morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

derived category automorphisms from mirror symmetry inspired by the homological mirror symmetry conjecture of kontsevich 30 we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi projective variety msc 2000 18e30 14j32

ozsvath szabo invariants and tight contact three manifolds ii let s 3 k be the oriented 3 manifold obtained by rational r surgery on a knot k s 3 using the contact ozsv ath szab o invariants we prove for a class of knots k containing all the algebraic knots that s 3 k carries positive tight contact structures for every r 6 g s k 1 where gs k is the slice genus of k this implies in particular that the brieskorn spheres 2 3 4 and 2 3 3 carry tight positive contact structures as an application of our main result we show that for each m2n there exists a seifert bered rational homology 3 sphere mm carrying at least m pairwise non isomorphic tight nonllable contact structures

on some examples and constructions of contact manifolds the first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3 manifold which is not smoothly a rational homology sphere contact embeds with trivial normal bundle inside a hypertight closed contact 5 manifold this uses known construction procedures by bourgeois on products with tori and geiges on branched covering spaces we pass from these procedures to definitions this allows to prove a uniqueness statement in the case of contact branched coverings and to study the global properties such as tightness and fillability of the results of both constructions without relying on any auxiliary choice in the procedures a second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book as introduced by giroux and of contact fiber bundle as introduced by lerman for instance we give a definition of bourgeois contact structures on flat contact fiber bundles which is local strictly includes the results of the bourgeois construction and allows to recover an isotopy class of supporting open books on the fibers this last point relies on a reinterpretation inspired by an idea by giroux of supporting open books in terms of pairs of contact vector fields

numerical solution for a nonlinear one dimensional problem of thermoelasticity a numerical solution for a nonlinear one dimensional boundary value problem of thermoelasticity for the elastic half space is presented the resulting equations are discussed and the numerical method is investigated for stability comparison with other existing numerical schemes is carried out the obtained results clearly indicate the process of shock formation the presented figures show the effects of different nonlinear coupling constants on the distributions of the mechanical displacement and temperature in the medium a special case is briefly discussed

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

symplectic homology product via legendrian surgery this research announcement continues the study of the symplectic homology of weinstein manifolds undertaken by the authors bourgeois f ekholm t eliashberg y 2009 arxiv 0911 0026 where the symplectic homology as a vector space was expressed in terms of the legendrian homology algebra of the attaching spheres of critical handles here we express the product and batalin vilkovisky operator of symplectic homology in that context

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

right veering diffeomorphisms of compact surfaces with boundary we initiate the study of the monoid of right veering diffeomorphisms on a compact oriented surface with nonempty boundary the monoid strictly contains the monoid of products of positive dehn twists we explain the relationship to tight contact structures and open book decompositions

almost contact 5 manifolds are contact the existence of a contact structure is proved in any homotopy class of almost contact structures on a closed 5 dimensional manifold

handlebody construction of stein surfaces the topology of stein surfaces and contact 3 manifolds is studied by means of handle decompositions a simple characterization of homeomorphism types of stein surfaces is obtained they correspond to open handlebodies with all handles of index 2 an uncountable collection of exotic l4 s is shown to admit stein structures new invariants of contact 3 manifolds are produced including a complete and computable set of invariants for determining the homotopy class of a 2 plane field on a 3 manifold these invariants are applicable to seiberg witten theory several families of oriented 3 manifolds are examined namely the seifert fibered spaces and all surgeries on various links in s3 and in each case it is seen that most members of the family are the

adaptive output feedback design for a class of nonlinear discrete time systems we present adaptive output feedback schemes for a class of discrete time systems whose continuous time counterparts are the output feedback canonical form we first develop a systematic design procedure for the deterministic case global boundedness and convergence are achieved with no restriction imposed on the nonlinearities then we proceed to develop an analogous design procedure for the adaptive case depending on the characteristics of the nonlinearities global boundedness and convergence can be achieved with various update laws a rank condition is required to ensure such properties for nonlinearities which are not globally lipschitz

legendrian contact homology in the boundary of a subcritical weinstein 4 manifold we give a combinatorial description of the legendrian contact homology algebra associated to a legendrian link in s1 s2 or any connected sum k s1 s2 viewed as the contact boundary of the weinstein manifold obtained by attaching 1 handles to the 4 ball in view of the surgery formula for symplectic homology this gives a combinatorial description of the symplectic homology of any 4 dimensional weinstein manifold and of the linearized contact homology of its boundary we also study examples and discuss the invariance of the legendrian homology algebra under deformations from both the combinatorial and the analytical perspectives

legendrian fronts for affine varieties in this article we study weinstein structures endowed with a lefschetz fibration in terms of the legendrian front projection first we provide a systematic recipe for translating from a weinstein lefschetz bifibration to a legendrian handlebody then we present several new applications of this technique to symplectic topology this includes the detection of flexibility and rigidity for several families of weinstein manifolds and the existence of closed exact lagrangian submanifolds in particular we prove that the koras russell cubic is stein deformation equivalent to c3 and we verify the affine parts of the algebraic mirrors of two weinstein 4 folds

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

geometric criteria for overtwistedness we establish geometric criteria to decide whether a contact manifold is over twisted starting with the original denition we rst relate the dierent overtwisted disks d 2n ot in each dimension and show that a manifold is overtwisted if the legendrian un knot is loose then we characterize overtwistedness in terms of open book decompositions and provide several applications

rational symplectic field theory for legendrian knots we construct a combinatorial invariant of legendrian knots in standard contact three space this invariant which encodes rational relative symplectic field theory and extends contact homology counts holomorphic disks with an arbitrary number of positive punctures the construction uses ideas from string topology

brief robust adaptive control of uncertain discrete time systems in this paper a robust adaptive controller for a class of nonlinear uncertain discrete time systems is developed by combining the backstepping procedures with a simple parameter estimator subject to parameter projection it is shown that the proposed controller can ensure boundedness of all signals in the overall adaptive systems in the presence of unmodelled dynamics and disturbances it can also guarantee that the tracking error is bounded by a function of the size of the unmodelled dynamics in the ideal case when there are no unmodelled dynamics and disturbances perfect tracking is ensured

global smooth solutions and asymptotic stability in one dimensional nonlinear thermoelasticity the global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one dimensional homogeneous reference configuration dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t is described

filtrations on the knot contact homology of transverse knots we construct a new invariant of transverse links in the standard contact structure on mathbb r 3 this invariant is a doubly filtered version of the knot contact homology differential graded algebra dga of the link see ekholm et al knot contact homology arxiv 1109 1542 2011 ng duke math j 141 2 365 406 2008 here the knot contact homology of a link in mathbb r 3 is the legendrian contact homology dga of its conormal lift into the unit cotangent bundle s mathbb r 3 of mathbb r 3 and the filtrations are constructed by counting intersections of the holomorphic disks of the dga differential with two conormal lifts of the contact structure we also present a combinatorial formula for the filtered dga in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

tight not semi fillable contact circle bundles extending our earlier results we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi fillable thus confirming a conjecture of ko honda

existence of almost everywhere solution for nonlinear hyperbolic parabolic system in this paper the existence theorem for an almost everywhere solution of a mixed problem for more complicated nonlinear hyperbolic parabolic nonlinear thermoelasticity system was proved using galerkin s method and suitable a priori estimates were obtained

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

self tuning minimum variance control of nonlinear systems of the hammerstein model self tuning is applied to the control of nonlinear systems represented by the hammerstein model wherein the nonlinearity is any odd order polynomial but control costing is not feasible in general initial relay control is employed to contain the deviations

transversality in elliptic morse theory for the symplectic action our goal in this paper is to settle some transversality question for the perturbed nonlinear cauchy riemann equations on the cylinder these results play a central role in the denition of symplectic floer homology and hence in the proof of the arnold conjecture there is currently no other reference to these transversality results in the open literature our approach does not require aronszajn s theorem instead we derive the unique continuation theorem from a generalization of the carleman similarity principle

contact structures on m s2 we show that if a manifold m admits a contact structure then so does m times s 2 our proof relies on surgery theory a theorem of eliashberg on contact surgery and a theorem of bourgeois showing that if m admits a contact structure then so does m times t 2

weak and strong fillability of higher dimensional contact manifolds for contact manifolds in dimension three the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent we extend these facts to higher dimensions in particular we define a natural generalization of weak fillings and prove that it is indeed weaker at least in dimension five while also being obstructed by all known manifestations of overtwistedness we also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense these depend on a higher dimensional analogue of giroux torsion which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary

a duality exact sequence for legendrian contact homology we establish a long exact sequence for legendrian submanifolds l p r where p is an exact symplectic manifold which admit a hamiltonian isotopy that displaces the projection of l to p off of itself in this sequence the singular homology h maps to linearized contact cohomology ch which maps to linearized contact homology ch which maps to singular homology in particular the sequence implies a duality between ker ch h and ch im h furthermore this duality is compatible with poincare duality in l in the following sense the poincare dual of a singular class which is the image of a ch maps to a class ch such that a 1 the exact sequence generalizes the duality for legendrian knots in r3 see 26 and leads to a refinement of the arnold conjecture for double points of an exact lagrangian admitting a legendrian lift with linearizable contact homology first proved in 7

differential algebra of legendrian links let the space r q p u be equipped with the standard contact form du pdq a link l r3 is called legendrian if the restriction of to l vanishes two legendrian links are said to be legendrian isotopic if they can be connected by a path in the space of legendrian embeddings consider the map r3 r2 q p u q p since the kernel of d is everywhere transverse to the kernel of the projection l of a legendrian link l is an immersed curve we say that a link l r3 is generic with respect to or generic if all self intersections of l are transverse double points given a generic legendrian link l its diagram is the curve l r2 at every crossing of which the overpassing branch the one with the larger value of u is marked it is well known that every smooth link type can be realized by a legendrian link it is shown in sect 11 that such a realization can be chosen to have a diagram with relatively few crossings the so called classical invariants of an oriented legendrian link l are defined as follows first of them is just the smooth isotopy type of l let l1 lk be the components of l the maslov number m li of li is twice the rotation number of li the thurston bennequin number li of li is the linking number between li and s li where s is a small shift along the u direction the orientation on r3 involved in the definition of the linking number is given by the form d dq dp du in terms of the diagram li is the number of double points of li counted with signs shown in fig 1 the thurston bennequin number does not depend on the orientation of li while the maslov number changes its sign when the orientation changes

legendrian contact homology in p times mathbb r a rigorous foundation for the contact homology of legendrian submanifolds in a contact manifold of the form p x r where p is an exact symplectic manifold is established the class of such contact manifolds includes 1 jet spaces of smooth manifolds as an application contact homology is used to provide smooth isotopy invariants of submanifolds ofr and more generally invariants of self transverse immersions into r n up to restricted regular homotopies when n 3 this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of ng

exact lagrangian immersions with one double point revisited we study exact lagrangian immersions with one double point of a closed orientable manifold k into mathbb c n we prove that if the maslov grading of the double point does not equal 1 then k is homotopy equivalent to the sphere and if in addition the lagrangian gauss map of the immersion is stably homotopic to that of the whitney immersion then k bounds a parallelizable n 1 manifold the hypothesis on the gauss map always holds when n 2k or when n 8k 1 the argument studies a filling of k obtained from solutions to perturbed cauchy riemann equations with boundary on the image f k of the immersion this leads to a new and simplified proof of some of the main results of ekholm and smith exact lagrangian immersions with a single double point 2011 which treated lagrangian immersions in the case n 2k by applying similar techniques to a lagrange surgery of the immersion as well as to an extension of these results to the odd dimensional case

explicit self tuning control for a class of non linear systems abstract this paper is concerned with adaptive control of discrete time linear systems having a polynomial input non linearity a new self tuning control scheme is presented to implement adaptively the suboptimal control law derived in the previous work by the on line adaptation of the compensators p q 1 and r q 1 the difficulty of cautiously choosing p q 1 and r q 1 a priori to ensure the stability of the closed loop system is thus circumvented a simple locally convergent self tuning algorithm and a globally convergent improved self tuning algorithm are given which provide an efficient way to the control of this class of non linear systems simulation results verify the effectiveness of the proposed algorithm

generalized gauss maps and integrals for three component links toward higher helicities for magnetic fields and fluid flows to each three component link in the 3 sphere we associate a generalized gauss map from the 3 torus to the 2 sphere and show that the pairwise linking numbers and milnor triple linking number that classify the link up to link homotopy correspond to the pontryagin invariants that classify its generalized gauss map up to homotopy we view this as a natural extension of the familiar situation for two component links in 3 space where the linking number is the degree of the classical gauss map from the 2 torus to the 2 sphere the generalized gauss map like its prototype is geometrically natural in the sense that it is equivariant with respect to orientation preserving isometries of the ambient space thus positioning it for application to physical situations when the pairwise linking numbers of a three component link are all zero we give an integral formula for the triple linking number analogous to the gauss integral for the pairwise linking numbers this new integral is also geometrically natural like its prototype in the sense that the integrand is invariant under orientation preserving isometries of the ambient space versions of this integral have been applied by komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields we have set this entire paper in the 3 sphere because our generalized gauss map is easiest to present here but in a subsequent paper we will give the corresponding maps and integral formulas in euclidean 3 space

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

global existence of small solutions to the initial value problem for nonlinear thermoelasticity abstract we prove the existence of global small solutions to the initial value problem in three dimensional nonlinear thermoelasticity with quadratic completely coupled nonlinear terms the asymptotic behaviour of the solution as t is also described

exact lagrangians in a_n surface singularities in this paper we classify lagrangian spheres in a n surface singularities up to hamiltonian isotopy combining with a result of ritter geom funct anal 20 3 779 816 2010 this yields a complete classification of exact lagrangians in a n surface singularities our main new tool is the application of a technique which we call ball swappings and its relative version

differential algebra of cubic planar graphs abstract in this article we associate a combinatorial differential graded algebra to a cubic planar graph g this algebra is defined combinatorially by counting binary sequences which we introduce and several explicit computations are provided in addition in the appendix by k sackel the f q rational points of its graded augmentation variety are shown to coincide with q 1 colorings of the dual graph

embeddings of danielewski surfaces a danielewski surface is defined by a polynomial of the form p x nz p y define also the polynomial p x nz r x p y where r x is a non constant polynomial of degree n 1 and r 0 1 we show that when n 2 and deg p y 2 the general fibers of p and p are not isomorphic as algebraic surfaces but that the zero fibers are isomorphic consequently for every non special danielewski surface s there exist non equivalent algebraic embeddings of s in 3 using different methods we also give non equivalent embeddings of the surfaces xz y d n 1 for an infinite sequence of integers d n we then consider a certain algebraic action of the orthogonal group mathcal o 2 on 4 which was first considered by schwarz and then studied by masuda and petrie who proved that this action could not be linearized this was done by comparing the strata of this action to those of the induced tangent space action inequivalent embeddings of a certain singular danielewski surface s in 3 are found we generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in 3 which are algebraically non isomorphic but holomorphically isomorphic

adaptive weighted suboptimal control for linear dynamic systems having a polynomial input introducing a cost term on the highest power of the nonlinear polynomial we derive a simple direct adaptive control weighted suboptimal control algorithm which is very useful for the control of all stable nonminimum phase and some unstable nonminimum phase linear systems having a polynomial input both stochastic and deterministic situations are considered and the global convergence of the proposed algorithm has been established

contact homology and virtual fundamental cycles we give a construction of contact homology in the sense of eliashberg givental hofer specifically we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo holomorphic curves the aim of this work is to provide a rigorous construction of contact homology an invariant of contact manifolds and exact symplectic cobordisms due to eliashberg givental hofer eli98 egh00 the contact homology of y is defined in terms of pseudo holomorphic curve counts in the sense of gromov gro85 in the symplectization r y unfortunately the moduli spaces of such curves often suffer from a severe lack of transversality and for this reason it has until now remained an important open problem to define the relevant curve counts in full generality we will use the framework developed in par15 to construct coherent virtual fundamental cycles on the moduli spaces in question thus giving rise to the desired curve counts our methods are quite general and should apply equally well to many other moduli spaces of interest previously a number of important special cases of contact homology and closely related invariants have been constructed rigorously using generic and or automatic transversality techniques cylindrical contact homology of some three manifolds was constructed by bao

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

adaptive control of a class of multivariable nonlinear systems and convergence analysis adaptive control is discussed of a class of multivariable nonlinear systems which can be characterized by a stochastic multivariable hammerstein model whose linear part possesses an arbitrary interactor matrix a simple suboptimal control law is derived which provides an efficient way to control a multivariable hammerstein model whose linear part is not necessarily minimum phase a direct adaption scheme is presented to implement the control law and the global convergence of the algorithm is established

stability analysis of a discrete time adaptive control algorithm having a polynomial input stability and convergence are shown for adaptive control of a cascade connection of a finite odd order polynomial followed by a linear system the main point is the establishment of the linear boundedness condition of the key technical lemma 7

cellular legendrian contact homology for surfaces part i abstract we give a computation of the legendrian contact homology lch dga for an arbitrary generic legendrian surface l in the 1 jet space of a surface as input we require a suitable cellular decomposition of the base projection of l a collection of generators is associated to each cell and the differential is given by explicit matrix formulas in the present article we prove that the equivalence class of this cellular dga does not depend on the choice of decomposition and in the sequel papers 35 36 we use this result to show that the cellular dga is equivalent to the usual legendrian contact homology dga defined via holomorphic curves extensions are made to allow legendrians with non generic cone point singularities we apply our approach to compute the lch dga for several examples including an infinite family and to give general formulas for dgas of front spinnings allowing for the axis of symmetry to intersect l

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

controller design oriented model identification method for hammerstein system abstract the paper proposes a new polynomial identification method for the hammerstein system by combining the results of parameter estimation theory and functions approximation theory the method circumvents many disadvantages of the earlier methods it is suitable for producing the controller design oriented model of hammerstein system consistency and convergency analysis are given for the obtained parameter estimates of the system s linear dynamics and approximating polynomial of its nonlinear memoryless element respectively simulation study verifies the effectiveness of the method