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

compactness of conformal metrics with integral bounds on curvature in this thesis we show that a sequence of conformal metrics on a compact n dimensional riemannian manifold n 4 which has an upper bound on volume and an upper bound on the lp norm of the curvature tensor for fixed p n 2 has a subsequence which converges in c if n 3 we have the same result if we assume in addition that the scalar curvature has an l2 bound as corollaries we have the compactness of a sequence of conformal metrics on a compact three manifold which are isospectral with respect to either the standard or conformal laplacian and the result of lelong ferrand that any compact manifold with non compact conformal group is conformally equivalent to the standard sphere

controllability and observability for affine nonlinear hamiltonian systems it is shown that an affine nonlinear hamiltonian system is controllable if and only if it is observable in the sense that strong accessibility implies local weak observability and vice versa furthermore it is shown that a nonminimal hamiltonian system can be reduced to a locally weakly observable and strongly accessible system in such a way that the reduced system is again hamiltonian

on the application of some of m g krein s results to the spectral analysis of sturm liouville operators abstract in one of his works m g krein discovered an analogy between polynomials orthogonal on the unit circle and generalized eigenfunctions of certain differential systems we used some ideas of this paper to obtain new results in spectral analysis of sturm liouville operators

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

a note on the theorems of m g krein and l a sakhnovich on continuous analogs of orthogonal polynomials on the circle abstract continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations introduced and studied by krein and recently generalized to matrix systems by sakhnovich we prove that the continuous analogs of the adjoint polynomials converge in the upper half plane in the case of l 2 coefficients but in general the limit can be defined only up to a constant multiple even when the coefficients are in l p for any p 2 the spectral measure is absolutely continuous and the szego kolmogorov krein condition is satisfied thus we point out that krein s and sakhnovich s papers contain an inaccuracy which does not undermine known implications from these results

on the continuous analog of rakhmanov s theorem for orthogonal polynomials abstract we obtain the continuous analogs of rakhmanov s theorem for polynomials orthogonal on the unit circle sturm liouville operators and krein systems are considered for a sturm liouville operator with bounded potential q we prove the following statement if the essential spectrum and absolutely continuous component of the spectral measure fill the whole positive half line then q decays at infinity in the certain integral sense

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

remarque sur le probleme de yamabe x g designant une variete localement conformement plate on etudie avec une approche toute nouvelle la fonction de green de l operateur laplacien conforme sur x les resultats obtenus nous permettent de montrer que les fonctions locales e r2 1 n2 prolongees par leurs constantes naturelles fournissent l inegalite 14n n 2 n2n du theoreme fondamental de aubin theoreme 1 page 280 de t aubin j funct anal 32 1979 148 174 let x g be a locally conformally flat manifold in this paper we study with a very new approach the green function of the conformal laplacian operator on x then with the results obtained we prove that the local functions e r2 1 n2 continued by their natural constants lead to the inequality 14n n 2 n2n of the fundamental theorem of aubin theorem 1 p 280 of t aubin j funct anal 32 1979 148 174

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

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

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

transformation conforme de la courbure scalaire sur une variete riemannienne compacte abstract if p is a c positive function on a compact riemannian manifold of dimension n 3 and metric g we define a conformal invariant v r and we prove that if v r is small enough r is the scalar curvature of the manifold endowed with a metric conformal to g then we are interested in the case of the sphere s n and finally in yamabe s problem

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator we consider 1 d schrodinger operators on l 2 r with slowly decaying potentials under some conditions on the potential related to the first integrals of the kdv equation we prove that the a c spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable examples show that for special classes of sparse potentials these results can not be improved

decomposable representations and lagrangian submanifolds of moduli spaces associated to surface groups the importance of explicit examples of lagrangian submanifolds of moduli spaces is revealed by papers such as dostoglou and salamon ann of math 2 139 3 581 640 1994 and salamon proceedings of the international congress of mathematicians vol 1 2 zurich 1994 pp 526 536 birkhauser basel 1995 given a 3 manifold m with boundary m dostoglou and salamon use such examples to obtain a proof of the atiyah floer conjecture relating the symplectic floer homology of the representation space hom 1 m u u associated to an explicit pair of lagrangian submanifolds of this representation space and the instanton homology of the 3 manifold m in the present paper we construct a lagrangian submanifold of the space of representations mathcal m g l hom mathcal c pi g l u u of the fundamental group g lof a punctured riemann surface g l into an arbitrary compact connected lie group u this lagrangian submanifold is obtained as the fixed point set of an anti symplectic involution hat beta defined on mathcal m g l we show that the involution hat beta is induced by a form reversing involution defined on the quasi hamiltonian space u times u g times mathcal c 1 times cdots times mathcal c l the fact that hat beta has a non empty fixed point set is a consequence of the real convexity theorem for group valued momentum maps proved in schaffhauser a real convexity theorem for quasi hamiltonian actions submitted 25 p 2007 http arxiv org abs math 0705 0858 the notion of decomposable representation provides a geometric interpretation of the lagrangian submanifold thus obtained

when is a symplectic quotient an orbifold let k be a compact lie group of positive dimension we show that for most unitary k modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold the quotient of a unitary module of a finite group when k is connected we show that even a symplectomorphism to a linear symplectic orbifold does not exist our results yield conditions that preclude the symplectic quotient of a hamiltonian k manifold from being locally isomorphic to an orbifold as an application we determine which unitary su2 modules yield symplectic quotients that are z graded regularly symplectomorphic to a linear symplectic orbifold we similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold

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 the existence of the absolutely continuous component for the measure associated with some orthogonal systems in this article we consider two orthogonal systems sturm liouville operators and krein systems for krein systems we study the behavior of generalized polynomials at the infinity for spectral parameters in the upper half plain that makes it possible to establish the presence of absolutely continuous component of the associated measure for sturm liouville operator on the half line with bounded potential q we prove that essential support of absolutely continuous component of the spectral measure is m if and q l 2 r that holds for all boundary conditions at zero this result partially solves one open problem stated recently by s molchanov m novitskii and b vainberg we consider also some other classes of potentials

nonlinear observability and duality developing the method introduced by hermann and krener 3 we construct quasi minimal realizations for complete systems solving the eventual non regularity in the c case the results obtained are dual of those known for the accessibility problem

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

reduction of hamiltonian systems with symmetry abstract this paper presents an alternative phase space reduction process for hamiltonian systems with symmetry using a lie algebra of infinitesimal symmetries instead of a lie group of symmetries this approach avoids the use of the various conditions on the action of the group or of the isotropy subgroup necessary for the usual construction of a reduced phase space quotient with a globally defined symplectic manifold structure the existence of such a reduced phase space is proved without recourse to the existence of an equivariant moment map instead it is assumed that the orbits of the action associated to the lie algebra can be described as the level sets of some analytic map an assumption that holds in many of the known examples the relation between the two reduction processes is discussed in particular some cases where the marsden weinstein process leads to a quotient space which is not a smooth manifold are shown to be amenable to the method now proposed the natural control theoretic interpretation of the techniques used is also discussed

about the mass of certain second order elliptic operators abstract let m g be a closed riemannian manifold of dimension n 3 and let f c m such that the operator p f g f is positive if g is flat near some point p and f vanishes around p we can define the mass of p f as the constant term in the expansion of the green function of p f at p in this paper we establish many results on the mass of such operators in particular if f n 2 4 n 1 s g i e if p f is the yamabe operator we show the following result assume that there exists a closed simply connected non spin manifold m such that the mass is non negative for every metric g as above on m then the mass is non negative for every such metric on every closed manifold of the same dimension as m

solutions nodales sur les varietes riemanniennes resume nodal solutions of nonlinear pde with critical growth are obtained on com pact manifolds the variational method and bifurcations theory are used to con struct a branch of nodal solutions consequences for other pde are suggested

a multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

demonstration de la conjecture de la masse positive abstract on a compact riemannian manifold v n g n 2 when the conformal laplacian l is invertible we show under necessary hypotheses that if the green function g l of l is of the form here r d p q g l p q 1 n 2 n 1 r n 2 h p q with h p q bounded on v then lim r 0 r 1 n b p r h p q d q 0 applications to the positive mass conjecture and to the c 0 compactness of the set of the solutions of the yamabe equation

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

singular angular momentum mappings we algebraically reduce the system consisting of a nonrelativistic particle moving in r with vanishing angular momentum after analyzing the conical structure of the constraint set 1 0 we use algebraic geometric techniques to explicitly construct the reduced poisson algebra of rotationally invariant observables this procedure enables us to completely identify the effects of the singularity in j 0 on the system we then group theoretically reduce the system and compare our results with those obtained algebraically

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

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