on simultaneous linearization of diffeomorphisms of the sphere let r1 r2 rm be rotations generating sod 1 d 2 and f1 f2 fm be their small smooth perturbations we show that f can be simultaneously linearized if and only if the associated random walk has zero lyapunov exponents as a consequence we obtain stable ergodicity of actions of random rotations in even dimensions 1 main results let f1 f2 fm be diffeomorphisms of s d 2 let i i n be a sequence of independent random variables uniformly distributed on 1 m consider the markov process on s 1 xn f nxn 1 if is an invariant measure for this process let 1 2 d be the lyapunov exponents of denote

the inverse problem of the calculus of variations the use of geometrical calculus in douglas s analysis the main objective of this paper is to work out a full scale application of the integrability analysis of the inverse problem of the calculus of variations as developed in recent papers by sarlet and crampin for this purpose the celebrated work of douglas on systems with two degrees of freedom is taken as the reference model it is shown that the coordinate free geometrical calculus used in sarlet and crampin s general theoretical developments provides effective tools also to do the practical calculations the result is not only that all subcases distinguished by douglas can be given a more intrinsic characterization but also that in most of the cases the calculations can be carried out in a more efficient way and often lead to sharper conclusions

holomorphic differentials thermostats and anosov flows we introduce a new family of thermostat flows on the unit tangent bundle of an oriented riemannian two manifold suitably reparametrised these flows include the geodesic flow of metrics of negative gauss curvature and the geodesic flow induced by the hilbert metric on the quotient surface of divisible convex sets we show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties in particular we prove that they admit a dominated splitting and we identify special cases in which the flows are anosov in the latter case we study when they admit an invariant measure in the lebesgue class and the regularity of the weak foliations

on the period spectrum of a symplectic mapping let m be a compact riemannian manifold it has been known for a long time that the singularities of the wave trace trace cos t are located at the periods of the closed geodesics do these singularities also contain information about the geometry of m in the neighborhood of a closed geodesic we prove that the birkhoff canonical form of the poincare map can be determined from the singularities of the wave trace

instability of high dimensional hamiltonian systems multiple resonances do not impede diffusion abstract we consider models given by hamiltonians of the form h i p q t e h i j 1 n 1 2 p j 2 v j q j e q i p q t e where i i r d t d p q r n t t 1 these are higher dimensional analogues both in the center and hyperbolic directions of the models studied in 28 29 43 and are usually called a priori unstable hamiltonian systems all these models present the large gap problem we show that for 0 e 1 under regularity and explicit non degeneracy conditions on the model there are orbits whose action variables i perform rather arbitrary excursions in a domain of size o 1 this domain includes resonance lines and hence large gaps among d dimensional kam tori this phenomenon is known as arnold diffusion the method of proof follows closely the strategy of 28 29 the main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions i i r d we show that since these multiple resonances happen in sets of codimension greater than one in the space of actions i they can be contoured this corresponds to the mechanism called diffusion across resonances in the physics literature the present paper however differs substantially from 28 29 on the technical details of the proofs we have taken advantage of the theory of the scattering map developed in 31 notably the symplectic properties which were not available when the above papers were written we have analyzed the conditions imposed on the resonances in more detail more precisely we have found that there is a simple condition on the melnikov potential which allows us to conclude that the resonances are crossed in particular this condition does not depend on the resonances so that the results are new even when applied to the models in 28 29

dynamical spectral rigidity among mathbb z _2 symmetric strictly convex domains close to a circle appendix b coauthored with h hezari

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

generalized symmetry integrability test for discrete equations on the square lattice we present an integrability test for discrete equations on the square lattice which is based on the existence of a generalized symmetry we apply this test to a number of equations obtained in different recent papers as a result we prove the integrability of seven equations which differ essentially from the qv equation introduced by viallet and thus from the adler bobenko suris list of equations contained therein

classical scattering theory in one dimension we discuss various scattering properties of the classical system ofn repelling particles on the real line in the integrable case i e if the asymptotic velocities are preserved under the scattering map the asymptotic phases behave as if the particles collided pairwise

smooth conjugacy and s r b measures for uniformly and non uniformly hyperbolic systems we give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional anosov systems we also show that if a homeomorphism conjugating two smooth low dimensional anosov systems is absolutely continuous then it is as smooth as the maps we furthermore prove generalizations of these facts for non uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions

asymptotics of trajectories for cone potentials abstract we study the asymptotics of trajectories of the classical hamiltonian dynamics for hamiltonians with cone potentials we have shown earlier that all trajectories are asymptotically free 5 i e the asymptotic velocities exist here we show that the generic trajectories are asymptotically uniform i e the asymptotic phases exist

method for searching higher symmetries for quad graph equations a generalized symmetry integrability test for discrete equations on the square lattice is studied integrability conditions are discussed a method for searching higher symmetries including non autonomous ones for quad graph equations is suggested based on characteristic vector fields

on the regional gradient observability of time fractional diffusion processes this paper for the first time addresses the concepts of regional gradient observability for the riemann liouville time fractional order diffusion system in an interested subregion of the whole domain without the knowledge of the initial vector and its gradient the riemann liouville time fractional order diffusion system which replaces the first order time derivative of normal diffusion system by a riemann liouville time fractional order derivative of order 0 1 is used to well characterize those anomalous sub diffusion processes the characterizations of the strategic sensors when the system under consideration is regional gradient observability are explored we then describe an approach leading to the reconstruction of the initial gradient in the considered subregion with zero residual gradient vector at last to illustrate the effectiveness of our results we present several application examples where the sensors are zone pointwise or filament ones

on state constraint representations and mesh dependent gradient projection convergence rates for optimal control problems two distinct nonlinear programming formulations are investigated for ode optimal control problems with pointwise state and control constraints the first formulation treats the differential equations of state as an equality constraint in the conventional manner the second formulation employs a different equality constraint entailing the integrated state transition map related convergence rate estimates are developed for augmented gradient projection methods and discrete time approximations to a large representative class of ode control problems in the first formulation the rate estimates are mesh dependent and the predicted number of inner loop gradient projection iterations needed to achieve a fixed small deviation from the optimal value of the augmented lagrangian is inversely proportional to the square of the mesh width in the second formulation the convergence rates and predicted iteration counts are mesh invariant the computational costs per iteration in the two formulations are comparable these estimates elucidate previously published numerical experiments with augmented gradient projection agp methods and constrained regulator problems

the helmholtz conditions revisited a new approach to the inverse problem of lagrangian dynamics deals with the general problem of finding a multiplier matrix that can give to a prescribed system of second order ordinary equations the structure of euler lagrange equations the approach is based on a generalisation of previous studies on linear systems the main result concerns a set of necessary and sufficient conditions for the existence of a multiplier which contains an infinite set of algebraic equations the coefficients of which can be used to derive necessary conditions involving only the given right hand sides of the differential equations an outline is given of interesting points for future studies and an example is presented for which all multipliers are explicitly constructed

generalized symmetry classification of discrete equations of a class depending on twelve parameters we carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square which belong to a twelve parametric class the direct result of this classification is a list of equations containing no new examples however as an indirect result of this work we find a number of integrable examples pretending to be new one of them has a nonstandard symmetry structure the others are analogues of the liouville equation in the sense that they are darboux integrable we also enumerate all equations of the class which are linearizable via a two point first integral and specify the nature of integrability of some known equations

the generalized symmetry method for discrete equations the generalized symmetry method is applied to a class of completely discrete equations including the adler bobenko suris list assuming the existence of a generalized symmetry we derive a few integrability conditions suitable for testing and classifying the equations of this class those conditions are used at the end to test for integrability discretizations of some well known hyperbolic equations

differentiability and analyticity of topological entropy for anosov and geodesic flows in this paper we investigate the regularity of the topological entropyh top forc k perturbations of anosov flows we show that the topological entropy varies almost as smoothly as the perturbation the results in this paper along with several related results have been announced in kkpw

nonhomogeneous locally free actions of the affine group we classify locally free smooth actions of the real affine group on closed three dimensional manifolds up to smooth conjugacy in particular we show that there are non homogeneous actions when the manifold is not a rational homology sphere and its fundamental group is not solvable we say two right actions 1 and 2 of a lie group g on manifolds m1 and m2 are c conjugate if there exist an isomorphism of g and a c diffeomorphism h from m1 to m2 such that h 1 p g 2 h p g for any g 2 g and p 2 m1 the map h is called a c conjugacy between 1 and 2 we also say 1 and 2 are c orbit equivalent if there exists a c

d alembert lagrange analytical dynamics for nonholonomic systems the d alembert lagrange principle dlp is designed primarily for dynamical systems under ideal geometric constraints although it can also cover linear velocity constraints its application to nonlinear kinematic constraints has so far remained elusive mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint on recognition that the commutation rule traditionally accepted for velocity displacements in lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements which can be adjoined via lagrange s multipliers to the d alembert lagrange equation to yield the equations of state and also new transpositional relations for nonholonomic systems the equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived in appendix a from the quite different gauss principle the present advance therefore solves a long outstanding problem on the application of dlp to ideal nonholonomic systems and as an aside provides validity to axioms as the chetaev rule previously left theoretically unjustified a more general transpositional form of the boltzmann hamel equation is also obtained

from nonlinear to hamiltonian via feedback mechanical control systems are an especially interesting and important class of nonlinear control systems they possess a rich mathematical structure and yet physical considerations are extremely important for the solution of a large class of control problems we broaden the applicability of design methodologies developed for mechanical control systems by rendering nonlinear control systems mechanical by a proper choice of feedback in particular we characterize which control systems can be transformed to hamiltonian control systems by a feedback transformation

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

on the inverse problem of calculus of variations for fourth order equations fels conditions fels m e 1996 trans amer math soc 348 5007 5029 doi 10 1090 s0002 9947 96 01720 5 ensure the existence and uniqueness of the lagrangian in the case of a fourth order equation we show that when fels conditions are satisfied the lagrangian can be derived from the jacobi last multiplier as in the case of a second order equation indeed we prove that if a lagrangian exists for an equation of any even order then it can be derived from the jacobi last multiplier two equations from a number theory paper by hall hall r r 2002 j number theory 93 235 245 doi 10 1006 jnth 2001 2719 one of the second and one of the fourth order will be used to exemplify the method the known link between jacobi last multiplier and lie symmetries is also exploited finally the lagrangians of two fourth order equations drawn from physics are determined with the same method

an eds approach to the inverse problem in the calculus of variations the inverse problem in the calculus of variations for a given set of second order ordinary differential equations consists of deciding whether their solutions are those of euler lagrange equations and whether the lagrangian if it exists is unique this paper discusses the exterior differential systems approach to this problem in particular it proposes an algorithmic procedure towards the construction of a certain differential ideal the emphasis is not so much on obtaining a complete set of integrability conditions for the problem but rather on producing a minimal set to expedite the differential ideal process

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

global and asymptotic convergence rate estimates for a class of projected gradient processes projected gradient processes of the goldstein levitin polyak type are considered for constrained minimization problems min omega f with omega a convex set in a hilbert space x and f x to mathbb r 1 a differentiable functional global and local convergence theorems are established for a large class of these processes including those generated with implicit step length rules proposed by bertsekas and goldstein in this analysis traditional uniform strong positivity conditions on the hessian nabla 2 f are replaced by weaker pseudoconvexity conditions and growth conditions on f when f has a unique minimizes in omega convergence rates are shown to depend on how rapidly the function gamma sigma inf r f x f xi mid x in omega x xi geqq sigma grows with increasing sigma 0 if gamma sigma geqq b sigma nu for some b 0 the processes f n in question converge to inf f like o n nu nu 2 linear

runge kutta methods in optimal control and the transformed adjoint system the convergence rate is determined for runge kutta discretizations of nonlinear control problems the analysis utilizes a connection between the kuhn tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle this connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function

subriemannian metric induced by single input locally accessible control affine systems abstract a subriemannian metric defined by a distribution and an inner product on the distribution on the state space of single input locally accessible control systems is constructed based on the vector fields of the control system and their higher order derivatives conditions for the existence of such a metric are derived based on a feedback invariant quantity and a normal form for this class of control systems is provided the construction is illustrated in a simple car control problem

kam tori and whiskered invariant tori for non autonomous systems abstract we consider non autonomous dynamical systems which converge to autonomous or periodic systems exponentially fast in time such systems appear naturally as models of many physical processes affected by external pulses we introduce definitions of non autonomous invariant tori and non autonomous whiskered tori and their invariant manifolds and we prove their persistence under small perturbations smooth dependence on parameters and several geometric properties if the systems are hamiltonian the tori are lagrangian manifolds we note that such definitions are problematic for general time dependent systems but we show that they are unambiguous for systems converging exponentially fast to autonomous the proof of persistence relies only on a standard implicit function theorem in banach spaces and it does not require that the rotations in the tori are diophantine nor that the systems we consider preserve any geometric structure we only require that the autonomous system preserves these objects in particular when the autonomous system is integrable we obtain the persistence of tori with rational rotational we also discuss fast and efficient algorithms for their computation the method also applies to infinite dimensional systems which define a good evolution e g pde s when the systems considered are hamiltonian we show that the time dependent invariant tori are isotropic hence the invariant tori of maximal dimension are lagrangian manifolds we also obtain that the un stable manifolds of whiskered tori are lagrangian manifolds we also include a comparison with the more global theory developed in blazevski and de la llave 2011

optimal control of the temperature in a catalytic converter abstract this paper is concerned with the optimization of the dynamics of the gas flow in an exhaust pipe as a prototypical question the start up heating of the catalytic converter is considered reaching the optimal temperature in the converter rapidly competes against the costs i e releasing too much unburnt fuel in the exhaust gases the underlying model is a one dimensional small mach number asymptotic gas dynamic model for a mixture of burnt and unburnt gases formulated on a network consisting of the various pieces of the exhaust tube for the optimization adjoint calculus is applied on the continuous level the resulting system is then discretized finally numerical examples show the validity of this approach

the inverse problem of the calculus of variations for scalar fourth order ordinary differential equations a simple invariant characterization of the scalar fourth order ordinary differential equations which admit a variational multiplier is given the necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth order equation through the application of cartan s equivalence method the solution to the inverse problem for fourth order scalar equations provides the solution to an equivalence problem for secondorder lagrangians as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated lagrangian

statistical regularities of self intersection counts for geodesics on negatively curved surfaces let s be a compact surface with constant negative curvature 1 from among all closed geodesics on of length t choose one at random and letnt be the number of its self intersections we prove that for a certain constant 0 the random variable nt t t has a limit distribution as t we conjecture that for surfaces of variable negative curvature the order of magnitude of typical variations is t rather than t we also prove analogous results for generic geodesics that is geodesics whose initial tangent vectors are chosen randomly according to normalized liouville measure date november 10 2011 1991 mathematics subject classification primary 57m05 secondary 53c22 37d40

livsic theorem for matrix cocycles we prove the liv sic theorem for arbitrary gl m r cocycles we con sider a hyperbolic dynamical system f x x and a holder continuous function a x gl m r we show that if a has trivial periodic data i e a f n 1 p a fp a p id for each periodic point p f n p then there exists a holder continuous function c x gl m r satisfying a x c fx c x 1 for all x2 x the main new ingredients in the proof are results of independent interest on relations between the periodic data lyapunov exponents and uniform estimates on growth of products along orbits for an arbitrary holder function a

jacobi s last multiplier and lagrangians for multidimensional systems we demonstrate that the formalism for the calculation of jacobi s last multiplier for a one degree of freedom system extends naturally to systems of more than one degree of freedom thereby extending results of whittaker dating from more than a century ago and rao proceedings of the benares mathematical society 2 53 1940 dating from almost 70 years ago we illustrate the theory with an application taken from the theory of coupled oscillators we indicate how many lagrangians can be obtained for such a system

time dependent kinetic energy metrics for lagrangians of electromagnetic type we extend the results obtained in sarlet and prince 2010 j phys a math theor 43 445204 about a class of lagrangian systems which admit alternative kinetic energy metrics to second order mechanical systems with explicit time dependence the main results are that a time dependent alternative metric will have constant eigenvalues and will give rise to a time dependent coordinate transformation which partially decouples the system

singular value analysis of nonlinear symmetric systems in this paper we introduce the notions of state space symmetry for nonlinear systems and of the cross operator as a nontrivial natural extension of the linear symmetric case in terms of the controllability and observability operators associated to it we give a characterization of a symmetric nonlinear system in terms of the cross operator and a coordinate transformation then we analyze the use of the cross operator for solving the hankel singular value problem of the system the result is a new and simpler characterization of the solutions of this problem in terms of the cross operator and a metric

extended hamiltonian learning on riemannian manifolds theoretical aspects this paper introduces a general theory of extended hamiltonian second order learning on riemannian manifolds as an instance of learning by constrained criterion optimization the dynamical learning equations are derived within the general framework of extended hamiltonian stationary action principle and are expressed in a coordinate free fashion a theoretical analysis is carried out in order to compare the features of the dynamical learning theory with the features exhibited by the gradient based ones in particular gradient based learning is shown to be an instance of dynamical learning and the classical gradient based learning modified by a momentum term is shown to resemble discrete time dynamical learning moreover the convergence features of gradient based and dynamical learning are compared on a theoretical basis this paper discusses cases of learning by dynamical systems on manifolds of interest in the scientific literature namely the stiefel manifold the special orthogonal group the grassmann manifold the group of symmetric positive definite matrices the generalized flag manifold and the real symplectic group of matrices

characterization of gradient control systems given a general nonlinear affine control system with outputs and a torsion free affine connection defined on its state space we investigate the gradient realization problem we give necessary and sufficient conditions under which the control system can be written as a gradient control system corresponding to some pseudo riemannian metric whose levi civita connection is equal to the given affine connection the results rely on a suitable notion of compatibility of the system with respect to the given affine connection and on the output behavior of the prolonged system and the gradient extension the symmetric product associated with an affine connection plays a key role throughout the discussion

on nonholonomic systems and variational principles we consider the compatibility of the equations of motion which follow from d alembert s principle in the case of a general autonomous nonholonomic mechanical system in n dimensions with those equations which follow for the same system by assuming the validity of a specific variational action principle in which the nonholonomic conditions are implemented by means of the multiplication rule in the calculus of variations the equations of motion which follow from the principle of d alembert are not identical to the equations which follow from the variational action principle we give a proof that the solutions to the equations of motion which follow from d alembert s principle do not in general satisfy the equations which follow from the action principle with nonholonomic constraints thus the principle of d alembert and the minimal action principle involving the multiplication rule are not compatible in the case of systems with nonholonomic constraints for simplicity the proof is given for autonomous syst

invariants for smooth conjugacy of hyperbolic dynamical systems ii we show that the eigenvalues of the derivatives at periodic points form a complete set of invariants for smooth local conjugacy of anosov diffeomorphisms oft2

the jacobi last multiplier for linear partial difference equations we present a discretization of the jacobi last multiplier with some applications to the computation of solutions of linear partial difference equations

the fried conjecture in small dimensions we study the twisted ruelle zeta function x s for smooth anosov vector fields x acting on flat vector bundles over smooth compact manifolds in dimension 3 we prove fried conjecture relating reidemeister torsion and x 0 in higher dimensions we show more generally that x 0 is locally constant with respect to the vector field x under a spectral condition as a consequence we also show fried conjecture for anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3 manifolds this gives the first examples of non analytic anosov flows and geodesic flows in variable negative curvature where fried conjecture holds true

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

integrable hamiltonians with exponential potential abstract we construct a large class of integrable hamiltonian systems with n degrees of freedom this class naturally extends the nonperiodic hamiltonians of toda lattice type

time dependent scattering theory for odes and applications to reaction dynamics we develop a time dependent scattering theory for general vector fields in euclidean space we give conditions that ensure that the wave maps exist are smooth invertible and depend smoothly on parameters we then discuss the intertwining relations and how they can be used to compute stable unstable manifolds for time dependent normally hyperbolic invariant manifolds the theory is particularly effective for hamiltonian mechanics we also give perturbative calculations of the scattering map that are analogous to fermi s golden rule in quantum mechanics we apply this theory to a problem in transition state theory concerning the exposure of molecules to a laser pulse for a short time we present a method to compute invariant manifolds for the laser driven henon heilies system and give perturbative calculations of the change in the branching ratio

a sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms we present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term the sensitivity analysis is based on shift variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations based on a first order approximation by shift variations in l1 we introduce the concept of shift differentiability which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking type functionals in the main part of the paper we show that entropy solutions are generically shift differentiable at almost all times t 0 with respect to the control hereby we admit shift variations for the initial data which allows us to use the shift differentiability result repeatedly over time slabs this is useful for the design of optimization methods with time domain decomposition our analysis especially of the shock sensitivity combines structural results by using generalized characteristics and an adjoint argument our adjoint based shock sensitivity analysis does not require us to restrict the richness of the shock structure a priori and admits shock generation points the analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one sided lipschitz condition as a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking type functionals

geometric properties of the scattering map of a normally hyperbolic invariant manifold given a normally hyperbolic invariant manifold for a map f whose sta ble and unstable invariant manifolds intersect transversally we consider its associated scattering map that is the map that given an asymptotic orbit in the past gives the asymptotic orbit in the future we show that when f and are symplectic resp exact symplectic then the scattering map is symplectic resp exact symplectic furthermore we show that in the exact symplectic case there are extremely easy formulas for the primitive function which have a variational interpretation as dierence of actions we use this geometric information to obtain ecien t perturbative calculations of the scattering map using deformation theory this perturbation theory generalizes and extends several results already obtained using the melnikov method analogous results are true for hamiltonian o ws the proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems hence the results can be used to obtain intersection properties of objects of any type we also reexamine the calculation of the scattering map in a geodesic o w perturbed by a quasi periodic potential we show that the geometric theory reproduces the results obtained in dls06b using methods of fast slow systems moreover the geometric theory allows to compute perturbatively the dependence on the slow variables which does not seem to be accessible to the previous methods

on invariant volumes of codimension one anosov flows and the verjovsky conjecture we show that any topologically transitive codimension one anosov flow on a closed manifold is topologically equivalent to a smooth anosov flow that preserves a smooth volume by a classical theorem due to verjovsky any higher dimensional codimension one anosov flow is topologically transitive recently simic showed that any higher dimensional codimension one anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an anosov diffeomorphism therefore our result gives a complete classification of codimension one anosov flows up to topological equivalence in higher dimensions

a geometrical version of the helmholtz conditions in time dependent lagrangian dynamics appropriate geometrical machinery for the study of time dependent lagrangian dynamics is developed it is applied to the inverse problem of the calculus of variations and a set of necessary and sufficient conditions for the existence of a lagrangian are given in terms of the existence of a 2 form with suitable properties which are exactly equivalent to the helmholtz conditions

analytic smoothing of geometric maps with applications to kam theory we show that finitely differentiable diffeomorphisms which are either symplectic volume preserving or contact can be approximated with analytic diffeomorphisms that are respectively symplectic volume preserving or contact we prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence in cr norms of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function as an application to this result we give a proof of the existence the local uniqueness and the bootstrap of regularity of kam tori for finitely differentiable symplectic maps the symplectic maps considered here are not assumed either to be written in action angle variables or to be perturbations of integrable systems our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non degeneracy condition and that is approximately invariant the symplectic volume preserving and contact forms are assumed to be analytic

on deformation boundary rigidity and spectral rigidity of riemannian surfaces with no focal points although the main results of the paper are obtained in the two dimensional case the problems under consideration can be posed in an arbitrary dimension let m g be a compact riemannian manifold with boundary the boundary rigidity problem consists of determining the riemannian metric g on m up to isometries which are the identity on the boundary by measuring the geodesic distance dg between boundary points it is easy to see that the answer to this question is no in general we can find metrics g such that there exists x0 m satisfying dist x0 m supx y mdg x y then we can change the metric g in a neighborhood of x0 without changing the boundary distance function a compact riemannian manifold m g with boundary is called boundary rigid if the metric g is uniquely determined by dg up to isometries which are the identity on the boundary there are a few classes of metrics which are known to be boundary rigid see 13 for references in this paper we consider the linearized problem let g be a one parameter family of riemannian metrics on m such that dg dg0 e e for a geodesic a b m of the metric g with a b m it is easy to see that

the discrete first second and thirty fourth painleve hierarchies the discrete first and second painleve equations dp and dp are integrable difference equations which have classical first second or third painleve equations p p or p as continuum limits dp and dp are believed to be integrable because they are discrete isomonodromy conditions for associated single valued linear problems an infinite hierarchy of integrable difference equations that share the same linear deformation problem as dp was shown to exist by cresswell and joshi in this paper we recall the results shown for dp and show how to deduce a hierarchy for dp each member of the respective hierarchies is shown to be generated by difference recursion operators furthermore we show that continuum limits of these difference hierarchies lead to the p p and p hierarchies finally we construct miura transformations of the dp hierarchy and show that these lead to the hierarchy of the discrete thirty fourth painleve equation

adaptive multilevel inexact sqp methods for pde constrained optimization we present a class of inexact adaptive multilevel trust region sqp methods for the efficient solution of optimization problems governed by nonlinear pdes the algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process 1 implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2 implementable accuracy requirements for iterative solvers of the linearized pde and adjoint pde on the current grid we prove global convergence to a stationary point of the infinite dimensional problem moreover we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equations numerical results are presented

realization theory for hamiltonian systems we prove that control systems of the hamiltonian type have a quasi minimal realization on a state space of minimal dimension of one of two types a hamiltonian system or the suspension of a time dependent hamiltonian system these realizations are unique up to a symplectomorphism in the first case or a canonical transformation in the second one when the lie algebra of the initial system sigma assumed to be analytic is finite dimensional we obtain a characterization of the state space and the dynamics of the quasi minimal realization in terms of the coadjoint actions of lie groups associated with sigma an example is given applying these results using control theoretic methods we prove the kostant kirillov souriau theorem the technique used for systems with finite dimensional lie algebra is closely related to the theory developed by those authors

absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian author s damanik d gorodetski a solomyak b abstract 2015 we prove for the square fibonacci hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants this is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact dimensional measures

configuration controllability of simple mechanical control systems in this paper we present a definition of configuration controllability for mechanical systems whose lagrangian is kinetic energy with respect to a riemannian metric minus potential energy a computable test for this new version of controllability is derived this condition involves an object which we call the symmetric product of particular interest is a definition of equilibrium controllability for which we are able to derive computable sufficient conditions examples illustrate the theory

adjoint and hamiltonian input output differential equations based on developments in the theory of variational and hamiltonian control systems by crouch and van der schaft 1987 this paper answers two questions given an input output differential equation description of a nonlinear system what is the adjoint variational system in input output differential form and what are the conditions for the system to be hamiltonian i e such that the variational and the adjoint variational systems coincide this resulting set of conditions is then used to generalize classical conditions such as the well known helmholtz conditions for the inverse problem in classical mechanics

invariant distributions beurling transforms and tensor tomography in higher dimensions in the recent articles paternain et al j differ geom 98 147 181 2014 invent math 193 229 247 2013 a number of tensor tomography results were proved on two dimensional manifolds the purpose of this paper is to extend some of these methods to manifolds of any dimension a central concept is the surjectivity of the adjoint of the geodesic ray transform or equivalently the existence of certain distributions that are invariant under the geodesic flow we prove that on any anosov manifold one can find invariant distributions with controlled first fourier coefficients the proof is based on subelliptic type estimates and a pestov identity we present an alternative construction valid on manifolds with nonpositive curvature based on the fact that a natural beurling transform on such manifolds turns out to be essentially a contraction finally we obtain uniqueness results in tensor tomography both on simple and anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified jacobi equation