the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

the hamiltonian structure of general relativistic perfect fluids we show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift are hamiltonian relative to a certain poisson structure for the fluid variables a lie poisson structure associated to the dual of a semi direct product lie algebra is used while the bracket for the gravitational variables has the usual canonical symplectic structure the evolution is governed by a hamiltonian which is equivalent to that obtained from a canonical analysis the relationship of our hamiltonian structure with other approaches in the literature such as clebsch potentials lagrangian to eulerian transformations and its use in clarifying linearization stability are discussed

high performance numerical algorithms and software for structured total least squares we present a software package for structured total least squares approximation problems the allowed structures in the data matrix are block toeplitz block hankel unstructured and exact combination of blocks with these structures can be specified the computational complexity of the algorithms is o m where m is the sample size we show simulation examples with different approximation problems application of the method for multivariable system identification is illustrated on examples from the database for identification of systems daisy

removal of boundary singularities of pseudo holomorphic curves with lagrangian boundary conditions we give a proof of the theorem of removing isolated singularities of pseudo holomorphic curves with lagrangian boundary conditions and bounded symplectic area the proof is a combination of some lp type estimates standard techniques of geometric p d e and some ideas from symplectic geometry and calibration theory

richardson number criterion for the nonlinear stability of three dimensional stratified flow with use of a method of arnol d we derive the necessary and sufficient conditions for the formal stability of a parallel shear flow in a three dimensional stratified fluid when the local richardson number defined with respect to density variations is everywhere greater than unity the equilibrium is formally stable under nonlinear pertrubations the essential physical content of the nonlinear stability result is that the total energy acts as a potential well for deformations of the fluid across constant density surfaces this well is required to have definite curvature to assure stability under these deformations

structured total least squares and l2 approximation problems abstract it is shown how structured and weighted total least squares and l 2 approximation problems lead to a nonlinear generalized singular value decomposition an inverse iteration scheme to find a local minimum is proposed the emphasis of the paper is not on the convergence analysis of the algorithm rather the purpose is to illustrate its use in numerous applications in systems and control including total least squares with relative errors and or fixed elements inverse singular value problems an errors in variables variant of the kalman filter impulse response realization from noisy data h 2 model reduction h 2 system identification and calculating the largest stability radius of uncertain linear systems several numerical examples are given

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

a symplectic fixed point theorem for cpn two symplectic diffeomorphisms 0 1 of a symplectic manifold x are said to be homologous if there exists a smooth homotopy 1 t 0 1 of symplectic diffeomorphisms between them such that the timedependent vector field t defined byd dt t t o t is a globally hamiltonian vector field for allt i e there exists a smooth real valued timedependent hamiltonian functionh x t onx x 0 1 such that t dh t whereht h x t

symplectic connections and the linearisation of hamiltonian systems this paper uses symplectic connections to give a hamiltonian structure to the first variation equation for a hamiltonian system along a given dynamic solution this structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the hamiltonian plus casimir as energy this structure is different from the well known and elementary tangent space construction our results are applied to systems with symmetry and to lie poisson systems in particular

cubic forms in 14 variables the result can be rephrased in geometric language to say that any projective cubic hypersurface defined over q of dimension at least 12 has a q point davenport s result was extended to arbitrary number fields by pleasants 9 and it would be interesting to know whether theorem 1 could similarly be extended these results can be seen as an attempt to extend the classical theorem of meyer 1884 from quadratic forms to cubic forms meyer showed that any indefinite quadratic from over z in 5 or more variables must represent zero indeed meyer s result was generalized by minkowski who showed that a quadratic form over z in any number of variables represents zero if and only if it represents zero over every completion of q it is a well known fact that this local condition is automatically satisfied for qp as soon as n 5 thus meyer s result requires only the condition of indefiniteness the analogous fact for cubic forms is that p adic zeros exist whenever n 10 see davenport 6 chapter 18 for example of course the condition for r holds for any n in this case thus it is natural to conjecture that theorem 1 should hold as soon as n 10 indeed for smaller values of n one might expect that it should suffice for the form c x to represent zero in each field qp however when n 3 or 4 it is possible for a cubic form to have zeros in every completion of q without there being a global zero this is shown by the examples

de rham hodge kodaira operator on loop groups we consider a based loop group le g over a compact lie groupg endowed with its pinned wiener measure the law of the brownian bridge ong and we shall calculate the ricci curvature for differentialn forms over le g a type of bochner weitzenbock formula for general differentialn forms or shigekawa identity will be established

proper fredholm submanifolds of hilbert space on considere des varietes et des sous varietes de hilbert riemanniennes on demontre qu une sous variete de fredholm propre a une structure de fredholm naturelle et on montre que les operateurs de forme sont compacts et que chaque fonction distance euclidienne fa satisfait la condition c de palais et smale on etudie la geometrie des sous varietes de fredholm propre a fibre normal plat

hamiltonian pseudo rotations of projective spaces the main theme of the paper is the dynamics of hamiltonian diffeomorphisms of mathbb c mathbb p n with the minimal possible number of periodic points equal to n 1 by arnold s conjecture called here hamiltonian pseudo rotations we prove several results on the dynamics of pseudo rotations going beyond periodic orbits using floer theoretical methods one of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points partially extending a theorem of le calvez and yoccoz and of franks and misiurewicz to higher dimensions the other is a strong variant of the lagrangian poincare recurrence conjecture for pseudo rotations we also prove the c 0 rigidity of pseudo rotations with exponentially liouville mean index vector this is a higher dimensional counterpart of a theorem of bramham establishing such rigidity for pseudo rotations of the disk

spatial variational principles in continuum mechanics in order to obtain balance equations for non simple continua particularly in cases where the suggestive power of newtonian dynamics fails it is customary to resort to variational techniques and principles the list of pertinent quotations could be long starting with the book by the brothers cosserat see section 98 of 1 for a review and for an outline of the work of toupin

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

quadratic forms and systems of forms in many variables let f 1 ldots f r be quadratic forms with integer coefficients in n variables when n ge 9r and the variety v f 1 ldots f r is a smooth complete intersection we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish which in particular implies the hasse principle for v f 1 ldots f r previous work in this direction required n to grow at least quadratically with r we give a similar result for r forms of degree d conditional on an upper bound for the number of solutions to an auxiliary inequality in principle this result may apply as soon as n d2 dr in the case that d ge 3 several strategies are available to prove the necessary upper bound for the auxiliary inequality in a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients

a quantum cup length estimate for symplectic fixed points a new lower bound for the number of fixed points of hamiltonian automorphisms of closed symplectic manifolds m is established the new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds we prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup length estimate holds true if the hofer energy of the hamiltonian automorphism is sufficiently small for arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup length based on the quantum cohomology ring of m providing a quantum cup length estimate

on the equivalence of constrained total least squares and structured total least squares several extensions of the total least squares tls method that are able to calculate a structured rank deficient approximation of a data matrix have been developed the main result of this article is the demonstration of the equivalence of two of these approaches namely the constrained total least squares ctls approach and the structured total least squares stls approach we also present a numerical comparison of both methods

degenerate relative equilibria curvature of the momentum map and homoclinic bifurcation abstract a fundamental class of solutions of symmetric hamiltonian systems is relative equilibria in this paper the nonlinear problem near a degenerate relative equilibrium is considered the degeneracy creates a saddle center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit the surprising result is that the curvature of the pullback of the momentum map to the lie algebra determines the normal form for the homoclinic bifurcation there is also an induced directional geometric phase in the homoclinic bifurcation the backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map the theory is constructive and generalities are given for symmetric hamiltonian systems on a vector space of dimension 2 n 2 with an n dimensional abelian symmetry group examples for n 1 2 3 are presented to illustrate application of the theory

hamiltonian structures and stability for rigid bodies with flexible attachments the dynamics of a rigid body with flexible attachments is studied a general framework for problems of this type is established in the context of poisson manifolds and reduction a simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration second the energy casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out the poisson structure and stability results take into account vibrations of the string rotations of the rigid body their coupling at the point of attachment and centrifugal and coriolis forces

variational methods for periodic orbits of reduced hamiltonian systems a variational method for periodic orbits is not easy to apply to a hamiltonian system when the symplectic form is not exact however if the hamiltonian system in question is a reduced one from a hamiltonian system on an exact symplectic manifold the variational method applies to the latter system in order to find periodic orbits of the reduced system this paper studies variational methods for periodic orbits in the systems reduced by the marsden weinstein and the orbit reduction procedures periodic orbits of the reduced systems are characterized as critical points of action functionals for loops in the original phase space together with lagrange multipliers

multiple delays estimation for chirp signals using structured total least squares abstract we employ the structured total least squares stls framework for estimating the delays associated with multiple reflections of a chirp signal the classical methods of using matched filter correlation processing originally aimed at estimating the delay of a single reflection have poor resolution and accuracy in the presence of multiple closely spaced reflections with overlapping correlation functions the stls framework offers enhanced resolution and accuracy for this problem we provide an overview of the stls framework for complex valued data and parameters showing validity of the riemannian singular value decomposition risvd approach which is a well established tool for real valued stls problems and also offer a new stls algorithm via alternating coordinates minimization acm characterized by guaranteed convergence and by an ability to account for errors in the equations thus potentially being more robust against model mismatch e g whenever the exact number of reflections is unknown we then turn to formulate the delays estimation problem in the complex valued stls framework and use simulation results to demonstrate and analyze the accuracy and convergence performance and associated trade offs of the proposed approach

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

constrained maximum likelihood solution of linear equations the total least squares tls is used to solve a set of inconsistent linear equations ax spl ap y when there are errors not only in the observations y but in the modeling matrix a as well the tls seeks the least squares perturbation of both y and a that leads to a consistent set of equations when y and a have a defined structure we usually want the perturbations to also have this structure unfortunately standard tls does not generally preserve the perturbation structure so other methods are required we examine this problem using a probabilistic framework and derive an approach to determining the most probable set of perturbations given an a priori perturbation probability density function while our approach is applicable to both gaussian and non gaussian distributions we show in the uncorrelated gaussian case that our method is equivalent to several existing methods our approach is therefore more general and can be applied to a wider variety of signal processing problems

shape from moments an estimation theory perspective this paper discusses the problem of recovering a planar polygon from its measured complex moments these moments correspond to an indicator function defined over the polygon s support previous work on this problem gave necessary and sufficient conditions for such successful recovery process and focused mainly on the case of exact measurements being given in this paper we extend these results and treat the same problem in the case where a longer than necessary series of noise corrupted moments is given similar to methods found in array processing system identification and signal processing we discuss a set of possible estimation procedures that are based on the prony and the pencil methods relate them one to the other and compare them through simulations we then present an improvement over these methods based on the direct use of the maximum likelihood estimator exploiting the above methods as initialization finally we show how regularization and thus maximum a posteriori probability estimator could be applied to reflect prior knowledge about the recovered polygon

ricci tensors on some infinite dimensional lie algebras abstract the ricci tensor has been computed in several infinite dimensional situations in this work we shall be interested in the case of the central extension of loop groups and in the asymptotic behaviour of the ricci tensor on free loop groups as the riemannian metric varies

morse theory for fixed points of symplectic diffeomorphisms on demontre le cas particulier suivant de la conjecture de arnold sur les points fixes d une deformation exacte d une variete symplectique compacte close p si 2 p 0 et tous les points fixes de sont non degeneres alors leur nombre est plus grand ou egal a la somme des nombres de betti de p par rapport aux coefficients z 2

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

structure preserving discretization of incompressible fluids the geometric nature of euler fluids has been clearly identified and extensively studied over the years culminating with lagrangian and hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume preserving diffeomorphisms and kelvin s circulation theorem is viewed as a consequence of noether s theorem associated with the particle relabeling symmetry of fluid mechanics however computational approaches to fluid mechanics have been largely derived from a numerical analytic point of view and are rarely designed with structure preservation in mind and often suffer from spurious numerical artifacts such as energy and circulation drift in contrast this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely eulerian form our approach approximates the group of volume preserving diffeomorphisms using a finite dimensional lie group and associated discrete euler equations are derived from a variational principle with non holonomic constraints the resulting discrete equations of motion yield a structure preserving time integrator with good long term energy behavior and for which an exact discrete kelvin s circulation theorem holds

the extended least squares criterion minimization algorithms and applications the least squares ls estimation criterion on one hand and the total ls tls constrained tls ctls and structured tls stls criteria on the other hand can be viewed as opposite limiting cases of a more general criterion which we term extended ls xls the xls criterion distinguishes measurement errors from modeling errors by properly weighting and balancing the two error sources in the context of certain models termed pseudo linear we derive two iterative algorithms for minimizing the xls criterion one is a straightforward alternating coordinates minimization and the other is an extension of an existing ctls algorithm the algorithms exhibit different tradeoffs between convergence rate computational load and accuracy the xls criterion can be applied to popular estimation problems such as identifying an autoregressive ar with exogenous noise arx system from noisy input output measurements or estimating the parameters of an ar process from noisy measurements we demonstrate the convergence properties and performance of the algorithms with examples of the latter

instantons and geometric invariant theory we show that the yang mills instantons can be described in terms of certain holomorphic bundles on the projective plane the proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries

a c 0 counterexample to the arnold conjecture the arnold conjecture states that a hamiltonian diffeomorphism of a closed and connected symplectic manifold m omega must have at least as many fixed points as the minimal number of critical points of a smooth function on m it is well known that the arnold conjecture holds for hamiltonian homeomorphisms of closed symplectic surfaces the goal of this paper is to provide a counterexample to the arnold conjecture for hamiltonian homeomorphisms in dimensions four and higher more precisely we prove that every closed and connected symplectic manifold of dimension at least four admits a hamiltonian homeomorphism with a single fixed point

on the hasse principle for quartic hypersurfaces we establish the hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to 28 defined over q

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

the maxwell vlasov equations in euler poincare form low s well known action principle for the maxwell vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of eulerian and lagrangian variables by imposing suitable constraints on the variations and analyzing invariance properties of the lagrangian as one does for the euler equations for the rigid body and ideal fluids we first transform this action principle into purely eulerian variables hamilton s principle for the eulerian description of low s action principle then casts the maxwell vlasov equations into euler poincare form for right invariant motion on the diffeomorphism group of position velocity phase space openface r 6 legendre transforming the eulerian form of low s action principle produces the hamiltonian formulation of these equations in the eulerian description since it arises from euler poincare equations this hamiltonian formulation can be written in terms of a poisson structure that contains the lie poisson bracket on the dual of a semidirect product lie algebra because of degeneracies in the lagrangian the legendre transform is dealt with using the dirac theory of constraints another maxwell vlasov poisson structure is known whose ingredients are the lie poisson bracket on the dual of the lie algebra of symplectomorphisms of phase space and the born infeld brackets for the maxwell field we discuss the relationship between these two hamiltonian formulations we also discuss the general kelvin noether theorem for euler poincare equations and its meaning in the plasma context

integration by parts formula and logarithmic sobolev inequality on the path space over loop groups the geometric stochastic analysis on the riemannian path space developed recently gives rise to the concept of tangent processes roughly speaking it is the infinitesimal version of the girsanov theorem using this concept we shall establish a formula of integration by parts on the path space over a loop group following the martingale method developed in capitaine hsu and ledoux we shall prove that the logarithmic sobolev inequality holds on the full paths as a particular case of our result we obtain the driver lohrenz s heat kernel logarithmic sobolev inequalities over loop groups the stochastic parallel transport introduced by driver will play a crucial role

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

the constrained total least squares technique and its applications to harmonic superresolution the constrained total least squares ctls method is a natural extension of tls to the case when the noise components of the coefficients are algebraically related the ctls technique is developed and some of its applications to superresolution harmonic analysis are presented the ctls problem is reduced to an unconstrained minimization problem over a small set of variables a perturbation analysis of the ctls solution is derived and from it the root mean square error rmse of the ctls solution which is valid for small noise levels is obtained in closed form the complex version of the newton method is derived and applied to determine the ctls solution it is also shown that the ctls problem is equivalent to a constrained parameter maximum likelihood problem the ctls technique is applied to estimate the frequencies of sinusoids in white noise and the angle of arrival of narrowband wavefronts at a linear uniform array in both cases the ctls method shows superior or similar accuracy to other advanced techniques

integration on loop groups i quasi invariant measures on a riemannian manifold m the brownian motion defines a wiener measure on the loops over m in the case of r quasi invariance means the cameron martin theorem s which fully describes the quasi invariance by translation of the wiener measure in order to state a generalization of this theorem we suppose that m is an homogeneous space under the action of a compact lie group the noncommutativity of this situation introduces a completely new phenomenon the hypoellipticity associated to the lie bracket of the basic vector fields curiously enough hypoellipticity which helps to get regularity in finite dimensions works against quasi invariance in infinite dimensions 18 we denote by g a compact connected lie group by h a connected subgroup of g and by m the homogeneous space g h we denote by y and 2 the corresponding lie algebras we choose a euclidean metric on 9 which is ad g invariant this metric induces a riemannian metric on g and on m we denote p g the paths space on g that is the space of continuous maps y from 0 l into g such that y o e we denote by l g the space of loops on g that is

complex trkalian fields and solutions to euler s equations for the ideal fluid we consider solutions to the complex trkalian equation vec nabla times vc vc where vc is a 3 component vector function with each component in the complex field and may be expressed in the form vc e ig vec nabla f with g real and f complex we find there are precisely two classes of solutions one where g is a cartesian variable and one where g is the spherical radial coordinate we consider these flows to be the simplest of all exact 3 d solutions to the euler s equation for the ideal incompressible fluid the novel approach we use in solving for these classes of solutions to these 3 dimensional vector pdes involves differential geometric techniques one may employ the method to generate solutions to other classes of vector pdes

total least squares for affinely structured matrices and the noisy realization problem structured rank deficient matrices arise in many applications in signal processing system identification and control theory the author discusses the structured total least squares stls problem which is the problem of approximating affinely structured matrices i e matrices affine in the parameters by similarly structured rank deficient ones while minimizing an l sub 2 error criterion it is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition which can be solved via an algorithm that is inspired by inverse iteration next the author concentrates on the so called l sub 2 optimal noisy realization problem which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional time invariant linear system of a given order this can be solved as a structured total least squares problem it is shown with some simple counter examples that classical algorithms such as the steiglitz mcbride 1965 iterative quadratic maximum likelihood and cadzow s 1988 iteration do not converge to the optimal l sub 2 solution despite misleading claims in the literature

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

on harmonic maps from r1 1 into hilbert loop groups motivated by the works of gu pure appl math 33 727 1980 ginibre and velo ann phys 142 393 1982 on classical solutions of a two dimensional model the global existence of harmonic maps from the minkowski plane r1 1 into hilbert loop groups is proved the construction of the harmonic maps strongly depends on some geometric properties of the loop groups

geometric investigations of a vorticity model equation abstract this article consists of a detailed geometric study of the one dimensional vorticity model equation t u x 2 u x 0 h u x t r x s 1 which is a particular case of the generalized constantin lax majda equation wunsch showed that this equation is the euler arnold equation on diff s 1 when the latter is endowed with the right invariant homogeneous h 1 2 metric in this article we prove that the exponential map of this riemannian metric is not fredholm and that the sectional curvature is locally unbounded furthermore we prove a beale kato majda type blow up criterion which we then use to demonstrate a link to our non fredholmness result finally we extend a blow up result of castro cordoba to the periodic case and to a much wider class of initial conditions using a new generalization of an inequality for hilbert transforms due to cordoba cordoba

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

periodic orbits of hamiltonian systems and symplectic reduction the notion of relative periodic orbits for hamiltonian systems with symmetry is discussed and a correspondence between periodic orbits of reduced and unreduced hamiltonian systems is established variational principles with symmetries are studied from the point of view of symplectic reduction of the space of loops leading to a characterization of reduced periodic orbits by means of the critical subsets of an action functional restricted to a submanifold of the loop space of the unreduced manifold finally as an application it is shown that if the symplectic form has finite integral rank then the periodic orbits of a hamiltonian system on the symplectic manifold admit a variational characterization

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

symplectic geometry and topology this is a review of the ideas underlying the application of symplectic geometry to hamiltonian systems the paper begins with symplectic manifolds and their lagrangian submanifolds covers contact manifolds and their legendrian submanifolds and indicates the first steps of symplectic and contact topology

logarithmic sobolev inequality on free loop groups for heat kernel measures associated with the general sobolev spaces abstract in this paper we will prove the logarithmic sobolev inequality on free loop groups for various heat kernel measures which p malliavin 1989 1991 in diffusion process and related problems in analysis m a pinsley ed vol i birkhauser basel constructed those measures are associated with the sobolev spaces of order s s 1 2 of the free loops in the lie algebra we will equipp the free loop groups with those metrics and will show that a formula of weitzenbock type holds which enables us to apply the method of driver and lohrenz 1996 j funct anal 146 381 448

string structures on loop bundles differential geometry and topology of principal loop bundles bundles of loop groups over loop spaces are investigated string structures defined as bundle extensions corresponding to the central extension of the structure group do not always exist various methods for deriving the obstruction to the existence of string structures are discussed

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

lin constraints clebsch potentials and variational principles the poisson bracket formulation of fluid plasma and rigid body type systems has undergone considerable recent development using techniques of symmetry group reduction the relationship between this approach and that using lin constraints and clebsch potentials is established the connection is made in the setting of abstract clebsch variables as well as that of variational principles on reduced spaces variational principles for both the clebsch and reduced form such as fluids in spatial representation are derived from the standard variational principle of hamilton in material lagrangian representation using reduction theory

solving quantum optimal control problems using clebsch variables and lin constraints clebsch variables and lin constraints are applied to the study of a class of optimal control problems for affine controlled quantum systems the optimal control problem will be modelled with controls defined on an auxiliary space where the dynamical group of the system acts freely the reciprocity between both theories the classical theory defined by the objective functional and the quantum system is established by using a suitable version of lagrange s multipliers theorem and a geometrical interpretation of the constraints of the system as defining a subspace of horizontal curves in an associated bundle it is shown how the solutions of the variational problem defined by the objective functional determine solutions of the quantum problem then a new way of obtaining explicit solutions for a family of optimal control problems for affine controlled quantum systems finite or infinite dimensional is obtained one of its main advantages is the the use of clebsch variables allows to compute such solutions from solutions of invariant problems that can often be computed explicitly this procedure can be presented as an algorithm that can be applied to a large class of systems finally some simple examples spin control a simple quantum hamiltonian with an elroy beanie type classical model and a controlled one dimensional quantum harmonic oscillator illustrating the main features of the theory will be discussed

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

existence of multiple periodic orbits on star shaped hamiltonian surfaces consider the hamiltonian system hs i 1 n here h c2 2n in this paper we investigate the existence of periodic orbits of hs on a given energy surface z 2n h z c c o is a constant the surface is required to verify certain geometric assumptions bounds a star shaped compact region and for some ellipsoid 2n o o depending in an explicit fashion on the lengths of the main axes of and one other geometrical parameter of such that if furthermore 2 2 1 then hs has at least n distinct geometric orbits on this result is shown to extend and unify several earlier works on this subject among them works by weinstein rabinowitz ekeland lasry and ekeland in proving this result we construct index theories for an s1 action from which we derive abstract critical point theorems for s1 invariant functionals we also derive an estimate for the minimal period of solutions to differential equatious