an extension of a bound for functions in sobolev spaces with applications to m s spline interpolation and smoothing given a function f on a bounded open subset of mathbb r n with a lipschitz continuous boundary we obtain a sobolev bound involving the values of f at finitely many points of overline omega this result improves previous ones due to narcowich et al math comp 74 743 763 2005 and wendland and rieger numer math 101 643 662 2005 we then apply the sobolev bound to derive error estimates for interpolating and smoothing m s splines in the case of smoothing noisy data as well as exact data are considered

blowing up kahler manifolds with constant scalar curvature ii in this paper we prove the existence of kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature kahler metric in the case where the manifold has nontrivial holomorphic vector fields with zeros we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature kahler metrics

spin holonomy of einstein manifolds berger s theorem classifies the linear holonomy groups of irreducible simply connected riemannian manifolds for physical applications however it is at least as important to have a classification of the possible spin holonomy groups defined by the parallel transport of spinors of non simply connected manifolds we establish a complete classification of the spin holonomy groups of all compact locally irreducible einstein riemannian spin manifolds of non negative scalar curvature

a new proof of the existence of kahler einstein metrics onk3 i on utilise la theorie des twisteurs pour donner une demonstration entierement differente de l existence de metriques de kahler einstein sur k 3

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

toric anti self dual einstein metrics via complex geometry using the twistor correspondence we give a classification of toric anti self dual einstein metrics each such metric is essentially determined by an odd holomorphic function this explains how the einstein metrics fit into the classification of general toric anti self dual metrics given in an earlier paper donaldson and fine in math ann 336 2 281 309 2006 the results complement the work of calderbank pedersen j differential geom 60 3 485 521 2002 who describe where the einstein metrics appear amongst the joyce spaces leading to a different classification taking the twistor transform of our result gives a new proof of their theorem

compact riemannian 7 manifolds with holonomy g sb 2 ii this is the second of two papers about metrics of holonomy g2 on compact 7manifolds in our first paper 15 we established the existence of a family of metrics of holonomy g2 on a single compact simply connected 7 manifold m using three general results theorems a b and c our purpose in this paper is to explore the theory of compact riemannian 7 manifolds with holonomy g2 in greater detail by relying on theorems a c we will be able to avoid the emphasis on analysis that characterized 15 so that this sequel will have a more topological flavour the paper has four chapters the first chapter consists of introductory material section 1 1 gives some elementary geometric and topological material on compact 7 manifolds with torsion free g2structures then 1 2 describes the holonomy groups su 2 and su 3 and 1 3 explains the concept of asymptotically locally euclidean riemannian manifolds shortened to ale spaces with special holonomy recall that in 15 a compact 7 manifold m was defined by desingularizing a quotient t of the 7 torus by a finite group of isometries z2 the subject of chapters 2 and 3 is a generalization of this idea chapter 2 defines a general construction for compact 7 manifolds with torsion free g2structures which works by desingularizing quotients t for finite groups the ale spaces with holonomy su 2 and su 3 discussed in 1 3 are an essential ingredient in performing this desingularization the central result of chapter 2 is theorem 2 2 3 which states that given a suitable finite group and certain other data one may construct a compact 7manifold m from t that admits torsion free g2structures this result is proved using theorems a c of 15 chapter 3 is devoted entirely to examples of this construction we give many examples of compact 7 manifolds with holonomy g2 and determine their basic topological invariants the betti numbers and fundamental group finally in chapter 4 we discuss some areas of interest and give a number of open problems this paper is not written to be read independently of 15 the language and results of 15 will be used freely in particular the introductory material in 15 1 1 for reference we reproduce here the model 3and 4forms defining the flat g2structure on r as given in 15 1 1

extremal metrics on blowups in this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of kahler manifolds which already carry an extremal metric as a particular case we construct extremal metrics on mathbb p 2 blown up k points in general position with k lt m 2

exponential stability for a class of boundary conditions on a euler bernoulli beam subject to disturbances via boundary control abstract this paper outlines the procedure for applying a sliding mode backstepping boundary control technique to a euler bernoulli beam subject to unknown bounded disturbances on the boundary for four different types of boundary conditions referred to in this paper as a class of boundary conditions these boundary conditions include all combinations of pinned and sliding types with control and disturbances on one boundary i e pinned pinned pinned sliding sliding pinned and sliding sliding with input on the second boundary the technique was developed for the specific case of a pinned pinned beam in the literature and in this paper the technique is generalized to the degree that it is possible to cover all four of the aforementioned cases furthermore a proof of the fully arbitrary exponential stability of the closed loop system is provided in this paper to this point only asymptotic stability was shown for the closed loop system for one case the technique outlined in this paper provides for a class of boundary conditions a controller that exponentially stabilizes the vibrations in a beam in the presence of bounded unknown disturbances the beam displacement returns to the origin and its decay is bounded by a known exponential decay function after a known finite time reaching phase

on the construction of some complete metrics with exceptional holonomy on construit explicitement trois metriques completes distinctes d holonomie egale a g 2 une metrique complete d holonomie egale a spin 7 et diverses metriques incompletes d holonomie exceptionnelle

infinite loop spaces and positive scalar curvature we study the homotopy type of the space of metrics of positive scalar curvature on high dimensional compact spin manifolds hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real k theory spectrum our main results concern the nontriviality of this map we prove that for 2n ge 6 the natural ko orientation from the infinite loop space of the madsen tillmann weiss spectrum factors up to homotopy through the space of metrics of positive scalar curvature on any 2n dimensional spin manifold for manifolds of odd dimension 2n 1 ge 7 we prove the existence of a similar factorisation when combined with computational methods from homotopy theory these results have strong implications for example the secondary index map is surjective on all rational homotopy groups we also present more refined calculations concerning integral homotopy groups to prove our results we use three major sets of technical tools and results the first set of tools comes from riemannian geometry we use a parameterised version of the gromov lawson surgery technique which allows us to apply homotopy theoretic techniques to spaces of metrics of positive scalar curvature secondly we relate hitchin s secondary index to several other index theoretical results such as the atiyah singer family index theorem the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem finally we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories the key new ingredient we use in this paper is the high dimensional analogue of the madsen weiss theorem proven by galatius and the third named author

a note on mirror symmetry for manifolds with spin 7 holonomy starting from the extended worldsheet superconformal algebras associated with g2 manifolds we extend the algebra to joyce s orbifolds with holonomy we show how the mirror symmetry in manifolds with holonomy arises as the automorphism in the extended sperconformal algebra in one class of joyce s orbifolds the automorphism of the superconformal algebra can be realized as 14 kinds of t dualities along the supersymmetric t4 toroidal fibrations in the manifolds with holonomy in this class of examples joyce s orbifolds are pairwise identified under the mirror symmetry we then discuss some interesting features of the mirror symmetry on the manifolds with exceptional holonomy and how it is different from the calabi yau mirror symmetry

sliding mode control of schrodinger equation ode in the presence of unmatched disturbances abstract in this paper we consider boundary stabilization for a cascade of schrodinger equation ode system with both matched and unmatched disturbances the backstepping method is first applied to transform the system into an equivalent target system where the target system is input to state stable to reject the matched disturbance the sliding mode control smc law is designed for the target system the well posedness of the closed loop system is proved and the reachability of the sliding manifold in finite time is justified by infinite dimensional system theory it is shown that the resulting closed loop system is input to state stable a numerical example illustrates the efficiency of the sliding mode design that reduces the ultimate bound of the closed loop system by rejecting the matched disturbance

compact 8 manifolds with holonomy spin 7 in berger s classification 4 of the possible holonomy groups of a nonsymmetric irreducible riemannian manifold there are two special cases the exceptional holonomy groups g2 in 7 dimensions and spin 7 in 8 dimensions bryant 6 proved that such metrics exist locally using the theory of exterior differential systems and gave some explicit examples later bryant and salamon 7 constructed explicit complete metrics with holonomy g2 and spin 7 in two previous papers 12 13 the author constructed many examples of compact riemannian 7 manifolds with holonomy g2 this paper will construct examples of compact riemannian 8 manifolds with holonomy spin 7 using similar methods we believe that these are the first examples known since metrics with holonomy spin 7 are ricci flat these are also new examples of compact ricci flat riemannian 8 manifolds let m be an 8 manifold a spin 7 structure on m can be encoded in a 4 form on m a special 4 form satisfying the condition that the stabilizer of at each point should be isomorphic to spin 7 by an abuse of notation we usually identify the spin 7 structure with its associated 4 form since spin 7 so 8 the spin 7 structure also induces a riemannian metric g and an orientation on m it turns out that the holonomy group hol g of g is a subgroup of spin 7 with spin 7 structure if and only if d 0 the quantity d is called the torsion of the spin 7 structure and is called torsion free if d 0 very briefly the plan of the paper splits into four steps as follows firstly a compact 8 manifold m is given secondly a spin 7 structure on m is found with small torsion i e d is small thirdly is deformed to a nearby spin 7 structure with d 0 thus is torsion free and if g is the associated metric then hol g spin 7 fourthly it is shown that hol g is spin 7 and not some proper subgroup the structure of the paper is designed around this division of the construction into four steps there are six chapters this first chapter is of introductory material the second chapter has the full statements of the main results in

toric self dual einstein metrics as quotients we use the quaternion kahler reduction technique to study old and new self dual einstein metrics of negative scalar curvature with at least a two dimensional isometry group and relate the quotient construction to the hyperbolic eigenfunction ansatz we focus in particular on the semi quaternion kahler quotients of semi quaternion kahler hyperboloids analysing the completeness and topology and relating them to the self dual einstein hermitian metrics of apostolov gauduchon and bryant

homogeneous contact manifolds and resolutions of calabi yau cones in the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds the main feature of our approach is to describe the cartan ehresmann connection gauge field for principal u 1 bundles over complex flag manifolds by using elements of representation theory of simple lie algebras this description allows us to compute explicitly the expression of the contact form for any boothby wang fibration over complex flag manifolds boothby and wang in ann math 68 721 734 1958 as well as their underlying sasaki structures by following conlon and hein duke math j 162 2855 2902 2013 van coevering math ann 2009 https doi org 10 1007 s00208 009 0446 1 and goto j math soc jpn 64 1005 1052 2012 as an application of our results we use the cartan remmert reduction grauert in math ann 146 331 368 1962 and the calabi ansatz technique calabi in ann sci ecole norm sup 4 12 269 294 1979 to provide many explicit examples of crepant resolutions of calabi yau cones with certain homogeneous sasaki einstein manifolds realized as links of isolated singularities these concrete examples illustrate the existence part of the conjecture introduced in martelli and sparks phys rev d 79 6 065009 2009

collapsing manifolds obtained by kummer type constructions we construct f structures on a bott manifold and on some other manifolds obtained by kummer type constructions we also prove that if m e x where e is a fiber bundle with structure group g and a fiber admitting a g invariant metric of non negative sectional curvature and x admits an f structure with one trivial covering then one can construct on m a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero i e vol k m 0 as a corollary we prove that all the elements in the spin cobordism ring can be represented by manifolds m with vol k m 0

a momentum construction for circle invariant kahler metrics examples of kahler metrics of constant scalar curvature are relatively scarce over the past two decades several workers in geometry and physics have used symmetry reduction to construct complete kahler metrics of constant scalar curvature by ode methods one fruitful idea the calabi ansatz is to begin with an hermitian line bundle p l h m g m over a kahler manifold and to search for kahler forms p m dd c f t in some disk subbundle where t is the logarithm of the norm function and f is a function of one variable our main technical result theorem a is the calculation of the scalar curvature for an arbitrary kahler metric g arising from the calabi ansatz this suggests geometric hypotheses which we call a constancy to impose upon the base metric g m and hermitian structure h in order that the scalar curvature of g be specified by solving an ode we show that constancy is necessary and sufficient for the calabi ansatz to work in the following sense under the assumption of constancy the disk bundle admits a one parameter family of complete kahler metrics of constant scalar curvature that restrict to g m on the zero section theorems b and d an analogous result holds for the punctured disk bundle theorem c a simple criterion determines when such a metric is einstein conversely in the absence of constancy the calabi ansatz yields at most one metric of constant scalar curvature in either the disk bundle or the punctured disk bundle theorem e many of the metrics constructed here seem to be new including a complete negative einstein kahler metric on the disk subbundle of a stable vector bundle over a riemann surface of genus at least two and a complete scalar flat kahler metric on c 2

bach flat asymptotically locally euclidean metrics we obtain a volume growth and curvature decay result for various classes of complete noncompact riemannian metrics in dimension 4 in particular our method applies to anti self dual or kahler metrics with zero scalar curvature and metrics with harmonic curvature similar results were obtained for einstein metrics in and89 bkn89 tia90 but our analysis differs from the einstein case in that 1 we consider more generally a fourth order system in the metric and 2 we do not assume any pointwise ricci curvature bound

an extended error analysis for a meshfree discretization method of darcy s problem recently d schrader and h wendland math comp 80 2011 pp 263 277 a new meshfree approximation method for darcy s problem has been introduced and analyzed this method is based on a symmetric collocation approach using radial basis functions producing solutions with an analytically divergence free velocity part however the error analysis provided in that paper works only for smooth solutions where the smoothness is intrinsically linked to the smoothness of the employed basis function in this paper we will extend the error analysis to less smooth functions showing that the approximation order for rougher solutions is determined by the smoothness of the solution rather than the smoothness of the basis function

on blowing up extremal kahler manifolds we show that the blowup of an extremal kahler manifold at a relatively stable point in the sense of git admits an extremal metric in kahler classes that make the exceptional divisor sufficiently small extending a result of arezzo pacard and singer we also study the k polystability of these blowups sharpening a result of stoppa in this case as an application we show that the blowup of a kahler einstein manifold at a point admits a constant scalar curvature kahler metric in classes that make the exceptional divisor small if it is k polystable with respect to these classes

discontinuous unit feedback control of uncertain infinite dimensional systems control systems driven by a discontinuous unit feedback in a hilbert space are studied the equation which describes a system motion taking place in the discontinuity manifold and further referred to as a sliding mode is derived by means of a special regularization technique based on the sliding mode equation the procedure of synthesis of a discontinuous unit control signal is developed restricted to a class of infinite dimensional systems with finite dimensional unstable part this procedure generates the control law which ensures desired dynamic properties as well as robustness of the closed loop system with respect to matched disturbances as an illustration of the capabilities of the procedure proposed a scalar unit controller of an uncertain exponentially minimum phase dynamic system is constructed and applied to heat processes and distributed mechanical oscillators

a high order analytically divergence free discretization method for darcy s problem we develop and analyze a meshfree discretization method for darcy s problem our approximation scheme is based upon optimal recovery which leads to a collocation scheme using divergence free positive definite kernels besides producing analytically incompressible flow fields our method can be of arbitrary order works in arbitrary space dimension and for arbitrary geometries after deriving the scheme we investigate the approximation error for smooth target functions and derive optimal approximation orders finally we illustrate the method by giving numerical examples

metrics of horowitz myers type with the negative constant scalar curvature we construct a one parameter family of complete metrics of horowitz myers type with the negative constant scalar curvature we also verify a positive energy conjecture of horowitz and myers for these metrics

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

lyapunov approach to output feedback stabilization for the euler bernoulli beam equation with boundary input disturbance we propose a boundary output feedback control law for a one dimensional euler bernoulli beam equation with general external disturbance entering the control end a galerkin approximation scheme is constructed to show the existence of solution to the closed loop system the exponential stability of the closed loop system is obtained by the lyapunov functional method numerical simulations are presented for illustration

counter examples to the generalized positive action conjecture we give examples of complete locally asymptotically flat riemannian 4 manifolds with zero scalar curvature and negative mass the generalized positive action conjecture of hawking and pope 5 is therefore false

ale scalar flat kahler metrics on non compact weighted projective spaces we construct new explicit toric scalar flat kahler ale metrics with orbifold singularities which we use to obtain smooth extremal kahler metrics on appropriate resolutions of orbifolds

calabi yau metrics on canonical bundles of complex flag manifolds abstract in the present paper we provide a description of complete calabi yau metrics on the canonical bundle of generalized complex flag manifolds by means of lie theory we give an explicit description of complete ricci flat kahler metrics obtained through the calabi ansatz technique we use this approach to provide several explicit examples of noncompact complete calabi yau manifolds these examples include canonical bundles of non toric flag manifolds e g grassmann manifolds and full flag manifolds

on conformally kahler einstein manifolds we prove that any compact complex surface with c1 0 admits an einstein metric which is conformally related to a kahler metric the key new ingredient is the existence of such a metric on the blow up cp2 2cp2 of the complex projective plane at two distinct points

exact controllability of the euler bernoulli equation with boundary controls for displacement and moment abstract we consider the euler bernoulli problem 1 1 in the solution w t with boundary controls g1 and g2 acting in the dirichlet traces for w and w respectively we show two main exact controllability results both in the spaces of maximal regularity of the solution at t t see i lasiecka and r triggiani regularity theory for a class of euler bernoulli equations a cosine operator approach boll unione mat ital 7 3 b 1989 and both without geometrical conditions on the open bounded domain except for smoothness of its boundary one with g1 l2 0 t l2 and g2 h1 0 t l2 for any t 0 arbitrarily short and one with g1 h01 0 t l2 and g2 l2 0 t l2 for all t 0 sufficiently large an interpolation result between these two cases is also presented a direct approach is given based on two main steps first by means of an operator model i lasiecka and r triggiani regularity theory for a class of euler bernoulli equations a cosine operator approach boll unione mat ital 7 3 b 1989 for problem 1 1 and a functional analytic approach the question of exact controllability is shown to be equivalent to an a priori inequality for the corresponding homogeneous problem next this a priori inequality is proved to hold true by means of multiplier techniques these are inspired by recent progress on the maximal regularity exact controllability and uniform stabilization questions for second order hyperbolic equations

existence of parallel spinors on nonsimply connected riemannian manifolds it is well known and important for applications that ricci flat riemannian manifolds of nongeneric holonomy always admit a parallel covariant constant spinor if they are simply connected the nonsimply connected case is much more subtle however we show that a parallel spinor can still be found in this case provided that the real dimension is not a multiple of four and provided that the spin structure is carefully chosen

vibration damping in beams via piezo actuation using optimal boundary control open loop optimal control theory is formulated and applied to damp out the vibrations of a beam where the control action is implemented using piezoceramic actuators the optimal control law is derived by using a maximum principle developed for one dimensional structures where the control function appears in the boundary conditions in the form of a moment the objective function is specified as a weighted quadratic functional of the displacement and velocity which is to be minimized at a specified terminal time using continuous piezoelectric actuators the expenditure of control force is included in the objective functional as a penalty term the explicit solution of the problem is developed for cantilever beams using eigenfunction expansions of the state and adjoint variables the effectiveness of the proposed control mechanism is assessed by plotting the displacement and velocity against time it is shown that both quantities are damped out substantially as compared to an uncontrolled beam and this reduction depends on the magnitude of the control moment the capabilities of piezo actuation are also investigated by means of control moment versus piezo and beam thickness graphs which indicate the required minimum level of voltage to be applied on piezo materials in relation to geometric dimensions of the combined active passive structure the graphs show the magnitude of the control moment which can be achieved using piezoceramics in terms of problem inputs such as voltage piezo and beam thicknesses

on the geometry and mass of static asymptotically ads spacetimes and the uniqueness of the ads soliton we prove two theorems announced in 6 for static spacetimes that solve einstein s equation with negative cosmological constant the first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity analogous to the structure theorems of cheeger and gromoll for manifolds of non negative ricci curvature for spacetimes with ricci flat conformal boundary the convexity condition is associated with negative mass the second theorem is a uniqueness theorem for the negative mass ads soliton spacetime this result lends support to the new positive mass conjecture due to horowitz and myers which states that the unique lowest mass solution which asymptotes to the ads soliton is the soliton itself this conjecture was motivated by a nonsupersymmetric version of the ads cft correspondence our results add to the growing body of rigorous mathematical results inspired by the ads cft correspondence conjecture our techniques exploit a special geometric feature which the universal cover of the soliton spacetime shares with familiar ground state spacetimes such as minkowski spacetime namely the presence of a null line or complete achronal null geodesic and the totally geodesic null hypersurface that it determines en route we provide an analysis of the boundary data at conformal infinity for the lorentzian signature static einstein equations in the spirit of the fefferman graham analysis for the riemannian signature case this leads us to generalize to arbitrary dimension a mass definition for static asymptotically ads spacetimes given by chru ciel and simon we prove equivalence of this mass definition with those of ashtekar magnon and hawking horowitz

two conjectures on ricci flat kahler metrics we propose two conjectures about ricci flat kahler metrics conjecture 1 a ricci flat projectively induced metric is flat conjecture 2 a ricci flat metric on an n dimensional complex manifold such that the a n 1 coefficient of the tyz expansion vanishes is flat we verify conjecture 1 see theorem 1 1 under the assumptions that the metric is radial and stable projectively induced and conjecture 2 see theorem 1 2 for complex surfaces whose metric is either radial or complete and ale we end the paper by showing by means of the simanca metric that the assumption of ricci flatness in conjecture 1 and in theorem 1 2 cannot be weakened to scalar flatness see theorem 1 3

critical metrics on connected sums of einstein four manifolds abstract we develop a gluing procedure designed to obtain canonical metrics on connected sums of einstein four manifolds the main application is an existence result using two well known einstein manifolds as building blocks the fubini study metric on cp 2 and the product metric on s 2 s 2 using these metrics in various gluing configurations toric invariant critical metrics are found on connected sums for a specific riemannian functional which depends on the global geometry of the factors furthermore using certain quotients of s 2 s 2 as one of the gluing factors critical metrics on several non simply connected manifolds are also obtained

generalized hermite interpolation via matrix valued conditionally positive definite functions in this paper we consider a broad class of interpolation problems for both scalarand vector valued multivariate functions subject to linear side conditions such as being divergence free where the data are generated via integration against compactly supported distributions we show that by using certain families of matrix valued conditionally positive definite functions such interpolation problems are well poised that is the interpolation matrices are invertible as a sample result we show that a divergence free vector field can be interpolated by a linear combination of convolutions of the data generating distributions with a divergence free 3 x 3 matrix valued conditionally positive definite function in addition we obtain norm estimates for inverses of interpolation matrices that arise in a class of multivariate hermite interpolation problems

arbitrary decay rate for euler bernoulli beam by backstepping boundary feedback we consider a problem of stabilization of the euler bernoulli beam the beam is controlled at one end using position and moment actuators and has the ldquoslidingrdquo boundary condition at the opposite end we design the controllers that achieve any prescribed decay rate of the closed loop system removing a long standing limitation of classical ldquoboundary damperrdquo controllers the idea of the control design is to use the well known representation of the euler bernoulli beam model through the schrodinger equation and then adapt recently developed backstepping designs for the latter in order to stabilize the beam we derive the explicit integral transformation and its inverse of the closed loop system into an exponentially stable target system the transformation is of a novel volterra fredholm type the design is illustrated with simulations

sobolev type approximation rates for divergence free and curl free rbf interpolants recently error estimates have been made available for divergence free radial basis function rbf interpolants however these results are only valid for functions within the associated reproducing kernel hilbert space rkhs of the matrix valued rbf functions within the associated rkhs also known as the native space of the rbf can be characterized as vector fields having a specific smoothness making the native space quite small in this paper we develop sobolev type error estimates when the target function is less smooth than functions in the native space

meshless collocation error estimates with application to dynamical systems in this paper we derive error estimates for generalized interpolation in particular collocation in sobolev spaces we employ our estimates in collocation problems using radial basis functions and extend and improve previously known results for elliptic problems finally we use meshless collocation to approximate lyapunov functions for dynamical systems

blowing up extremal kahler manifolds ii this is a continuation of the work of arezzo pacard singer and the author on blowups of extremal kahler manifolds we prove the conjecture stated in szekelyhidi duke math j 161 8 1411 1453 2012 and we relate this result to the k stability of blown up manifolds as an application we prove that if a kahler manifold m of dimension 2 admits a constant scalar curvature csck metric then the blowup of m at a point admits a csck metric if and only if it is k stable as long as the exceptional divisor is sufficiently small

brief paper sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties this paper considers the stabilization problem of a one dimensional unstable heat conduction system rod modeled by a parabolic partial differential equation pde powered with a dirichlet type actuator from one of the boundaries by applying the volterra integral transformation a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties the associated lyapunov function is used for designing an infinite dimensional sliding manifold on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances it is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero a continuous control law satisfying the reaching condition is obtained by passing a discontinuous signum signal through an integrator

the active disturbance rejection and sliding mode control approach to the stabilization of the euler bernoulli beam equation with boundary input disturbance in this paper we are concerned with the boundary feedback stabilization of a one dimensional euler bernoulli beam equation with the external disturbance flowing to the control end the active disturbance rejection control adrc and sliding mode control smc are adopted in investigation by the adrc approach the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed loop it is shown that the external disturbance can be attenuated in the sense that the resulting closed loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity in the second part we use the smc to reject the disturbance by removing the condition in adrc that the derivative of the disturbance is supposed to be bounded the existence and uniqueness of the solution for the closed loop via smc are proved and the monotonicity of the reaching condition is presented without the differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system the numerical simulations validate the effectiveness of both methods

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

parallel spinors and holonomy groups in this paper we complete the classification of spin manifolds admitting parallel spinors in terms of the riemannian holonomy groups more precisely we show that on a given n dimensional riemannian manifold spin structures with parallel spinors are in one to one correspondence with lifts to spinn of the riemannian holonomy group with fixed points on the spin representation space in particular we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors

a mass for alf manifolds we prove a positive mass theorem on alf manifolds i e complete noncompact manifolds that are asymptotic to a circle fibration over a euclidean base with fibers of asymptotically constant length

mass in kahler geometry we prove a simple explicit formula for the mass of any asymptotically locally euclidean ale kahler manifold assuming only the sort of weak fall off conditions required for the mass to actually be well defined for ale scalar flat kahler manifolds the mass turns out to be a topological invariant depending only on the underlying smooth manifold the first chern class of the complex structure and the kahler class of the metric when the metric is actually ae asymptotically euclidean our formula not only implies a positive mass theorem for kahler metrics but also yields a penrose type inequality for the mass

equivariant connected sums of compact self dual manifolds due to a recent theorem of taubes self dual conformal structures on compact four dimensional manifolds exist in abundance 43 therefore finding examples in general is no longer an issue in this subject to deepen our understanding of self dual conformal geometry we take the path of trying to understand the possible symmetry groups of this geometry i e the groups of orientation preserving conformal transformations the first systematic approach which constructs a large collection of compact self dual manifolds with a non trivial group of symmetries is lebrun s hyperbolic ansatz 22 the condition of this ansatz is that the symmetry group contains an s l u 1 which acts on the manifold semi freely meaning that the isotropy group at any point of the manifold is either trivial or the entire group s i with this condition on the group action and some favourable topological conditions on the underlying manifold the geometrical equations of self duality on the given four dimensional manifold are reduced to the s 1 monopole equations on hyperbolic 3 space with this setup lebrun explicitly constructed a large number of compact self dual manifolds with semi free s l symmetry including the fubinistudy metric on the complex projective plane ci 2 and a one parameter family of self dual metrics on c c first found in 38 we shall call the self dual metrics on nci 2 n 1 constructed by lebrun s hyperbolic ansatz the lebrun metrics after this very successful ansatz it is natural to try to find new examples which are beyond the scope of that method

dynamic boundary control of a euler bernoulli beam a flexible beam clamped to a rigid base at one end and free at the other end is considered to stabilize the beam vibrations a dynamic boundary force control and a dynamic boundary torque control applied at the free end of the beam are proposed it is proved that with the proposed controls the beam vibrations decay exponentially the proof uses a lyapunov functional based on the energy functional of the system

backstepping approach to the arbitrary decay rate for euler bernoulli beam under boundary feedback in this article we are concerned with the boundary stabilisation of the euler bernoulli beam equation for which all eigenvalues of the control free system are located on the imaginary axis of the complex plane the fourth order system in spacial variable is transformed into a coupled heat like system this enables us to make a natural backstepping transformation in vector form to transform the system into a target system which has arbitrary decay rate the state feedback is thus designed it is shown that the original closed loop system is exponentially stable with the given arbitrary decay rate

gluing eguchi hanson metrics and a question of page in 1978 gibbons pope and page proposed a physical picture for the ricci flat kahler metrics on the k3 surface based on a gluing construction in this construction one starts from a flat torus with 16 orbifold points and resolves the orbifold singularities by gluing in 16 small eguchi hanson manifolds that all have the same orientation this construction was carried out rigorously by topiwala lebrun singer and donaldson in 1981 page asked whether the above construction can be modified by reversing the orientations of some of the eguchi hanson manifolds this is a subtle question if successful this construction would produce einstein metrics that are neither kahler nor self dual in this paper we focus on a configuration of maximal symmetry involving eight small eguchi hanson manifolds of each orientation that are arranged according to a chessboard pattern by analyzing the interactions between eguchi hanson manifolds with opposite orientation we identify a nonvanishing obstruction to the gluing problem thereby destroying any hope of producing a metric of zero ricci curvature in this way using this obstruction we are able to understand the dynamics of such metrics under ricci flow as long as the eguchi hanson manifolds remain small in particular for the configuration described above we obtain an ancient solution to the ricci flow with the property that the maximum of the riemann curvature tensor blows up at a rate of t 1 2 while the maximum of the ricci curvature converges to 0 2016 wiley periodicals inc

sliding mode boundary control of an euler bernoulli beam subject to disturbances in this paper a control technique is presented for an undamped pinned pinned euler bernoulli beam with control inputs and bounded disturbances on one boundary the control strategy drives the system to its origin at an arbitrary exponential rate in the presence of the disturbances this is achieved in two main steps first a backstepping transformation is used to convert the marginally stable euler bernoulli beam system to a new form that has an exponentially stable homogeneous form control inputs are needed to fully convert the system to this form however since they are distorted by unknown bounded disturbances the next step implements a sliding mode controller to account for them the proposed sliding manifolds require a combination of classical and second order techniques in order to avoid discontinuous chattering on the physical system therefore the continuous sliding mode controllers developed return the beam to its origin at an arbitrary exponential rate and do so in the presence of unknown bounded disturbances on the boundary the main contributions of this paper with respect to previous backstepping designs for the euler bernoulli beam are that all three of the following goals are accomplished together i steady state position is the origin ii decay rate has no theoretical restrictions and iii is robust to bounded disturbances

einstein metrics and complex singularities this paper is concerned with the construction of special metrics on non compact 4 manifolds which arise as resolutions of complex orbifold singularities our study is close in spirit to the construction of the hyperkahler gravitational instantons but we focus on a different class of singularities we show that any resolution x of an isolated cyclic quotient singularity admits a complete scalar flat kahler metric which is hyperkahler if and only if k x is trivial and that if k x is strictly nef then x also admits a complete non kahler self dual einstein metric of negative scalar curvature in particular complete self dual einstein metrics are constructed on simply connected non compact 4 manifolds with arbitrary second betti number

selfdual einstein metrics with torus symmetry it is well known that any 4 dimensional hyperkahler metric with two commuting killing fields may be obtained explicitly via the gibbons hawking ansatz from a harmonic function invariant under a killing field on 3 in this paper we find all selfdual einstein metrics of nonzero scalar curvature with two commuting killing fields they are given explicitly in terms of a local eigenfunction of the laplacian on the hyperbolic plane we discuss the relation of this construction to a class of selfdual spaces found by joyce and some einstein weyl spaces found by ward and then show that certain multipole hyperbolic eigenfunctions yield explicit formulae for the quaternion kahler quotients of pm 1 by an m 2 torus studied by galicki and lawson as a consequence we are able to place the well known cohomogeneity one metrics the quaternion kahler quotients of p2 and noncompact analogues and the more recently studied selfdual einstein hermitian metrics in a unified framework and give new complete examples

non minimal scalar flat kahler surfaces and parabolic stability a new construction is presented of scalar flat kahler metrics on non minimal ruled surfaces the method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank 2 parabolically stable holomorphic bundles this rather general construction is shown also to give new examples of low genus in particular it is shown that mathbb cp 2 blown up at 10 suitably chosen points admits a scalar flat kahler metric this answers a question raised by claude lebrun in 1986 in connection with the classification of compact self dual 4 manifolds

hamiltonian constructions of kahler einstein metrics and kahler metrics of constant scalar curvature assuming the existence of a real torus acting through holomorphic isometries on a kahler manifold we construct an ansatz for kahler einstein metrics and an ansatz for kahler metrics with constant scalar curvature using this hamiltonian approach we solve the differential equations in special cases and find in particular a family of constant scalar curvature kahler metrics describing a non linear superposition of the bergman metric the calabi metric and a higher dimensional generalization of the lebrun kahler metric the superposition contains kahler einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spacepn we also build such kahler geometries on kahler quotients of higher cohomogeneity