the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

emden fowler equations involving critical exponents on considere le probleme y t k y 2 y 2k 3 0 t 0 pour t 2 0 0 soit t inf t 0 y 0 sur t alors t 0 0 on etudie comment la position t du premier zero de y t varie avec

critical dimension of a hessian equation involving critical exponent and a related asymptotic result abstract let s k 2 u 1 k n 2 be the k th hessian operator we study the problem formula where k is the critical exponent for s k and is a ball results generalizing those obtained by brezis nirenberg for the special case k 1 are established a discussion 8 on the asymptotic behavior of solutions of the problem as 0 is also included

interconnection and damping assignment passivity based control of port controlled hamiltonian systems passivity based control pbc is a well established technique that has shown to be very powerful to design robust controllers for physical systems described by euler lagrange el equations of motion for regulation problems of mechanical systems which can be stabilized shaping only the potential energy pbc preserves the el structure and furthermore assigns a closed loop energy function equal to the difference between the energy of the system and the energy supplied by the controller thus we say that stabilization is achieved via energy balancing unfortunately these nice properties of el pbc are lost when used in other applications which require shaping of the total energy for instance in electrical or electromechanical systems or even some underactuated mechanical devices our main objective in this paper is to develop a new pbc theory which extends to a broader class of systems the aforementioned energy balancing stabilization mechanism and the structure invariance towards this end we depart from the el description of the systems and consider instead port controlled hamiltonian models which result from the network modelling of energy conserving lumped parameter physical systems with independent storage elements and strictly contain the class of el models

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

on the synthesis of boundary control laws for distributed port hamiltonian systems this paper is concerned with the energy shaping of 1 d linear boundary controlled port hamiltonian systems the energy casimir method is first proposed to deal with power preserving systems it is shown how to use finite dimensional dynamic boundary controllers and closed loop structural invariants to partially shape the closed loop energy function and how such controller finally reduces to a state feedback when dissipative port hamiltonian systems are considered the casimir functions do not exist anymore dissipation obstacle and the immersion via a dynamic controller reduction through invariants method cannot be applied the main contribution of this paper is to show how to use the same ideas and state functions to shape the closed loop energy function of dissipative systems through direct state feedback i e without relying on a dynamic controller and a reduction step in both cases the existence of solution and the asymptotic stability by additional damping injection of the closed loop system are proven the general theory and achievable closed loop performances are illustrated with the help of a concluding example the boundary stabilization of a longitudinal beam vibrations

the duistermaat heckman integration formula on flag manifolds an exposition is given of various geometrical properties of flag manifolds and of the duistermaat heckman integration formula as applied to flag manifolds

uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball we establish a new pohozaev type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div u m 2emsp14 u f u 0 inb andu 0 on b wherebis a finite ball in rn n 3 and 1 0 and 1 p q n 2 n 2 has a unique positive solution whenn 6 this gives a complete answer to an open problem raised by brezis and nirenberg in 1983 in the casen 6 we shall also derive some partial results to the open problem in the casesn 3 4 5

uniqueness of positive solutions of a quasilinear dirichlet problem with exponential nonlinearity uniqueness of positive solutions of the following equation has been obtained where b n is the ball of unit radius and 0 moreover if n 2 then the solutions are nondegenerate

bounded solutions of a k hessian equation involving a weighted nonlinear source abstract we consider the problem 1 s k d 2 u x 1 u q in b u 0 in b u 0 on b where b denotes the unit ball in r n n 2 k k n 0 q k and 0 we study the existence multiplicity and uniqueness of radially symmetric bounded solutions of 1 the methodology for obtaining our results is based on a dynamical systems approach for this we introduce a new transformation which reduces problem 1 to an autonomous two dimensional generalized lotka volterra system

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

wall crossing for symplectic vortices and quantum cohomology we derive a wall crossing formula for the symplectic vortex invariants of toric manifolds as an application we give a proof of batyrev s formula for the quantum cohomology of a monotone toric manifold with minimal chern number at least two

uniqueness theorems for negative radial solutions of k hessian equations in a ball abstract in this paper we apply a pohozaev type identity and the monotone separation techniques to prove the uniqueness of negative radial solutions of k hessian equations in a finite ball

exact number of solutions of a prescribed mean curvature equation abstract we study the exact number of positive solutions of the dirichlet problem for the one dimensional prescribed mean curvature equation u 1 u 2 u p u q u 0 u 1 0 where 0 is a parameter and p q satisfy either 1 p q or 0 p q 1

structure of positive radial solutions of semilinear elliptic equations abstract we study the positive radial solutions of a semilinear elliptic equation u f u 0 wheref u has a supercritical growth order for smallu 0 and a subcritical growth order for largeu by showing the uniqueness of positive solutions behaving likeo x 2 n at infinity we give an almost complete description for the structure of positive radial solutions as a consequence we also prove the uniqueness of positive solutions of the nonlinear dirichlet problem for the equation in a finite ball

uniqueness of positive radial solutions for δu u up 0 on an annulus abstract we prove uniqueness of positive radial solutions to the semilinear elliptic equation u u u p 0 p 1 subject to the dirichlet boundary condition on an annulus in r n n 3 as a by product our argument also provides a much simpler if not the simplest new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space r n

existence and uniqueness of nontrivial radial solutions for k hessian equations abstract we prove existence and uniqueness of nontrivial radial solutions to the k hessian problem s k d 2 u f u in u 0 on where s k d 2 u is the k hessian operator of u is the open unit ball in r n f is a continuous function and may have k sublinear growth at with possible k superlinear growth at 0 and is a large parameter

the kirwan map equivariant kirwan maps and their kernels consider a hamiltonian action of a compact lie group k on a compact symplectic manifold we find descriptions of the kernel of the kirwan map corresponding to a regular value of the moment map k we start with the case when k is a torus t we determine the kernel of the equivariant kirwan map defined by goldin in go corresponding to a generic circle s t and show how to recover from this the kernel of t as described by tolman and weitsman in to we in the situation when the fixed point set of the torus action is finite similar results have been obtained in our previous papers je je ma for a compact nonabelian lie group k we will use the non abelian localization formula of je ki1 and je ki2 to establish relationships some of them obtained by tolman and weitsman in to we between ker k and ker t where t k is a maximal torus in the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of t

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

passivity based control of a class of hamiltonian systems with nonholonomic constraints this paper is concerned with state and output feedback stabilization of a class of port hamiltonian systems with nonholonomic constraints first we study canonical forms for port hamiltonian systems with nonholonomic constraints second we give a new state feedback stabilization method by using non smooth hamiltonian functions via generalized canonical transformations third we propose a dynamic output feedback stabilization method without measuring the velocity based on the corresponding state feedback result numerical examples demonstrate the effectiveness of the proposed method

exact number of solutions of a one dimensional prescribed mean curvature equation with concave convex nonlinearities abstract in this paper an exact number of solutions is obtained for the one dimensional prescribed mean curvature equation with concave convex nonlinearities in the form of u 1 u 2 u p u q u x 0 0 x 1 u 0 u 1 0 where 0 is a parameter and p q satisfy 0 p 1 q the arguments are based upon a time map method

complex equivariant intersection excess normal bundles and bott chern currents the purpose of this paper is to establish an intersection formula in equivariant complex geometry in the presence of an excess normal bundle the contribution of the excess normal bundle to the formula appears through an additive genus k r in a forthcoming paper an infinite dimensional analogue of this formula will be shown to be the result of bismut lebeau on the behaviour of quillen metrics under complex immersions

index theorem and equivariant cohomology on the loop space in this paper extending ideas of witten and atiyah we describe some relations of equivariant cohomology on the loop space of a manifold to the path integral representation of the index of the dirac operator on a twisted spin complex in particular the natural extension of the chern character to the whole loop space is described also it is shown that the non zero free homotopy classes of the loop space have 0 measure for the index measures associated to the index problem

a local index theorem for non kahler manifolds on considere des conditions suffisantes pour qu un theoreme d indice local soit vrai pour un operateur de dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de levi civita l

equivariant de rham torsions the purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de rham theory this difference is the sum of the integral of a chern simons current and of a new invariant the v invariant of an odd dimensional manifold equipped with an action of a compact lie group the v invariant localizes on the critical manifolds of invariant morse bott functions the results in this paper are shown to be compatible with results of bunke and also our with previous results on analytic torsion forms

exact multiplicity for semilinear elliptic dirichlet problems involving concave and convex nonlinearities let b be the unit ball in r n n 3 let 0 p q n 2 n 2 in 1994 ambrosetti et al found that the semilinear elliptic dirichlet problem admits at least two solutions for small 0 and no solution for large in this paper we prove that there is a critical number 0 such that this problem has exactly two solutions for 0 exactly one solution for and no solution for

existence and multiplicity for negative solutions of k hessian equations abstract in this paper we give some existence results for k hessian equations by sub sup method and discuss the multiplicity of k hessian equations especially for the radial solutions in a ball

an elementary proof of the uniqueness of positive radial solutions of a quasilinear dirichlet problem we consider the quasilinear elliptic equation text div left nabla u m 2 nabla u right u p lambda u m 2 u in in b where b is a ball or an annulus in n 1 m n p is a positive real number and e using a generalized poho aev type variational identity of ni serrin or pucci and serrin and an elementary calculus lemma we establish uniqueness of positive radial solutions for the dirichlet boundary condition if either left text i right b text is a ball lambda underset raise0 3em hbox smash scriptscriptstyle geqslant text 0 1 p underset raise0 3em hbox smash scriptscriptstyle leqslant frac mn n m text for m n text and 1 p infty text for m n text or left text ii right b text is an annulus lambda in mathbb r text and p frac mn n m

dirac structures on hilbert spaces and boundary control of distributed port hamiltonian systems abstract aim of this paper is to show how the dirac structure properties can be exploited in the development of energy based boundary control laws for distributed port hamiltonian systems usually stabilization of non zero equilibria has been achieved by looking at or generating a set of structural invariants namely casimir functions in closed loop since this approach fails when an infinite amount of energy is required at the equilibrium dissipation obstacle this paper illustrates a novel approach that enlarges the class of stabilizing controllers the starting point is the parametrization of the dynamics provided by the image representation of the dirac structure that is able to show the effects of the boundary inputs on the state evolution in this way energy balancing and control by state modulated source methodologies are extended to the distributed parameter scenario and a geometric interpretation of these control techniques is provided the theoretical results are discussed with the help of a simple but illustrative example i e a transmission line with an rlc load in both serial and parallel configurations in the latter case energy balancing controllers are not able to stabilize non zero equilibria because of the dissipation obstacle the problem is solved thanks to a boundary state modulated source

boundary blow up solutions to the k hessian equation with a weakly superlinear nonlinearity abstract by constructing new sub and super solutions we are concerned with determining values of for which there exist k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 and is a smooth bounded strictly convex domain in r n n 2 we suppose that the nonlinearity behaves like u k ln u as u which is more complex and difficult to deal with than the nonlinearity grows like u p with p k or faster at infinity further several new results of nonexistence global estimates and estimates near the boundary for the solutions are also given

on representations and integrability of mathematical structures in energy conserving physical systems in the present paper we elaborate on the underlying hamiltonian structure of interconnected energy conserving physical systems it is shown that a power conserving interconnection of port controlled generalized hamiltonian systems leads to an implicit generalized hamiltonian system and a power conserving partial interconnection to an implicit port controlled hamiltonian system the crucial concept is the notion of a generalized dirac structure defined on the space of energy variables or on the product of the space of energy variables and the space of flow variables in the port controlled case three natural representations of generalized dirac structures are treated necessary and sufficient conditions for closedness or integrability of dirac structures in all three representations are obtained the theory is applied to implicit port controlled generalized hamiltonian systems and it is shown that the closedness condition for the dirac structure leads to strong conditions on the input vector fields

the genus and a regularization of an s1 equivariant euler class we show that a new multiplicative genus in the sense of hirzebruch can be obtained by generalizing a calculation due to atiyah and witten we introduce this as the genus compute its value for some examples and highlight some of its interesting properties we also indicate a connection with the study of multiple zeta values which gives an algebraic interpretation for our proposed regularization procedure

achievable casimirs and its implications on control of port hamiltonian systems in this paper we extend results on interconnections of port hamiltonian systems to infinite dimensional port hamiltonian systems and to mixed finite and infinite dimensional port hamiltonian systems the problem of achievable dirac structures is now studied for systems with dissipation in the finite dimensional infinite dimensional and the mixed finite and infinite dimensional case we also characterize the set of achievable casimirs and study its application for the control of port hamiltonian systems

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

existence for a k hessian equation involving supercritical growth abstract in this paper we use variational techniques to give existence results for the problem s k u f x u in u 0 in u 0 on where s k u is the k hessian operator and f x u is a supercritical nonlinearity in the sense introduced by k tso ann inst henri poincare 1990 using some ideas from a celebrated article by brezis and nirenberg we show existence of a positive solution considering supercritical nonlinearities which is surprising given the validity of the pohozaev identity

bubbling nodal solutions for a large perturbation of the moser trudinger equation on planar domains in this work we study the existence of nodal solutions for the problem begin aligned delta u lambda u e u 2 u p quad text in quad omega quad u 0 quad text on partial omega end aligned where omega subseteq mathbb r 2 is a bounded smooth domain and p rightarrow 1 if omega is a ball it is known that the case p 1 defines a critical threshold between the existence and the non existence of radially symmetric sign changing solutions in this work we construct a blowing up family of nodal solutions to such problem as p rightarrow 1 when omega is an arbitrary domain and lambda is small enough as far as we know this is the first construction of sign changing solutions for a moser trudinger critical equation on a non symmetric domain

extremal functions for trudinger moser inequalities of adimurthi druet type in dimension two abstract combining carleson chang s result 9 with blow up analysis we prove existence of extremal functions for certain trudinger moser inequalities in dimension two this kind of inequality was originally proposed by adimurthi and o druet 1 extended by the author to high dimensional case and riemannian surface case 40 41 generalized by c tintarev to wider cases including singular form 36 and by m de souza and j m do o 14 to the whole euclidean space r 2 in addition to the euclidean case we also consider the riemannian surface case the results in the current paper complement that of l carleson and a chang 9 m struwe 35 m flucher 16 k lin 19 and adimurthi and o druet 1 our previous ones 26 41 and part of c tintarev 36

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

modeling and control of the timoshenko beam the distributed port hamiltonian approach the purpose of this paper is to show how the timoshenko beam can be fruitfully described within the framework of distributed port hamiltonian dph systems so that rather simple and elegant considerations can be drawn regarding both the modeling and control of this mechanical system after the dph model of the beam is introduced the control problem is discussed in particular it is shown how control approaches already presented in the literature can be unified and a new control methodology is presented and discussed this control methodology relies on the generalization to infinite dimensions of the concept of structural invariant casimir function and on the extension to distributed systems of the so called control by interconnection methodology in this way finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy i e by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipative effect is introduced

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

uniqueness and exact multiplicity of solutions for a class of dirichlet problems abstract we apply the monotone separation of graphs technique of l a peletier and j serrin l a peletier j serrin uniqueness of positive solutions of semilinear equations in r n arch ration mech anal 81 2 1983 181 197 l a peletier j serrin uniqueness of nonnegative solutions of semilinear equations in r n j differential equations 61 3 1986 380 397 as developed further by l erbe and m tang l erbe m tang structure of positive radial solutions of semilinear elliptic equations j differential equations 133 2 1997 179 202 to the question of exact multiplicity of positive solutions for a class of semilinear equations on a unit ball in r n we also observe that using p pucci and j serrin p pucci j serrin uniqueness of ground states for quasilinear elliptic operators indiana univ math j 47 2 1998 501 528 improvement of a certain identity of l erbe and m tang l erbe m tang structure of positive radial solutions of semilinear elliptic equations j differential equations 133 2 1997 179 202 produces a short proof of l erbe and m tang l erbe m tang structure of positive radial solutions of semilinear elliptic equations j differential equations 133 2 1997 179 202 result on the uniqueness of positive solution of 1 p q n 2 n 2 u u p u q 0 for x 1 u 0 when x 1

holomorphic morse inequalities for orbifolds we prove that demailly s holomorphic morse inequalities hold true for complex orbifolds by using a heat kernel method then we introduce the class of moishezon orbifolds and as an application of our inequalties we give a geometric criterion for a compact connected orbifold to be a moishezon orbifolds thus generalizing siu s and demailly s answers to the grauert riemenschneider conjecture to the orbifold case

uniqueness of positive solutions of a n laplace equation in a ball in rn with exponential nonlinearity abstract let n 2 and r n be a bounded domain then by trudinger moser embedding w 0 1 n is embedded in an orlicz space consisting of exponential functions consider the corresponding semilinear n laplace equation with critical or sub critical exponential nonlinearity in a ball b r with dirichlet boundary condition in this paper we prove that under suitable growth conditions on the nonlinearity there exists an 0 0 and a corresponding r 0 0 0 such that for all 0 r r 0 the problem admits a unique nondegenerate positive radial solution u with u 0

extremal for a k hessian inequality of trudinger moser type we consider k hessian operators s k u in bounded domains varomega in mathbb r n such that partial varomega is k 1 convex for so called k admissible functions u in varphi 0 k one has sobolev type inequalities of the form begin aligned vert u vert l p varomega le c vert u vert varphi 0 k end aligned where vert u vert varphi 0 k k 1 int varomega u s k u dx and 1 le p le k frac n k 1 n 2k the case n 2k is a borderline case of trudinger moser type and it was recently shown by tian wang that a corresponding inequality of exponential type holds begin aligned sup vert u vert varphi 0 k le 1 int varomega left mathrm e alpha u frac n 2 n sum j 0 k 1 frac alpha j u j frac n 2 n j right mathrm d x le c end aligned for alpha le alpha n n left frac omega n 1 k n 1 atopwithdelims k 1 right 2 n in this article we prove an analogue to the famous result of carleson chang namely that for varomega b r 0 the above supremum is attained also in the limiting case alpha alpha n

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

uniqueness of non negative solutions of semilinear equations in rn on etudie l unicite des solutions positives classiques u x du probleme u f u 0 dans r n u x 0 quand x avec n 1 et x x 1 x n

energy shaping of distributed parameter port hamiltonian systems based on finite element approximation the main contribution of this paper is a procedure for the control by energy shaping via casimir generation of infinite dimensional port hamiltonian systems based on a particular finite element approximation the proposed approach is justified by the fact that the adopted spatial discretization technique is able to preserve casimir functions in the closed loop system when going from the distributed to the approximated lumped parameter system besides the intrinsic difficulties related to the large number of state variables the finite element model is generally given in terms of a dirac structure and is completely a causal which implies that the plant dynamics is not given in standard input state output form but as a set of daes consequently the classical energy casimir method has to be extended in order to deal with dynamical systems with constraints usually appearing in the form of lagrangian multipliers the general methodology is illustrated with the help of an example in which the distributed parameter system is a lossless transmission line

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

canonical partition functions of hamiltonian systems and the stationary phase formula we show that there is a symplectic action of a torus associated to harmonic flows on the cotangent bundle of a semi simple lie algebra this allows to obtain a completely classical proof of the gallavotti marchioro formula by the method of the stationary phase

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

control by interconnection of mixed port hamiltonian systems in this note the regulation problem for mixed finite and infinite dimensional port hamiltonian systems m ph systems is discussed a m ph system results from the power conserving interconnection of finite and infinite dimensional systems in port hamiltonian form in particular the system given by the interconnection of two finite dimensional systems one of which is the controller by means of an infinite dimensional connection is studied the proposed control methodology is a generalization to the infinite dimensional case of a well established passivity based control technique for finite dimensional port hamiltonian systems the control by interconnection and energy shaping according to which the open loop energy function is shaped so that a minimum in the desired configuration is introduced this procedure is possible once the state variable of the controller is related to the state variable of the plant by constraining the state of the closed loop system on a structural invariant defined by a set of casimir functions in this way the energy function of the controller which is freely assignable becomes a function of the configuration of the plant and then it can be easily shaped in order to solve the regulation problem

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

uniqueness of positive radial solutions for n laplacian dirichlet problems we prove a theorem on the uniqueness of positive radial solutions to a dirichlet problem of the n laplacian in a finite ball of r n our proofs use only elementary analysis based on an identity due to erbe and tang the result can be applied to a large class of nonlinearities including some polynomials and functions with exponential growth in particular the one recently studied by adimurthi

existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations we prove the existence and uniqueness of fast decay solutions and clarify the structure of positive radial solutions of the quasilinear elliptic equation mu f u 0 in rn where m denotes the m laplace operator and f has a supercritical growth for small u 0 and a subcritical growth for large u our proofs use only elementary arguments based on several variational identities and a maximum principle of peletier and serrin

on certain infinite dimensional aspects of arakelov intersection theory letk y x be an embedding of compact complex manifolds bismut and lebeau have calculated the quillen norm of the canonical isomorphism identifying the determinant of the cohomology of a holomorphic vector bundle overy and the determinant of the cohomology of a resolution by a complex of holomorhic vector bundles overx the purpose of this paper is to show that the formula of bismut lebeau can be viewed as an equivariant intersection formula over the loop space of the considered manifolds in the presence of an infinite dimensional excess normal bundle this excess normal bundle is responsible for the appearance of the additive genusr of gillet and soule in the formula of bismut and lebeau