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

adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

numerical investigations of the structure around an invariant kam torus using the frequency map analysis abstract the hierarchical structure around kam tori is studied numerically on the standard map as a model problem applying the laskar frequency map analysis the theoretical prediction recently given by morbidelli and giorgilli j stat phys 78 1994 about the existence of a neighborhood almost completely full of slave tori around a chief torus are recovered and extended to the near critical regime

birkhoff normal form for some nonlinear pdes we consider the problem of extending to pdes birkhoff normal form theorem on hamiltonian systems close to nonresonant elliptic equilibria as a model problem we take the nonlinear wave equation with dirichlet boundary conditions on 0 g is an analytic skewsymmetric function which vanishes for u 0 and is periodic with period 2 in the x variable we prove under a nonresonance condition which is fulfilled for most g s that for any integer m there exists a canonical transformation that puts the hamiltonian in birkhoff normal form up to a reminder of order m the canonical transformation is well defined in a neighbourhood of the origin of a sobolev type phase space of sufficiently high order some dynamical consequences are obtained the technique of proof is applicable to quite general semilinear equations in one space dimension

on the performance of affine policies for two stage adaptive optimization a geometric perspective we consider two stage adjustable robust linear optimization problems with uncertain right hand side mathbf b b belonging to a convex and compact uncertainty set mathcal u u we provide an a priori approximation bound on the ratio of the optimal affine in mathbf b b solution to the optimal adjustable solution that depends on two fundamental geometric properties of mathcal u u a the symmetry and b the simplex dilation factor of the uncertainty set mathcal u u and provides deeper insight on the power of affine policies for this class of problems the bound improves upon a priori bounds obtained for robust and affine policies proposed in the literature we also find that the proposed a priori bound is quite close to a posteriori bounds computed in specific instances of an inventory control problem illustrating that the proposed bound is informative

spectral monodromy of non self adjoint operators in the present paper we build a combinatorial invariant called the spectral monodromy from the spectrum of a single non self adjoint h pseudodifferential operator with two degrees of freedom in the semi classical limit our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self adjoint h pseudodifferential operators given by s vu ngoc quantum monodromy in integrable systems commun math phys 203 2 465 479 1999 the first simple case that we treat in this work is a normal operator in this case the discrete spectrum can be identified with the joint spectrum of an integrable quantum system the second more complex case we propose is a small perturbation of a self adjoint operator with a classical integrability property we show that the discrete spectrum in a small band around the real axis also has a combinatorial monodromy the main difficulty in this case is that we do not know the description of the spectrum everywher

deriving robust counterparts of nonlinear uncertain inequalities in this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters we use convex analysis support functions conjugate functions fenchel duality and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints it turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function conveniently these two computations are completely independent this approach has several advantages first it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities second it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived we also study some cases where the inequality is nonconcave in the uncertain parameters

a tight characterization of the performance of static solutions in two stage adjustable robust linear optimization in this paper we study the performance of static solutions for two stage adjustable robust linear optimization problems with uncertain constraint and objective coefficients and give a tight characterization of the adaptivity gap computing an optimal adjustable robust optimization problem is often intractable since it requires to compute a solution for all possible realizations of uncertain parameters feige et al in lect notes comput sci 4513 439 453 2007 on the other hand a static solution is a single here and now solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently for most dynamic optimization problems we show that for a fairly general class of uncertainty sets a static solution is optimal for the two stage adjustable robust linear packing problems this is highly surprising in view of the usual perception about the conservativeness of static solutions furthermore when a static solution is not optimal for the adjustable robust problem we give a tight approximation bound on the performance of the static solution that is related to a measure of non convexity of a transformation of the uncertainty set we also show that our bound is at least as good and in many case significantly better as the bound given by the symmetry of the uncertainty set bertsimas and goyal in math methods oper res 77 3 323 343 2013 bertsimas et al in math oper res 36 1 24 54 2011

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

the graphical evaluation of explanatory variables in proportional hazard regression models summary a graphical technique for assessing explanatory variables in cox s proportional hazards regression model is examined the method arises from consideration of the cumulative hazard transformation and the partial likelihood score function and consists of permuting the observed rank statistic to account for the effects of explanatory variables fitted to the model an example is provided

analysis of multi type recurrent events in longitudinal studies application to a skin cancer prevention trial we consider the statistical modeling and analysis of replicated multi type point process data with covariates such data arise when heterogeneous subjects experience repeated events or failures which may be of several distinct types the underlying processes are modeled as nonhomogeneous mixed poisson processes with random subject and fixed covariate effects the method of maximum likelihood is used to obtain estimates and standard errors of the failure rate parameters and regression coefficients score tests and likelihood ratio statistics are used for covariate selection a graphical test of goodness of fit of the selected model is based on generalized residuals measures for determining the influence of an individual observation on the estimated regression coefficients and on the score test statistic are developed an application is described to a large ongoing randomized controlled clinical trial for the efficacy of nutritional supplements of selenium for the prevention of two types of skin cancer

an abstract birkhoff normal form theorem and exponential type stability of the 1d nls we study stability times for a family of parameter dependent nonlinear schrodinger equations on the circle close to the origin imposing a suitable diophantine condition first introduced by bourgain we prove a rather flexible birkhoff normal form theorem which implies e g exponential and sub exponential time estimates in the sobolev and gevrey class respectively

approximate case influence for the proportional hazards regression model with censored data a method is presented for approximating the influence of individual cases upon regression coefficient estimates obtained from the cox proportional hazards model observations can thus be identified which may greatly influence statistical inferences regarding the effects of prognostic factors upon survival time an example from a cancer clinical trial is given

the stability of full dimensional kam tori for nonlinear schrodinger equation abstract in this paper it is proved that the full dimensional invariant tori obtained by bourgain j funct anal 229 2005 no 1 62 94 is stable in a very long time for 1d nonlinear schrodinger equation with periodic boundary conditions

super exponential stability for generic real analytic elliptic equilibrium points we consider the dynamics in a neighborhood of an elliptic equilibrium point with a diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability generically both in a topological and measure theoretical sense any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point

supermodularity and affine policies in dynamic robust optimization this paper considers a particular class of dynamic robust optimization problems where a large number of decisions must be made in the first stage which consequently fix the constraints and cost structure underlying a one dimensional linear dynamical system we seek to bridge two classical paradigms for solving such problems namely 1 dynamic programming dp and 2 policies parameterized in model uncertainties also known as decision rules obtained by solving tractable convex optimization problems we show that if the uncertainty sets are integer sublattices of the unit hypercube the dp value functions are convex and supermodular in the uncertain parameters and a certain technical condition is satisfied then decision rules that are affine in the uncertain parameters are optimal we also derive conditions under which such rules can be obtained by optimizing simple i e linear objective functions over the uncertainty sets our results suggest new modeling paradigms for dynamic robust optimiz

estimates on invariant tori near an elliptic equilibrium point of a hamiltonian system abstract we give a precise statement for the kam theorem in a neighbourhood of an elliptic equilibrium point of a hamiltonian system if the frequencies of the elliptic point are nonresonant up to a certain order k 4 and a nondegeneracy condition is fulfilled we get an estimate for the measure of the complement of the kam tori in a neighbourhood of given radius moreover if the frequencies satisfy a diophantine condition with exponent we show that in a neighbourhood of radius r the measure of the complement is exponentially small in 1 r 1 1 we also give a related result for quasi diophantine frequencies which is more useful for practical purposes the results are obtained by putting the system in birkhoff normal form up to an appropiate order and the key point relies on giving accurate bounds for its terms

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

robust dynamic programming in this paper we propose a robust formulation for discrete time dynamic programming dp the objective of the robust formulation is to systematically mitigate the sensitivity of the dp optimal policy to ambiguity in the underlying transition probabilities the ambiguity is modeled by associating a set of conditional measures with each state action pair consequently in the robust formulation each policy has a set of measures associated with it we prove that when this set of measures has a certain rectangularity property all of the main results for finite and infinite horizon dp extend to natural robust counterparts we discuss techniques from nilim and el ghaoui 17 for constructing suitable sets of conditional measures that allow one to efficiently solve for the optimal robust policy we also show that robust dp is equivalent to stochastic zero sum games with perfect information

on a connection between kam and nekhoroshev s theorems abstract using the iteration of nekhoroshev s theorem as a basic tool we point out the existence of a hierarchic structure of nested domains underlying the phenomenon of diffusion at each level we find that the diffusions speed is exponentially small with respect to the previous level the set of kam tori is the domain characterized by a zero diffusion speed

topologically crossing heteroclinic connections to invariant tori we consider transition tori of arnold which have topologically crossing heteroclinic connections we prove the existence of shadowing orbits to a bi inflnite sequence of tori and of symbolic dynamics near a flnite collec tion of tori topological crossing intersections of stable and unstable manifolds of tori can be found as non trivial zeroes of certain melnikov functions our treatment relies on an extension of easton s method of correctly aligned win dows due to zgliczy

pareto efficiency in robust optimization this paper formalizes and adapts the well known concept of pareto efficiency in the context of the popular robust optimization ro methodology for linear optimization problems we argue that the classical ro paradigm need not produce solutions that possess the associated property of pareto optimality and we illustrate via examples how this could lead to inefficiencies and suboptimal performance in practice we provide a basic theoretical characterization of pareto robustly optimal pro solutions and extend the ro framework by proposing practical methods that verify pareto optimality and generate solutions that are pro critically important our methodology involves solving optimization problems that are of the same complexity as the underlying robust problems hence the potential improvements from our framework come at essentially limited extra computational cost we perform numerical experiments drawn from three different application areas portfolio optimization inventory management and project management which demonstrate that pro solutions have a significant potential upside compared with solutions obtained using classical ro methods this paper was accepted by gerard p cachon optimization

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

hyperbolicity versus partial hyperbolicity and the transversality torsion phenomenon abstract in this paper we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrees of freedom hamiltonian systems the transversality torsion phenomenon

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

worst case robust decisions for multi period mean variance portfolio optimization abstract in this paper we extend the multi period mean variance optimization framework to worst case design with multiple rival return and risk scenarios our approach involves a min max algorithm and a multi period mean variance optimization framework for the stochastic aspects of the scenario tree multi period portfolio optimization entails the construction of a scenario tree representing a discretised estimate of uncertainties and associated probabilities in future stages the expected value of the portfolio return is maximized simultaneously with the minimization of its variance there are two sources of further uncertainty that might require a strengthening of the robustness of the decision the first is that some rival uncertainty scenarios may be too critical to consider in terms of probabilities the second is that the return variance estimate is usually inaccurate and there are different rival estimates or scenarios in either case the best decision has the additional property that in terms of risk and return performance is guaranteed in view of all the rival scenarios the ex ante performance of min max models is tested using historical data and backtesting results are presented

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

numerical computation of normal forms around some periodic orbits of the restricted three body problem in this paper we introduce a general methodology for computing numerically the normal form around a periodic orbit of an autonomous analytic hamiltonian system the process follows two steps first we expand the hamiltonian in suitable coordinates around the orbit and second we perform a standard normal form scheme based on the lie series method this scheme is carried out up to some finite order and neglecting the remainder we obtain an accurate description of the dynamics in a small enough neighbourhood of the orbit in particular we obtain the invariant tori that generalize the elliptic directions of the periodic orbit on the other hand bounding the remainder one obtains lower estimates for the diffusion time around the orbit this procedure is applied to an elliptic periodic orbit of the spatial restricted three body problem the selected orbit belongs to the lyapunov family associated to the vertical oscillation of the equilibrium point l 5 the mass parameter mu has been chosen such that l 5 is unstable but the periodic orbit is still stable this allows to show the existence of regions of effective stability near l 5 for values of mu bigger that the routh critical value the computations have been done using formal expansions with numerical coefficients

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

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

optimality of affine policies in multistage robust optimization in this paper we prove the optimality of disturbance affine control policies in the context of one dimensional constrained multistage robust optimization our results cover the finite horizon case with minimax worst case objective and convex state costs plus linear control costs we develop a new proof methodology which explores the relationship between the geometrical properties of the feasible set of solutions and the structure of the objective function apart from providing an elegant and conceptually simple proof technique the approach also entails very fast algorithms for the case of piecewise affine state costs which we explore in connection with a classical inventory management application

asymptotic distribution theory for cox type regression models with general relative risk form the theory and application of the cox 1972 failure time regression model has almost without exception assumed an exponential form for the dependence of the hazard function on regression variables other regression forms may be more natural or descriptive in some applications for example a linear relative risk regression model provides a convenient framework for studying epidemiologic risk factor interactions this note uses the counting process formulation of andersen and gill 1982 to develop asymptotic distribution theory for a class of intensity function regression models in which the usual exponential regression form is relaxed to an arbitrary non negative twice differentiable form some stability and regularity conditions beyond those of andersen and gill are required to show the consistency of the observed information matrix which in general need not be positive semidefinite

branching of cantor manifolds of elliptic tori and applications to pdes we consider infinite dimensional hamiltonian systems we prove the existence of cantor manifolds of elliptic tori of any finite higher dimension accumulating on a given elliptic kam torus then close to an elliptic equilibrium we show the existence of cantor manifolds of elliptic tori which are branching points of other cantor manifolds of higher dimensional tori we also answer to a conjecture of bourgain proving the existence of invariant elliptic tori with tangential frequency along a pre assigned direction the proofs are based on an improved kam theorem its main advantages are an explicit characterization of the cantor set of parameters and weaker smallness conditions on the perturbation we apply these results to the nonlinear wave equation

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

symbolic dynamics and arnold diffusion we consider hyperbolic tori of three degrees of freedom initially hyperbolic hamiltonian systems we prove that if the stable and unstable manifold of a hyperbolictorus intersec t transversaly then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a bernoulli shift the proof is based on a new geometricodynamical feature of partially hyperbolic systems the transversality torsion phenomenon which produces complete hyperbolicity from partial hyperbolicity we deduce the existence of infinitely many hyperbolicperiodicorbits near the given torus the relevanc e of these results for the instability of near integrable hamiltonian systems is then discussed for a given transition chain we construct chain of hyperbolic periodic orbits then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolicsets r 2002 elsevier science usa all rights reserved

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

assessment of local influence summary statistical models usually involve some degree of approximation and therefore are nearly always wrong because of this inexactness an assessment of the influence of minor perturbations of the model is important we discuss a method for carrying out such an assessment the method is not restricted to linear regression models and it seems to provide a relatively simple unified approach for handling a variety of problems

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

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

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

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

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

superexponential stability of kam tori we study the dynamics in the neighborhood of an invariant torus of a nearly integrable system we provide an upper bound to the diffusion speed which turns out to be of superexponentially small size exp exp 1 being the distance from the invariant torus we also discuss the connection of this result with the existence of many invariant tori close to the considered one

erratum to existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems physica d 114 1998 3 80 abstract as was pointed out by gallavotti gentile and mastropietro physica d 137 2000 202 204 theorem 2 1 of rudnev and wiggins physica d 114 1998 3 80 is false this theorem claimed an exponentially small estimate 67 for the fourier coefficients of the splitting distance function 20 of rudnev and wiggins 1998 further it was used in order to arrive at theorems 2 2 and 2 3 thus invalidating the proofs of these theorems as given the present note is contingent upon the new results m rudnev s wiggins regular chaot dyn 4 1999 39 58 m rudnev s wiggins regular chaot dyn 5 2000 227 242 and vindicates the estimate in question this is achieved via redefining the splitting distance function in accordance with the approach developed in these papers such a definition is provided by 11 of this note despite the fact that 11 and the definition 20 given by rudnev and wiggins 1998 differ only by a near identity reparameterization the former affords the estimate 13 of this note which implies 67 of rudnev and wiggins 1998 theorem b of this note gives the correct statement of theorem 2 1 of rudnev and wiggins 1998 hence theorems 2 2 and 2 3 therein stand as they are modulo a few modifications compiled at the end of this note

on the power and limitations of affine policies in two stage adaptive optimization we consider a two stage adaptive linear optimization problem under right hand side uncertainty with a min max objective and give a sharp characterization of the power and limitations of affine policies where the second stage solution is an affine function of the right hand side uncertainty in particular we show that the worst case cost of an optimal affine policy can be omega m 1 2 delta times the worst case cost of an optimal fully adaptable solution for any 0 where m is the number of linear constraints we also show that the worst case cost of the best affine policy is o sqrt m times the optimal cost when the first stage constraint matrix has non negative coefficients moreover if there are only k m uncertain parameters we generalize the performance bound for affine policies to o sqrt k which is particularly useful if only a few parameters are uncertain we also provide an o sqrt k approximation algorithm for the general case without any restriction on the constraint matrix but the solution is not an affine function of the uncertain parameters we also give a tight characterization of the conditions under which an affine policy is optimal for the above model in particular we show that if the uncertainty set mathcal u subseteq mathbb r m is a simplex then an affine policy is optimal however an affine policy is suboptimal even if mathcal u is a convex combination of only m 3 extreme points only two more extreme points than a simplex and the worst case cost of an optimal affine policy can be a factor 2 worse than the worst case cost of an optimal fully adaptable solution for any 0

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

effective stability for gevrey and finitely differentiable prevalent hamiltonians for perturbations of integrable hamiltonian systems the nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time provided the integrable part satisfies a steepness condition and the system is analytic this fundamental result has been extended in two distinct directions the first one is due to niederman who showed that under the analyticity assumption the result holds true for a prevalent class of integrable systems which is much wider than the steep systems the second one is due to marco sauzin but it is limited to quasi convex integrable systems for which they showed exponential stability if the system is assumed to be only gevrey regular if the system is finitely differentiable we showed polynomial stability still in the quasi convex case the goal of this work is to generalize all these results in a unified way by proving exponential or polynomial stability for gevrey or finitely differentiable perturbations of prevalent integrable hamiltonian systems

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

high order control for symplectic maps abstract we revisit the problem of introducing an a priori control for devices that can be modeled via a symplectic map in a neighborhood of an elliptic equilibrium using a technique based on lie transform methods we produce a normal form algorithm that avoids the usual step of interpolating the map with a flow the formal algorithm is completed with quantitative estimates that bring into evidence the asymptotic character of the normal form transformation then we perform an heuristic analysis of the dynamical behavior of the map using the invariant function for the normalized map finally we discuss how control terms of different orders may be introduced so as to increase the size of the stable domain of the map the numerical examples are worked out on a two dimensional map of henon type

fractional kinetic model for chaotic transport in nonintegrable hamiltonian systems we propose a kinetic model of transport in nonintegrable hamiltonian systems based on a fractional kinetic equation with spatially dependent diffusion coefficient the diffusion coefficient is estimated from the remainder of the optimal normal form for the given region of the phase space after partitioning the phase space into building blocks a separate equation can be constructed for each block solving the kinetic equations approximately and estimating the diffusion time scales we convolve the solutions to get the description of the macroscopic behavior we show that in the limit of infinitely many blocks one can expect an approximate scaling relation between the lyapunov time and the diffusion or escape time which is either an exponential or a power law we check our results numerically on over a dozen hamiltonians and find a good agreement

a proof of the isoenergetic kam theorem from the ordinary one a proof is given of the isoenergetic kam theorem for hamiltonian systems using the ordinary kam theorem and a transversality argument

diophantine tori and spectral asymptotics for nonselfadjoint operators we study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint h pseudodifferential operators in dimension 2 assuming that the classical flow of the unperturbed part possesses several invariant lagrangian tori enjoying a diophantine property we get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases in the first case we assume that the strength e of the perturbation is o hd for some d 0 and is bounded from below by a fixed positive power of h in the second case e is assumed to be sufficiently small but independent of h and we describe the eigenvalues completely in a fixed h independent domain in the complex spectral plane

when are static and adjustable robust optimization problems with constraint wise uncertainty equivalent adjustable robust optimization aro generally produces better worst case solutions than static robust optimization ro however aro is computationally more difficult than ro in this paper we provide conditions under which the worst case objective values of aro and ro problems are equal we prove that when the uncertainty is constraint wise the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact then robust solutions are also optimal for the corresponding aro problem furthermore we prove that if some of the uncertain parameters are constraint wise and the rest are not then under a similar set of assumptions there is an optimal decision rule for the aro problem that does not depend on the constraint wise uncertain parameters also we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non constraint wise uncertain parameters finally we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems

a diagnostic statistic for the score test abstract this article provides exact and approximate expressions for the change in the score test on deletion of observations deletions of individuals and of entire risk sets from matched case control and survival studies are discussed in detail a simple formula is derived which measures the contribution of individual strata to score tests based on combining several contingency tables the diagnostic is illustrated by application to the stanford heart transplant data

existence of kam tori in degenerate hamiltonian systems abstract this article is devoted to the proof that invariant tori still exist in perturbed integrable hamiltonian systems with n degrees of freedom h p q n p p p q if the hypothesis of nondegeneracy is replaced by a much weaker one that the rank of the mapping n p can be any positive integer smaller than n provided the image of any open set under the mapping includes a cn 2 curved one dimensional submanifold

measuring the mixing efficiency in a simple model of stirring some analytical results and a quantitative study via frequency map analysis we prove the existence of invariant curves for a t periodic hamiltonian system which models a fluid stirring in a cylindrical tank when t is small and the assigned stirring protocol is piecewise constant furthermore using the numerical analysis of the fundamental frequency of laskar we investigate numerically the break down of invariant curves as t increases and we give a quantitative estimate of the efficiency of the mixing

effective stability for a hamiltonian system near an elliptic equilibrium point with an application to the restricted three body problem abstract we consider an n degrees of freedom hamiltonian system near an elliptic equilibrium point the system is put in normal form up to an arbitrary order and with respect to some resonance module and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form a bound to the rate of diffusion is thus found and by optimizing the order of normalization exponential estimates of nekhoroshev s type are obtained this provides explicit estimates for the stability properties of the elliptic point and leads in some cases to effective stability i e stability up to finite but long times an application to the stability of the triangular libration points in the spatial restricted three body is also given

detection of arnold diffusion in hamiltonian systems we detect diffusion along resonances in a quasi integrable system at small values of the perturbing parameter the diffusion coefficient goes to zero as the perturbation parameter goes to zero faster than a power law typical of chirikov diffusion and is compatible with an exponential law expected in the nekhoroshev theorem

logistic regression diagnostics a maximum likelihood fit of a logistic regression model and other similar models is extremely sensitive to outlying responses and extreme points in the design space we develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit the elements of the fitting process which constitute the usual output parameter estimates standard errors residuals etc will be used for this purpose with a properly designed computing package for fitting the usual maximum likelihood model the diagnostics are essentially free for the asking in particular good data analysis for logistic regression models need not be expensive or time consuming

high order curvatures and harmonicity regression we consider curvatures of all orders as defined by the generalized frenet serret formulae along the trajectories of a classical hamiltonian system with n degrees of freedom in the spirit of previous experiments on the first two of them time averages are numerically computed for the curvatures up to fifth order and for the microcanonical density in a typical anharmonic system the fpu quartic chain with checks in other models neat breakdowns of harmonic like behaviour define thresholds to anharmonicity for every at distinct values of the order parameter the energy density u the threshold at fixed order i is independent of the total n and it rapidly decreases as i grows however all curvatures are simultaneously sensitive or not to the initial conditions for or respectively confirming the previous identification of as an efficient indicator of the strong stochasticity transition this phenomenology which is discussed within the weak strong stochasticity problem gives a new insight into the progressive enforcement of a harmonic like structure as u decreases

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

effective stability and kam theory abstract the two main stability results for nearly integrable hamiltonian systems are revisited nekhoroshev theorem concerning exponential lower bounds for the stability time effective stability and kam theorem concerning the preservation of a majority of the nonresonant invariant tori perpetual stability to stress the relationship between both theorems a common approach is given to their proof consisting of bringing the system to a normal form constructed through the lie series method the estimates obtained for the size of the remainder rely on bounds of the associated vectorfields allowing one to get the optimal stability exponent in nekhoroshev theorem for quasiconvex systems on the other hand a direct and complete proof of the isoenergetic kam theorem is obtained moreover a modification of the proof leads to the notion of nearly invariant torus which constitutes a bridge between kam and nekhoroshev theorems

kam tori are very sticky rigorous lower bounds on the time to move away from an invariant lagrangian torus with linear flow abstract hamiltonian perturbation theory is used to derive rigorous nekhoroshev type lower bounds on the time it takes to move away from an invariant lagrangian torus when the flow on the torus is conjugate to a highly non resonant linear flow in particular these results give bounds on the time to move away from a kam torus if r is the distance from the torus then the lower bound on the time to double the distance from the torus is c r e k r where c and k are constants 1 n 2 n is the number of degrees of freedom and is a measure of the irrationality of the flow we thus prove that nearby trajectories take an extremely long time to move away from such an invariant torus and that the more irrational the linear flow the stickier the torus will be these estimates do not require the steepness or quasi convexity assumptions of nekherosev s theorem