity parameter h 0 these tables are ellipses for 0 and circumferences in the limit h 0 elliptic billiard tables are integrable and have four separatrices which break up when 0 we conjecture based on numerical experiments that as h 0 the area of the main lobes of the resulting turnstile which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories behaves like an exponential term e 2 h times an asymptotic series j 0 j h 2 j such that 0 0 this series is gevrey 1 of type 1 2 2 so that its borel transform is convergent on a disk of radius 2 2 in the limit 0 the series j 0 j 0 h 2 j is an analytic function which can be explicitly computed with a discrete melnikov method the asymptotic series j 0 j h 2 j associated to the second exponential term e 2 2 h has the same properties finally we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space our computations have been performed in multiple precision arithmetic namely with several thousands decimal digits and rely strongly on the expansion of the local invariant curves up to very high orders namely with several hundreds taylor coefficients our programs have been written using the pari system

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems abstract in this paper we study a generalization of arnold s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori that has subsequently been referred to as arnold diffusion in case when the number of degrees of freedom n 3 namely we consider a widely studied model of a pendulum weakly coupled with n 1 rotors with the degeneracies in the unperturbed hamiltonian corresponding to different timescales existing in the problem using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation thus improving the estimate of gallavotti 1994 which he calls quasiflat and generalizing the analogous recent estimate of delshams et al 1996 for the rapidly quasiperiodically forced pendulum to a much larger class of hamiltonian systems in particular the exponentially small upper bound for the transversality measure of the splitting applies when the hamiltonian has extra degeneracies namely when the frequencies on a torus become near resonant in fact we show that in such a case the quantity in question becomes smaller which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger nekhoroshev exponent nekhoroshev exponent nevertheless we emphasize that getting uniform estimates for an arbitrary n 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector although recent developments show that the first order of canonical perturbation theory given by melnikov integrals generally cannot be accepted as the leading order answer for the splitting distance for the case of more than two degrees of freedom including the rapidly quasiperiodically forced pendulum problem we suggest an analytic perturbation the majority of whose fourier components are strictly nondashzero for which melnikov integrals can be vindicated as the leading order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions this allows us to bound the splitting distance from below furthermore having such a perturbation for the case of three degrees of freedom we use a simple number theoretical argument to find the asymptotics of the fourier series with exponentially small coefficients involved this enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure and by proving the domineering contribution of the first order of perturbation theory for the transversality measure to suggest a leading order answer for this quantity which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies we elucidate the numerous arithmetic issues that obstruct getting a compact leading order approximation for the splitting size most of which can be overcome in the case of three degrees of freedom as our example shows these obstacles can be also possibly avoided in the same fashion for an arbitrary n 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n 3 or n 2

exponentially small splitting of separatrices in a weakly hyperbolic case abstract we validate the poincare melnikov method in the singular case of high frequency periodic perturbations of the hamiltonian h 0 x y 1 2 y 2 x 3 x 4 under appropriate conditions which among other things imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one the splitting is exponentially small

dynamics of 4d symplectic maps near a double resonance abstract we study the dynamics of a family of 4 d symplectic mappings near a doubly resonant elliptic fixed point we derive and discuss algebraic properties of the resonances required for the analysis of a takens type normal form in particular we propose a classification of the double resonances adapted to this problem including cases of both strong and weak resonances around a weak double resonance a junction of two resonances of two different orders both being larger than 4 the dynamics can be described in terms of a simple in general non integrable hamiltonian model the non integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder we use a 4 d generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4 d symplectic map we evaluate numerically the volume of a 4 d parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively in good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2 d analog different numerical studies point out that double resonances play a key role to understand arnold diffusion this paper has to be seen also as a first step in this direction

volumes of complex analytic subvarieties of hermitian symmetric spaces we give lower bounds of volumes of k dimensional complex analytic subvarieties of certain naturally defined domains in n dimensional complex space forms of constant positive zero or negative holomorphic sectional curvature for each 1 k n the lower bounds are sharp in the sense that these bounds are attained by k dimensional complete totally geodesic complex submanifolds such lower bounds are obtained by constructing singular potential functions corresponding to blow ups of the kahler metrics involved similar lower bounds are also obtained in the case of hermitian symmetric spaces of noncompact type in this case the lower bounds are sharp for those values of k at which the hermitian symmetric space contains k dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature

a fast optimization method for general binary code learning hashing or binary code learning has been recognized to accomplish efficient near neighbor search and has thus attracted broad interests in recent retrieval vision and learning studies one main challenge of learning to hash arises from the involvement of discrete variables in binary code optimization while the widely used continuous relaxation may achieve high learning efficiency the pursued codes are typically less effective due to accumulated quantization error in this paper we propose a novel binary code optimization method dubbed discrete proximal linearized minimization dplm which directly handles the discrete constraints during the learning process specifically the discrete thus nonsmooth nonconvex problem is reformulated as minimizing the sum of a smooth loss term with a nonsmooth indicator function the obtained problem is then efficiently solved by an iterative procedure with each iteration admitting an analytical discrete solution which is thus shown to converge very fast in addition the proposed method supports a large family of empirical loss functions which is particularly instantiated in this paper by both a supervised and an unsupervised hashing losses together with the bits uncorrelation and balance constraints in particular the proposed dplm with a supervised ell 2 loss encodes the whole nus wide database into 64 b binary codes within 10 s on a standard desktop computer the proposed approach is extensively evaluated on several large scale data sets and the generated binary codes are shown to achieve very promising results on both retrieval and classification tasks

the break up of a heteroclinic connection in a volume preserving mapping abstract we consider a family of three dimensional volume preserving maps depending on a small parameter e as e 0 these maps asymptote to flows which attain a heteroclinic connection we show that for small e the heteroclinic connection breaks up and that the splitting between its components scales with e like e exp e we estimate using the singularities of the e 0 heteroclinic orbit in the complex plane we then estimate using linearization about orbits in the complex plane while these estimates are not proven they are well supported by our numerical calculations the work described here is a special case of the theory derived by amick et al which applies to q dimensional volume preserving mappings

stability of the phase motion in race track microtrons we model the phase oscillations of electrons in race track microtrons by means of an area preserving map with a fixed point at the origin which represents the synchronous trajectory of a reference particle in the beam we study the nonlinear stability of the origin in terms of the synchronous phase the phase of the synchronous particle at the injection we estimate the size and shape of the stability domain around the origin whose main connected component is enclosed by an invariant curve we describe the evolution of the stability domain as the synchronous phase varies we also clarify the role of the stable and unstable invariant curves of some hyperbolic fixed or periodic points

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

exponentially small splitting of separatrices under fast quasiperiodic forcing we consider fast quasiperiodic perturbations with two frequencies 1 epsiv of a pendulum where is the golden mean number the complete system has a two dimensional invariant torus in a neighbourhood of the saddle point we study the splitting of the three dimensional invariant manifolds associated to this torus provided that the perturbation amplitude is small enough with respect to and some of its fourier coefficients the ones associated to fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the melnikov function

dynamics around simple resonant tori in nearly integrable hamiltonian systems abstract by an application of the k a m theory we derive an accurate normal form valid in the vicinity of partially hyperbolic tori which arise close to simple resonances in nearly integrable hamiltonian systems this normal form allows to detect orbits homoclinic to a persistent torus moreover it also gives precise estimates on the times of transition around the stable and unstable manifolds of these tori hence we provide an efficient tool to compute the speed of drift of orbits shadowing a chain of hyperbolic tori associated to a simple resonant curve in the action space

hashing with angular reconstructive embeddings large scale search methods are increasingly critical for many content based visual analysis applications among which hashing based approximate nearest neighbor search techniques have attracted broad interests due to their high efficiency in storage and retrieval however existing hashing works are commonly designed for measuring data similarity by the euclidean distances in this paper we focus on the problem of learning compact binary codes using the cosine similarity specifically we proposed novel angular reconstructive embeddings are method which aims at learning binary codes by minimizing the reconstruction error between the cosine similarities computed by original features and the resulting binary embeddings furthermore we devise two efficient algorithms for optimizing our are in continuous and discrete manners respectively we extensively evaluate the proposed are on several large scale image benchmarks the results demonstrate that are outperforms several state of the art methods

oscillatory motions for the restricted planar circular three body problem the restricted three body problem models the motion of a massless body under the influence of the newtonian gravitational force caused by two other bodies called the primaries when they move along circular keplerian orbits and the third body moves in the same plane one has the restricted planar circular three body problem rpc3bp in suitable coordinates it is a hamiltonian system of two degrees of freedom the conserved energy is usually called the jacobi constant llibre and simo math ann 248 2 153 184 1980 proved the existence of oscillatory motions for this system that is orbits which leave every bounded region but which return infinitely often to some fixed bounded region to prove their existence they had to assume the ratio between the masses of the primaries to be small enough in this paper we prove the existence of such motions for any value of the mass ratio mu closing the problem of existence of oscillatory motions in the rpc3bp to obtain such motions we restrict ourselves to the level sets of the jacobi constant we show that for any value of the mass ratio and for large values of the jacobi constant there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets these transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits the main achievement is to prove the existence of these orbits without assuming the mass ratio mu small when mu is not small this transversality can not be checked by means of classical perturbation theory since our method is valid for all values of mu we are able to detect a curve in the parameter space formed by mu and the jacobi constant where cubic homoclinic tangencies between the invariant manifolds of infinity appear

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

melnikov method and exponentially small splitting of separatrices abstract we study the splitting of separatrices for high frequency perturbations of a planar hamiltonian system it is well known that in the analytic case the splitting of separatrices is exponentially small with respect to the perturbation period 2 e and that the constant in the exponent is related to the position of singularities of the unperturbed separatrix in the complex time plane on the other hand in many examples the melnikov function can be calculated explicitly and is exponentially small with an exponent very close but larger than the exponent from the upper estimates so the melnikov function appears to be smaller than the upper estimate as well as the standard estimate of the approximation error we found a class of hamiltonians such that the melnikov function gives actual asymptotics for the splitting provided the perturbation amplitude is less than 0ep for some p 0

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

on the transcendence degree of the differential field generated by siegel modular forms 1 statement of the results it is a classical fact that the elliptic modular function 10 00 satisfies an algebraic differential equation of order 3 this goes back to jacobi s fundamenta nova and none of lower order cf ra m in this paper we show how these properties generalize to siegel modular functions of arbitrary degree some notations are necessary before we can state our main results let g be a positive integer called indifferently degree or genus let k be an algebraically closed subfield of c and set hg siegel half space of degree g the q vector group zg formed by symmetric matrices of order g has dimension

width of stochastic layers in near integrable two dimensional symplectic maps abstract width of stochastic layers around separatrix eight is studied for two dimensional symplectic maps close to integrable we prove that under rather general assumptions the relation w d 1 log holds where w is the width of the stochastic layer d is the width of a lobe domain and is a multiplier at the corresponding hyperbolic point

definability of restricted theta functions and families of abelian varieties we consider some classical maps from the theory of abelian varieties and their moduli spaces and we prove their definability on restricted domains in the o minimal structure ran exp in particular we prove that the projective embedding of the moduli space of the principally polarized abelian variety sp 2g z hg is definable in ran exp when restricted to siegel s fundamental set fg we also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces

separatrix splitting for systems with three time scales an exact expression for the determinant of the splitting matrix is derived for three degrees of freedom systems with three time scales it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in ann inst h poincare b 60 1 1994 the asymptotic validity of mel nikov s integrals is proved for the class of models considered which are polynomial perturbations the technique for exhibiting cancellations is inspired by renormalization theory in quantum electrodynamics and uses an analogue of dyson s equations to prove an infinite family of identities due to symmetries that remind us of ward s identities

a refined formula for the separatrix splitting for the standard map abstract we study the standard map for small values of a parameter e an asymptotic formula is found which describes the size of the splitting of the stable and unstable separatrices of the origin considered as a hyperbolic fixed point the formula contains an exponentially decreasing expression times an asymptotic series on powers of a small parameter the coefficients in the asymptotic expansion are to be found in the process of solving numerically a nonlinear recurrent system of difference equations in the complex domain

on the reducibility of linear differential equations with quasiperiodic coefficients the system x a eq t x in rd is considered where a is a constant matrix and q a quasiperiodic analytic matrix with r basic frequencies the eigenvalues of a are arbitrary including the purely imaginary case suppose that the set formed by the eigenvalues of a and the basic frequencies of q satisfies a nonresonant condition then there is a positive measure cantorian set e such that for e e the system is reducible to constant coefficients by means of a quasiperiodic change of variables provided a nondegeneracy condition holds this condition prevents locking at resonance

arnold diffusion in a priori chaotic symplectic maps we assume that a symplectic real analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder we prove that generically in the real analytic category the boundaries of the invariant cylinder are connected by trajectories of the map

diophantine problems and p adic period mappings we give an alternative proof of faltings s theorem mordell s conjecture a curve of genus at least two over a number field has finitely many rational points our argument utilizes the set up of faltings s original proof but is in spirit closer to the methods of chabauty and kim we replace the use of abelian varieties by a more detailed analysis of the variation of p adic galois representations in a family of algebraic varieties the key inputs into this analysis are the comparison theorems of p adic hodge theory and explicit topological computations of monodromy by the same methods we show that in sufficiently large dimension and degree the set of hypersurfaces in projective space with good reduction away from a fixed set of primes is contained in a proper zariski closed subset of the moduli space of all hypersurfaces this uses in an essential way the ax schanuel property for period mappings recently established by bakker and tsimerman

rapidly forced planar vector fields and splitting of separatrices abstract we prove that for analytic rapidly forced two dimensional systems the melnikov function is exponentially small with respect to the period we provide an example of non validity of melnikov s method and we obtain upper bounds for conservative and analytic planar systems for big amplitude periodic rapidly forcing

robust discrete code modeling for supervised hashing abstract recent years have witnessed the promising efficacy and efficiency of hashing also known as binary code learning for retrieving nearest neighbor in large scale data collections particularly with supervision knowledge e g semantic labels we may further gain considerable performance boost nevertheless most existing supervised hashing schemes suffer from the following limitations 1 severe quantization error caused by continuous relaxation of binary codes 2 disturbance of unreliable codes in subsequent hash function learning and 3 erroneous guidance derived from imprecise and incomplete semantic labels in this work we propose a novel supervised hashing approach termed as robust discrete code modeling rdcm which directly learns high quality discrete binary codes and hash functions by effectively suppressing the influence of unreliable binary codes and potentially noisily labeled samples rdcm employs l2 p norm which is capable of inducing sample wise sparsity to jointly perform code selection and noisy sample identification moreover we preserve the discrete constraint in rdcm to eliminate the quantization error an efficient algorithm is developed to solve the discrete optimization problem extensive experiments conducted on various real life datasets show the superiority of the proposed rdcm approach as compared to several state of the art hashing methods

ax lindemann for mathcal a _g we prove the ax lindemann theorem for the coarse moduli space a g of principally polarized abelian varieties of dimension g 1 we affirm the andre oort conjecture unconditionally for a g for g 6 and under grh for all g

learning discriminative binary codes for large scale cross modal retrieval hashing based methods have attracted considerable attention for efficient cross modal retrieval on large scale multimedia data the core problem of cross modal hashing is how to learn compact binary codes that construct the underlying correlations between heterogeneous features from different modalities a majority of recent approaches aim at learning hash functions to preserve the pairwise similarities defined by given class labels however these methods fail to explicitly explore the discriminative property of class labels during hash function learning in addition they usually discard the discrete constraints imposed on the to be learned binary codes and compromise to solve a relaxed problem with quantization to obtain the approximate binary solution therefore the binary codes generated by these methods are suboptimal and less discriminative to different classes to overcome these drawbacks we propose a novel cross modal hashing method termed discrete cross modal hashing dch which directly learns discriminative binary codes while retaining the discrete constraints specifically dch learns modality specific hash functions for generating unified binary codes and these binary codes are viewed as representative features for discriminative classification with class labels an effective discrete optimization algorithm is developed for dch to jointly learn the modality specific hash function and the unified binary codes extensive experiments on three benchmark data sets highlight the superiority of dch under various cross modal scenarios and show its state of the art performance

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

sequential compact code learning for unsupervised image hashing effective hashing for large scale image databases is a popular research area attracting much attention in computer vision and visual information retrieval several recent methods attempt to learn either graph embedding or semantic coding for fast and accurate applications in this paper a novel unsupervised framework termed evolutionary compact embedding ece is introduced to automatically learn the task specific binary hash codes it can be regarded as an optimization algorithm that combines the genetic programming gp and a boosting trick in our architecture each bit of ece is iteratively computed using a weak binary classification function which is generated through gp evolving by jointly minimizing its empirical risk with the adaboost strategy on a training set we address this as greedy optimization by embedding high dimensional data points into a similarity preserved hamming space with a low dimension we systematically evaluate ece on two data sets sift 1m and gist 1m showing the effectiveness and the accuracy of our method for a large scale similarity search

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

nonresonant limit for sequences of resonant orbits the case of holomorphic maps the approximation of a nonresonant orbit with a sequence of resonant orbits is considered for the holomorphic maps of the complex plane the problem is motivated by hamiltonian dynamics greene s conjecture and we consider a complexified hamiltonian map in the region far from the section of real dynamics where it can be reduced to a holomorphic map of a single complex variable for a sequence of maps in normal form with linear resonant frequencies the limit to a linear map with nonresonant diophantine frequency has a simple interpretation the flower like resonant orbits become circles due to the increase of the number of petals and the freezing of radial motion a similar non trivial result is proved for small perturbations of the normal forms by investigating the behaviour of the conjugation functions

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

on homoclinic splitting problems abstract counterexamples to the homoclinic splitting claimed in the paper existence of exponentially small separatrix settings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems by rudnev and wiggins physica d 114 1998 3 80 are given these counterexamples invalidate the proofs of the main results of that paper

a proof of the exponentially small transversality of the separatrices for the standard map in 1984 v f lazutkin laz84 lst89 obtained an asymptotic formula for the separatrix splitting angle for the standard map the difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter lazutkin s proof was based on two conjectures probably the original form of those conjectures was incorrect but lazutkin s method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices the consequent works laz91 laz92 gls94 and gel96 prepared the base to fill all the gaps of the original proof the present paper contains a complete and self contained proof of a refined version of the original formula formula 1 7 of the present paper in this form the formula was obtained in gls94 the proof is inspired by the ideas of lazutkin s original paper laz84

on the conservation of adiabatic invariants for a system of coupled rotators abstract we study the accuracy of the conservation of adiabatic invariants in a model of n weakly coupled rotators most attention is devoted to n 2 and frequency 1 2 with 2 1 quadratic irrational we apply a heuristic approximation scheme going back to jeans and to landau and teller and perform a very accurate numerical check of the result observing a quite remarkable agreement

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

modeling the shape of the scene a holistic representation of the spatial envelope in this paper we propose a computational model of the recognition of real world scenes that bypasses the segmentation and the processing of individual objects or regions the procedure is based on a very low dimensional representation of the scene that we term the spatial envelope we propose a set of perceptual dimensions naturalness openness roughness expansion ruggedness that represent the dominant spatial structure of a scene then we show that these dimensions may be reliably estimated using spectral and coarsely localized information the model generates a multidimensional space in which scenes sharing membership in semantic categories e g streets highways coasts are projected closed together the performance of the spatial envelope model shows that specific information about object shape or identity is not a requirement for scene categorization and that modeling a holistic representation of the scene informs about its probable semantic category

splitting of separatrices and symbolic dynamics for some degenerate standard like maps we consider standard like maps of class c1 with parabolic fixed points and such that the second derivative has a jump discontinuity at the fixed points an asymptotic formula for the splitting of the separatrices is obtained the structure of certain chaotic set in the stochastic layer around the separatrices is described in terms of symbolic dynamics it turns out in particular that swinging and rotating regimes of motion may alternate in time arbitrarily analogous results are stated for the so called separatrix mapping and for the more smooth standard like maps

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

on stochastic sea of the standard map consider a generic one parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism we show that for many parameters residual subset in an open set approaching the critical value the corresponding diffeomorphism has a transitive invariant set of full hausdorff dimension the set is a topological limit of hyperbolic sets and is accumulated by elliptic islands

the rational points of a definable set let x r n be a set that is definable in an o minimal structure over r this article shows that in a suitable sense there are very few rational points of x which do not lie on some connected semialgebraic subset of x of positive dimension

a sparse embedding and least variance encoding approach to hashing hashing is becoming increasingly important in large scale image retrieval for fast approximate similarity search and efficient data storage many popular hashing methods aim to preserve the knn graph of high dimensional data points in the low dimensional manifold space which is however difficult to achieve when the number of samples is big in this paper we propose an effective and efficient hashing approach by sparsely embedding a sample in the training sample space and encoding the sparse embedding vector over a learned dictionary to this end we partition the sample space into clusters via a linear spectral clustering method and then represent each sample as a sparse vector of normalized probabilities that it falls into its several closest clusters this actually embeds each sample sparsely in the sample space the sparse embedding vector is employed as the feature of each sample for hashing we then propose a least variance encoding model which learns a dictionary to encode the sparse embedding feature and consequently binarize the coding coefficients as the hash codes the dictionary and the binarization threshold are jointly optimized in our model experimental results on benchmark data sets demonstrated the effectiveness of the proposed approach in comparison with state of the art methods

a field theory approach to lindstedt series for hyperbolic tori in three time scales problems interacting systems consisting of two rotators and a pendulum are considered in a case in which the uncoupled systems have three very different characteristic time scales the abundance of unstable quasiperiodic motions in phase space is studied via the lindstedt series regarded as a sum of feynman graphs and studied with renormalization group techniques based on eliasson s work on kam tori the result is a strong improvement compared to our previous results on the domain of validity of bounds that imply existence of invariant tori large homoclinic angles long heteroclinic chains and drift diffusion in phase space

fast similarity search for learned metrics we introduce a method that enables scalable similarity search for learned metrics given pairwise similarity and dissimilarity constraints between some examples we learn a mahalanobis distance function that captures the examples underlying relationships well to allow sublinear time similarity search under the learned metric we show how to encode the learned metric parameterization into randomized locality sensitive hash functions we further formulate an indirect solution that enables metric learning and hashing for vector spaces whose high dimensionality makes it infeasible to learn an explicit transformation over the feature dimensions we demonstrate the approach applied to a variety of image data sets as well as a systems data set the learned metrics improve accuracy relative to commonly used metric baselines while our hashing construction enables efficient indexing with learned distances and very large databases

splitting of a small separatrix loop near the saddle center bifurcation in area preserving maps abstract when the saddle center bifurcation occurs in an analytic family of area preserving maps a parabolic fixed point first appears at the origin and then this point bifurcates creating an elliptic fixed point and a hyperbolic one separatrices of the hyperbolic fixed point form a small loop around the elliptic point in general the separatrices intersect transversely and the splitting is exponentially small with respect to the perturbation parameter we derive an asymptotic formula which describes the splitting and study the properties of the pre exponential factor

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

study of the double mathematical pendulum ii investigation of exponentially small homoclinic intersections we consider the double mathematical pendulum in the limit when the ratio of pendulum masses is close to zero and the ratio of pendulum lengths is close to infinity we found that the limit system has a hyperbolic periodic trajectory whose invariant manifolds intersect transversally and the intersections are exponentially small in this case we obtain an asymptotic formula of the homoclinic invariant for the limit system

stick slip chaos detection in coupled oscillators with friction abstract we consider two coupled oscillators with negative duffing type stiffness which are self due to friction and externally harmonically excited the fundamental solutions of the homoclinic orbit are constructed then the melnikov gruendler approach is used to define the melnikov s function including smooth and stick slip chaotic behaviour theoretical considerations are supported by numerical examples

exponentially small splitting a direct approach abstract in this paper we go beyond what was proposed in theory by melnikov 15 to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time periodic equations not only we derive integral formula for splitting distances of all orders but also we develop an analytic theory to evaluate the acquired multiple integrals we reveal that the dominance of the exponentially small poincare melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order melnikov integrals this symmetry is beheld by many non hamiltonian equations

asymptotic expansion of homoclinic structures in a symplectic mapping the phenomenon of separatrix splitting and complicated homoclinic structures constitute a symptom of chaos as pointed out by poincare more than a century ago taking a time discrete dynamical system with a double well potential as an example this interesting feature is demonstrated analytically on the basis of asymptotic expansions beyond all orders violent undulations of the unstable manifold in the extreme vicinity of hyperbolic fixed points are recovered excellently comparison with results for standard and henon maps is also made

a practical use of the melnikov homoclinic method using cutoff functions and periodic extensions we prove that the melnikov homoclinic method gives a criterium to show that for a finite time interval t t with t arbitrarily large the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions the method is applied to specific perturbations of the pendulum and of the gylden s problem

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

dynamics in chaotic zones of area preserving maps close to separatrix and global instability zones abstract the purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically first of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out the well known separatrix map defined on a figure eight when needed is used to determine the location of rotational invariant curves r i c inside and outside the resonance the interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices to produce theoretical estimates of the width of the stochastic zone distance to the r i c existence of tiny islands close to the separatrices in every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations describing the analogy when possible after this study we focus on the formation of larger domains without r i c e g birkhoff domains to this end we introduce the biseparatrix map model although this is a qualitative model the mechanism of destruction of the last r i c and hence the process of creation of zones without r i c is clarified by means of this simple model several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions

algebraic differential equations from covering maps abstract let y be a complex algebraic variety g y an action of an algebraic group on y u y c a complex submanifold g c a discrete zariski dense subgroup of g c which preserves u and u x c an analytic covering map of the complex algebraic variety x expressing x c as u we note that the theory of elimination of imaginaries in differentially closed fields produces a generalized schwarzian derivative y z where z is some algebraic variety expressing the quotient of y by the action of the constant points of g under the additional hypothesis that the restriction of to some set containing a fundamental domain is definable in an o minimal expansion of the real field we show as a consequence of the peterzil starchenko o minimal gaga theorem that the prima facie differentially analytic relation 1 is a well defined differential constructible function the function nearly inverts in the sense that for any differential field k of meromorphic functions if a b x k then a b if and only if after suitable restriction there is some g c with 1 a b

hashing on nonlinear manifolds learning based hashing methods have attracted considerable attention due to their ability to greatly increase the scale at which existing algorithms may operate most of these methods are designed to generate binary codes preserving the euclidean similarity in the original space manifold learning techniques in contrast are better able to model the intrinsic structure embedded in the original high dimensional data the complexities of these models and the problems with out of sample data have previously rendered them unsuitable for application to large scale embedding however in this paper how to learn compact binary embeddings on their intrinsic manifolds is considered in order to address the above mentioned difficulties an efficient inductive solution to the out of sample data problem and a process by which nonparametric manifold learning may be used as the basis of a hashing method are proposed the proposed approach thus allows the development of a range of new hashing techniques exploiting the flexibility of the wide variety of manifold learning approaches available it is particularly shown that hashing on the basis of t distributed stochastic neighbor embedding outperforms state of the art hashing methods on large scale benchmark data sets and is very effective for image classification with very short code lengths it is shown that the proposed framework can be further improved for example by minimizing the quantization error with learned orthogonal rotations without much computation overhead in addition a supervised inductive manifold hashing framework is developed by incorporating the label information which is shown to greatly advance the semantic retrieval performance

o minimality and the andre oort conjecture for mathbb c n we give an unconditional proof of the andre oort conjecture for arbitrary products of modular curves we establish two generalizations the first includes the manin mumford conjecture for arbitrary products of elliptic curves defined over q as well as lang s conjecture for torsion points in powers of the multiplicative group the second includes the manin mumford conjecture for abelian varieties defined over q our approach uses the theory of o minimal structures a part of model theory and follows a strategy proposed by zannier and implemented in three recent papers a new proof of the manin mumford conjecture by pila zannier a proof of a special but new case of pink s relative manin mumford conjecture by masser zannier and new proofs of certain known results of andre oort manin mumford type by pila 2010 mathematics subject classification 11g18 03c64