applications of the ashtekar gravity to four dimensional hyperkahler geometry and yang mills instantons the ashtekar jacobson smolin mason newman equations are used to construct the hyperkahler metrics on four dimensional manifolds these equations are closely related to anti self dual yang mills equations of the infinite dimensional gauge lie algebras of all volume preserving vector fields several examples of hyperkahler metrics are presented through the reductions of anti self dual connections for any gauge group anti self dual connections on hyperkahler manifolds are constructed using the solutions of both nahm and laplace equations

toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions in the present work some examples of toric hyperkahler metrics in eight dimensions are constructed first it is described how the calderbank pedersen metrics arise as a consequence of the joyce description of selfdual structures in four dimensions the jones tod correspondence and a result due to tod and przanowski it is also shown that any quaternionic kahler metric with t2 isometry is locally isometric to a calderbank pedersen one the swann construction of hyperkahler metrics in eight dimensions is applied to them to find hyperkahler examples with u 1 u 1 isometry the connection with the pedersen poon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation related to eigenfunctions of a certain linear equation these hyperkahler examples are lifted to solutions of the d 11 supergravity and type iia and iib backgrounds are found by use of dualities as before all the description is achieved in terms of a single eigenfunction f some explicit f are found together with the toda structure corresponding to the trajectories of the killing vectors of the calderbank pedersen bases

a class of nonlinear wave equations containing the continuous toda case we consider a nonlinear field equation which can be derived from a binomial lattice as a continuous limit this equation containing a perturbative friction like term and a free parameter reproduces the toda case in the absence of the friction like term and other equations of physical interest by choosing particular values of we apply the symmetry and the approximate symmetry approach and the prolongation technique our main aim is to check the limits of validity of different analytical methods in the study of nonlinear field equations we show that the equation under investigation with the friction like term is characterized by a finite dimensional lie algebra admitting a realization in terms of boson annhilation and creation operators in the absence of the friction like term the equation is linearized and connected with bessel type equations examples of exact solutions are displayed and the algebraic structure of the equation is discussed

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

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

optimal design of elastic plates with a constraint on the slope of the thickness function abstract a constraint on the maximum absolute value of the slope of the thickness function is found to have a profound influence on the problem of optimal design of axisymmetric elastic plates by securing a unique regular solution it is also shown that except for the case of axisymmetric deformation the constraint is active everywhere and the optimal design is a plate subdivided into annular regions of constant slope equal to the maximum value permitted

metric based mathematical derivation of efficient airfoil design variables within an aerodynamic shape optimization framework an efficient shape parameterization and deformation scheme is critical to allow flexible deformation of the surface with the maximum possible design space coverage numerous approaches have been developed for the geometric representation of airfoils a fundamental approach is considered here from the geometric perspective and a method is presented to allow the derivation of efficient generic and orthogonal airfoil geometric design variables this is achieved by the mathematical decomposition of a training library the resulting geometric modes are independent of a parameterization scheme surface and volume mesh and flow solver thus they are generally applicable however these modes are dependent on the training library and so a benchmark performance measure called the airfoil technology factor has also been incorporated into the scheme to allow intelligent metric based filtering or design space reduction of the training library to ensure eff

efficient and exact mesh deformation using multiscale rbf interpolation abstract radial basis function rbf interpolation is popular for mesh deformation due to robustness and generality but the cost scales with the number of surface points sourcing the deformation as o n s 3 hence there have been numerous works investigating efficient methods using reduced datasets however although reduced data methods are efficient they require a secondary method to treat an error vector field to ensure surface points not included in the primary deformation are moved to the correct location and the volume mesh moved accordingly a new method is presented which captures global and local motions at multiple scales using all the surface points and so no correction stage is required all surface points are used and a single interpolation built but the cost and conditioning issues associated with rbf methods are eliminated moreover the sparsity introduced is exploited using a wall distance function to further reduce the cost the method is compared to an efficient greedy method and it is shown mesh quality is always comparable with or better than with the greedy method and cost is comparable or cheaper at all stages surface mesh preprocessing is the dominant cost for reduced data methods and this cost is reduced significantly here greedy methods select points to minimise interpolation error requiring repeated system solution and cost o n r e d 4 to select n r e d points the multiscale method has no error and the problem is transferred to a geometric search with cost o n s log n s resulting in an eight orders of magnitude cost reduction for three dimensional meshes furthermore since the method is dependent on geometry not deformation it only needs to be applied once prior to simulation as the mesh deformation is decoupled from the point selection process

non moishezon twistor spaces of 4 mathbf cp 2 with non trivial automorphism group we show that a twistor space of a self dual metric on 4cp 2 with u 1 isometry is not moishezon iff there is a c orbit biholomorphic to a smooth elliptic curve where the c action is the complexification of the u 1 action on the twistor space it follows that the u 1 isometry has a two sphere whose isotropy group is z 2 we also prove the existence of such twistor spaces in a strong form to show that a problem of campana and kreusler is affirmative even though a twistor space is required to have a non trivial automorphism group

cross validation of aerodynamic shape optimization methodologies for aircraft wing body optimization this paper presents the cross validation of two aerodynamic shape optimization methodologies in order to validate characterize and compare the two methodologies both methodologies use gradient b

anti self dual four manifolds with a parallel real spinor antiselfdual metrics in the signature that admit a covariantly constant real spinor are studied it is shown that finding such metrics reduces to solving a fourthorder integrable partial differ

shape optimal design using b splines abstract shape optimal design of an elastic structure is formulated using a design element technique it is shown that bezier and b spline curves typical of the cad philosophy are well suited to the definition of design elements complex geometries can be described in a very compact way by a small set of design variables and a few design elements because of the b splines flexibility it is no longer necessary to piece design elements together in order to agree with the shape complexity nor to restrict the shape variations moreover the additional optimization constraints that are most often needed to avoid unrealistic designs when the shape variables are the nodal coordinates of a finite element mesh are automatically taken into account in the new formulation an analytical derivation of the sensitivity analysis will be established giving rise to numerical efficiency it will be seen that the resulting optimization problem does not involve highly nonlinear functions with respect to the shape variables so that simple mathematical programming algorithms can be applied to solve it some numerical examples are offered to demonstrate the power and generality of the new approach presented in this paper

algebraic dimension of twistor spaces si l espace twisteur d une variete compacte autoduale est moishezon la classe conforme autoduale contient une metrique de courbure scalaire positive

on twistor spaces of anti self dual hermitian surfaces we consider a complex surface m with anti self dual hermitian metric h and study the holomorphic properties of its twistor space z we show that the naturally defined divisor line bundle x is isomorphic to the 1 2 power of the canonical bundle of z if and only if there is a kahler metric of zero scalar curvature in the conformal class of h this has strong consequences on the geometry of m which were also found by c boyer 3 using completely different methods we also prove the existence of a very close relation between holomorphic vector fields on m and z in the case that m is compact and kahler

the vertex morphing method for node based shape optimization abstract the vertex morphing method a consistent surface control approach for shape optimization is presented this method introduces a control field and a map that relates it to the geometry field the optimization problem is solved in the control space and the map is defined such that both surface smoothness and mesh regularity criteria are satisfied simultaneously and synchronously in the geometry field the control field is discretized with the same discretization as the one of the geometry which leads to the richest design space possible for optimization of a discrete geometry the capabilities of the vertex morphing method are evaluated within an adjoint based cfd shape optimization workflow several engineering cases mainly from automotive industry are tested the method has shown significant success in performance efficiency and robustness even for problems with extreme geometrical complexities and millions of design parameters for applications in structural as well as fluid mechanics and their combination

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

moduli spaces of critical riemannian metrics in dimension four abstract we obtain a compactness result for various classes of riemannian metrics in dimension four in particular our method applies to anti self dual metrics kahler metrics with constant scalar curvature and metrics with harmonic curvature with certain geometric non collapsing assumptions the moduli space can be compactified by adding metrics with orbifold like singularities similar results were obtained for einstein metrics in j amer math soc 2 3 1989 455 invent math 97 2 1989 313 invent math 101 1 1990 101 but our analysis differs substantially from the einstein case in that we do not assume any pointwise ricci curvature bound

aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement an efficient gradient based algorithm for aerodynamic shape optimization is presented the algorithm consists of several components including a novel integrated geometry parameterization and mesh movement a parallel newton krylov flow solver and an adjoint based gradient evaluation to integrate geometry parameterization and mesh movement generalized b spline volumes are used to parameterize both the surface and volume mesh the volume mesh of b spline control points mimics a coarse mesh a linear elasticity mesh movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically using this approach mesh movement time is reduced by two to three orders of magnitude relative to a node based movement the mesh adjoint system also becomes smaller and is thus amenable to complex step derivative approximations when solving the flow adjoint equations using restarted krylov subspace methods a nested subspace strategy is shown to be more robust than truncating the entire subspace optimization is accomplished using a sequential quadratic programming algorithm the effectiveness of the complete algorithm is demonstrated using a lift constrained induced drag minimization that involves large changes in geometry

singular hyperbolic monopoles we study the twistor theory of singular hyperbolic su 2 monopoles following the approach taken by kronheimer 9 in the euclidean case we use our results to show that the moduli space of charge 1 monopoles possesses a natural 2 sphere of scalar flat kahler metrics in the zero mass limit the metrics reduce to a class of metrics first studied by lebrun in 10

twistor cr manifolds and three dimensional conformal geometry a cr i e partially complex 5 manifold is contructed as a sphere bundle over an arbitrary 3 manifold with conformal metric this so called twistor cr manifold is show to capture completely the original geometry and necessary and sufficient conditions are given for an abstract cr manifold to arise via the construction the above correspondence is then used to prove that a twistor cr manifold is locally imbeddable as a real hypersurface in c3 only if it is real analytic with respect to a suitable atlas introduction there is a very beautiful interplay between the conformal geometry of riemannian manifolds and complex analysis examples of which are found both in the classical theorem asserting the existence of isothermal coordinates on a surface and in the theory of self dual riemannian 4 manifolds penrose 1976 atiyah et al 1978 intermediating between these two theories is a relationship between 3dimensional conformal geometry and cr partially complex manifolds of dimension 5 in this paper we will explain this relationship focusing on the imbedding problem when can these partially complex manifolds be realized as real hypersurfaces in complex 3 manifolds the idea of associating a cr manifold with a conformal riemannian 3 manifold originated with penrose 1975 1983 who starting with a real analytic space like hypersurface in a real analytic lorentzian 4 manifold constructed a real hypersurface in a complex 3 manifold if the space like hypersurface is totally geodesic or merely all umbilic penroses s cr manifold coincides in our terminology with the twistor cr manifold of the conformal structure induced on the space like hypersurface after establishing notation and conventions in 0 principal results on cr manifolds are reviewed and the notions of cr vector bundle and cr contact form are introduced in 1 a general machine for constructing involutive distributions is produced and in 2 this construction is used to associate to every conformal riemannian 3 manifold a cr 5 manifold the cr manifolds arising in this way are characterized abstractly in 4 in terms of properties presented in 3 then in 5 it is shown that these cr manifolds are isomorphic to real hypersurfaces of complex received by the editors march 15 1983 and in revised form july 8 1983 1980 mathematics subject classificcation primary 32f25 secondary 53a30 83c99 work supported in part by nsf grant mcs 812 0790 001984 american mathematical society 0002 9947 84 1 00 25 per page

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

double solid twistor spaces the case of arbitrary signature in a recent paper 9 we constructed a series of new moishezon twistor spaces which are a kind of variant of the famous lebrun twistor spaces in this paper we explicitly give projective models of another series of moishezon twistor spaces on ncp2 for arbitrary n 3 which can be regarded as a generalization of the twistor spaces of double solid type on 3cp2 studied by kreusler kurke poon and the author similarly to the twistor spaces of double solid type on 3cp2 projective models of the present twistor spaces have a natural structure of double covering of a cp2 bundle over cp1 we explicitly give a defining polynomial of the branch divisor of the double covering whose restriction to fibers is degree four if n 4 these are new twistor spaces to the best of the author s knowledge we also compute the dimension of the moduli space of these twistor spaces differently from 9 the present investigation is based on analysis of pluri half anticanonical systems of the twistor spaces

deformation spaces associated to compact hyperbolic manifolds in this paper we take a first step toward understanding representations of cocompact lattices in so n 1 into arbitrary lie groups by studying the deformations of rational representations see proposition 5 1 for a rather general existence result this proposition has a number of algebraic applications for example we remark that such deformations show that the margulis super rigidity theorem see 30 cannot be extended to the rank 1 case we remark also that if so n 1 is one of the standard arithmetic examples described in section 7 then has a faithful representation in so n 1 the galois conjugate of the uniformization representation and proposition 5 1 may be used to deform the direct sum of and the trivial representation in so n 2 thereby constructing non trivial families of irreducible orthogonal representations of however most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain

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

gradient limiting shape control for efficient aerodynamic optimization local shape control methods such as b spline surfaces are well conditioned such that they allow high fidelity design optimization however this comes at the cost of degraded optimization converg

mobius transformations in noncommutative conformal geometry we study the projective linear group pgl 2 a associated with an arbitrary algebra a and its subgroups from the point of view of their action on the space of involutions in a this action formally resembles mobius transformations known from complex geometry by specifying a to be an algebra of bounded operators in a hilbert space h we rediscover the mobius group ev m defined by connes and study its action on the space of fredholm modules over the algebra a there is an induced action on the k homology of a which turns out to be trivial moreover this action leads naturally to a simpler object the polarized module underlying a given fredholm module and we discuss this relation in detail any polarized module can be lifted to a fredholm module and the set of different lifts forms a category whose morphisms are given by generalized mobius tranformations we present an example of a polarized module canonically associated with the differentiable structure of a smooth manifold v using our lifting procedure we obtain a class of fredholm modules characterizing the conformal structures on v fredholm modules obtained in this way are a special case of those constructed by connes sullivan and teleman

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

shape optimisation with multiresolution subdivision surfaces and immersed finite elements abstract we develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness during shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information by virtue of the multiresolution editing semantics updating the coarse control mesh vertex coordinates leads to large scale geometry changes and conversely updating the fine control mesh coordinates leads to small scale geometry changes in our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum this approach effectively prevents the appearance of non physical boundary geometry oscillations and control mesh pathologies like inverted elements independent of the fineness of the control mesh used for optimisation on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution with the immersed finite element method there is no need to maintain an analysis suitable domain mesh in some of the presented two and three dimensional elasticity examples the topology derivative is used for introducing new holes inside the domain the merging or removing of holes is not considered

generalized kaluza klein monopole quadratic algebras and ladder operators we present a generalized kaluza klein monopole system we solve this quantum superintegrable system on a euclidean taub nut manifold using the separation of variables of the corresponding schr odinger equation in spherical and parabolic coordinates we present the integrals of motion of this system the quadratic algebra generated by these integrals the realization in terms of a deformed oscillator algebra using the daskaloyannis construction and the energy spectrum the structure constants and thecasimir operator are functions not only of the hamiltonian but also of other two integrals commuting with all generators of the quadratic algebra and forming an abelian subalgebra we present another algebraic derivation of the energy spectrum of this system using the factorization method and ladder operators

a new approach to optimal designs for correlated observations this paper presents a new and efficient method for the construction of optimal designs for regression models with dependent error processes in contrast to most of the work in this field which starts with a model for a finite number of observations and considers the asymptotic properties of estimators and designs as the sample size converges to infinity our approach is based on a continuous time model we use results from stochastic anal ysis to identify the best linear unbiased estimator blue in this model based on the blue we construct an efficient linear estimator and corresponding optimal designs in the model for finite sample size by minimizing the mean squared error between the opti mal solution in the continuous time model and its discrete approximation with respect to the weights of the linear estimator and the optimal design points in particular in the multi parameter case in contrast to previous work on the subject the resulting estimators and corresponding optimal designs are very efficient and easy to implement this means that they are practi cally not distinguishable from the weighted least squares estimator and the corresponding optimal designs which have to be found numerically by non convex discrete optimization the advantages of the new approach are illustrated in several numerical examples

comparison of b spline surface and free form deformation geometry control for aerodynamic optimization two aerodynamic shape optimization geometry control methods b spline surface control and free form deformation are applied to three optimization problems and compared on the bases of optimal shap

optimal design for linear models with correlated observations in the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated a necessary condition for the optimality of a given design is provided which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data if the regression functions are eigenfunctions of an integral operator defined by the covariance kernel it is shown that the corresponding measure defines a universally optimal design for several models universally optimal designs can be identified explicitly in particular it is proved that the uniform distribution is universally optimal for a class of trigonometric regression models with a broad class of covariance kernels and that the arcsine distribution is universally optimal for the polynomial regression model with correlation structure defined by the logarithmic potential to the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations which are not restricted to the location scale model

a geometric comparison of aerofoil shape parameterisation methods a comprehensive review of aerofoil shape parameterization methods that can be used for aerodynamic shape optimization is presented seven parameterization methods are considered for a range of desi

two level free form and axial deformation for exploratory aerodynamic shape optimization an intuitive shape parameterization and control technique suitable for high fidelity aerodynamic shape optimization is presented it relies on the principles of free form and axial deformation enabling thorough exploration of the design space while keeping the number of design variables manageable surface sensitivities to the design variables are readily available their inclusion in a highly efficient and robust adjoint based optimization methodology involving linearly elastic volume mesh deformation and a newton krylov solver for the euler equations is described the flexibility of the proposed approach is demonstrated through the exploratory shape optimization of a three pronged feathered winglet leading to a span efficiency of 1 19 under a height to span ratio constraint of 0 1 and an optimization of a regional jet wing at transonic speed where a winglet is allowed to develop starting from a planar wingtip extension leading to an 18 8 reduction in drag

measures for designs in experiments with correlated errors summary in this paper we consider optimal design of experiments in the case of correlated obser vations we use and further develop the concept of design measures introduced by paizman mtiller 1998 for the construction of a simple quick and elegant design algorithm we support the construction of this algorithm for a general correlation structure by an interpretation in terms of norms examples demonstrate that our results are useful for generating exact designs by sampling from the obtained design measures most of the literature on experimental design operates under the assumption of uncorre lated errors and by exploiting the concept of a design measure introduced by kiefer 1959 in this setting an optimal design measure is usually computed by an iterative algorithm and then an exact design with independent replications approximately proportional to that measure is employed in the correlated case when replications are not allowed the implementation of design measures is not so straightforward recently however by adding virtual design dependent noise to the process paizman muiller 1998 have introduced a way of using this popular concept albeit requiring a very different interpretation furthermore paizman muller 2000 showed that solving minimisation problems in terms of these measures corresponds to computation of exact designs these measures serve as building blocks for the methods presented in this paper which provides a general numerical tool for screening designs by a different type of smoothing of some important nondifferentiable terms in the algorithm and without imposing restric tions on the number of support points we obtain a design measure the relative magnitude of this measure for each support point reflects the importance of its inclusion in any exact design this algorithm can also be interpreted in terms of certain information norms as pre

twistors kahler manifolds and bimeromorphic geometry i by considering deformations of the moishezon twistor spaces of cp2 cp2 constructed in 20 we showthat theblowupof c2 at n points in general position admits an asymptotically flat scalar flat kahler metric in each kahler class at least provided that the given points are nearly collinear department of mathematics suny at stony brook stony brook new york 11794 e mail address claude math sunysb edu this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 58 00 utc all use subject to http about jstor org terms

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

killing vectors in self dual euclidean einstein spaces using the formalism of complex h spaces we show that all real euclidean self dual spaces that admit at least one killing vector may be gauged so that only two distinct types of killing vectors appear in kahler coordinates these are the generators of a translational or a rotational symmetry we give explicit forms both for the killing vectors and for the constraint on the kahler potential function which allows for such a killing vector in the translational case we show how all such spaces are determined by the general solution of the three dimensional flat laplace s equation and how these are related to the multi taub nut metrics of gibbons and hawking in the rotational case we simplify the equation determining but this is not sufficient to obtain the general solution

einstein metrics spinning top motions and monopoles on considere la construction de solutions pour l equation d einstein autoduale on applique la construction de lebrun a la sphere de berger la 3 sphere de metrique 1 2 2 2 i 3 3 2 ou 1 2 3 est une base de 1 formes invariante a gauche satisfaisant d i j k e ijk j k et i 3 est une constante

an index theorem for anti self dual orbifold cone metrics abstract recently atiyah and lebrun proved versions of the gauss bonnet and hirzebruch signature theorems for metrics with edge cone singularities in dimension four which they applied to obtain an inequality of hitchin thorpe type for einstein edge cone metrics interestingly many natural examples of edge cone metrics in dimension four are anti self dual or self dual depending upon choice of orientation on such a space there is an important elliptic complex called the anti self dual deformation complex whose index gives crucial information about the local structure of the moduli space of anti self dual metrics in this paper we compute the index of this complex in the orbifold case and give several applications

a gradient based parameter free approach to shape optimization in shape optimization the independent node movement approach wherein finite element node coordinates are used directly as design variables allows the most freedom for shape change and avoids the time consuming parameterization preprocess however this approach lacks a length scale control that is necessary to ensure a well posed shape optimization problem and avoid numerical instability motivated by the success of filtering techniques that impose minimum length scales in topology optimization we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach

unified fluid structure interpolation and mesh motion using radial basis functions a multivariate interpolation scheme using radial basis functions is presented which results in a completely unified formulation for the fluid structure interpolation and mesh motion problems the method has several significant advantages primarily all volume mesh structural mesh and flow solver type dependence is removed and all operations are performed on totally arbitrary point clouds of any form hence all connectivity and user input requirements are removed from the computational fluid dynamics computational structural dynamics cfd csd coupling problem as only point clouds are required to determine the coupling also it may equally well be applied to structured and unstructured grids or structural and aerodynamic grids that intersect again because no connectivity information is required furthermore no expensive computations are required during an unsteady simulation just matrix vector multiplications since the required dependence relations are computed only once prior to any simulation and then remain constant this property means that the method is both perfectly parallel since only the data relevant to each structured block or unstructured partition are required to move those points and totally independent from the flow solver hence a completely generic black box tool can be developed which is ideal for use in an optimization approach aeroelastic behaviour of the brite euram mdo wing is analysed in terms of both static deflection and dynamic responses and it is demonstrated that responses are strongly dependent on the exact cfd csd interpolation used mesh quality is also examined during the motion resulting from a large surface deformation global grid quality is shown to be preserved well with local grid orthogonality also being maintained well particularly at and near the moving surface where the original orthogonality is retained copyright 2007 john wiley sons ltd

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

reduced surface point selection options for efficient mesh deformation using radial basis functions previous work by the authors has developed an efficient method for using radial basis functions rbfs to achieve high quality mesh deformation for large meshes for volume mesh deformation driven by surface motion the rbf system can become impractical for large meshes due to the large number of surface control points and so a particularly effective data reduction scheme has been developed to vastly reduce the number of surface points used the method uses a chosen error function on the surface mesh to select a reduced subset of the surface points this subset contains a sufficiently small number of points so as to make the volume deformation fast and a correction function is used to correct those surface points not included hence the scheme is split such that both parts are working on appropriate problems rbfs are an excellent way of finding smooth orthogonality preserving global deformations but are less suitable for enforcing an exact geometry for a large number of points while a simpler approach is ideal for diffusing small changes evenly but has quality and possibly expense drawbacks if used for the entire volume however alternatives exist for the error function used to select the reduced data set so here a comparison is made between three different options the surface error function the unit function and the power function tests run on structured and unstructured meshes show that the surface error function gives the lowest errors but this also requires a deformed surface shape to be known in advance of the simulation the unit and power functions both avoid the need for a deformed surface and the unit function is shown to be superior

index energy estimates for yang mills connections and einstein metrics we prove a conformally invariant estimate for the index of schrodinger operators acting on vector bundles over four manifolds related to the classical cwikel lieb rozenblum estimate applied to yang mills connections we obtain a bound for the index in terms of its energy which is conformally invariant and captures the sharp growth rate furthermore we derive an index estimate for einstein metrics in terms of the topology and the einstein hilbert energy lastly we derive conformally invariant estimates for the betti numbers of an oriented four manifold with positive scalar curvature

non abelian infinite algebra of generalized symmetries for the sdiff 2 toda equation we determine the non abelian algebra of generalized symmetries for the sdiff 2 toda equation a pde for a single function of three independent variables the solutions of which determine self dual vacuum solutions of the einstein field equations this algebra is a realization of two copies of the abstract algebra sdiff 2 along with an additional pair of elements that have derivation like properties on both of the copies it contains as a subalgebra the doubly infinite abelian algebra equivalent to the infinite hierarchy of higher flows found by takasaki and takebe an infinite prolongation of the jet bundle for the original pde to include all the variables allowed in their hierarchy is required for the presentation of this generalization because these symmetries have non zero commutators they generate a recursion relation allowing the generation and description of the entire algebra

compact scalar flat indefinite kahler surfaces with hamiltonian s1 symmetry the existence of scalar flat indefinite kahler metrics on compact complex surfaces is discussed in particular a compact scalar flat indefinite kahler surface admitting a hamiltonian s1 symmetry is proved to be biholomorphic to the product of two complex projective lines with the help of a generalization of the bando calabi futaki character in fact it is shown that none of such metrics exist on hirzebruch surfaces of positive degree on the other hand by employing an analogue of lebrun s hyperbolic ansatz we construct a wealth of explicit scalar flat indefinite kahler metrics on the product of complex projective lines and also prove that these explicit metrics provide infinitely many different isometry classes by examining a necessary and sufficient condition for these metrics to be isometric to each other

a new approach to optimal design for linear models with correlated observations we consider the problem of designing experiments for regression in the presence of correlated observations with the location model as the main example for a fixed correlation structure approximate optimal designs are determined explicitly and it is demonstrated that under the model assumptions made by bickel and herzberg 1979 for the determination of asymptotic optimal design the designs derived in this article converge weakly to the measures obtained by these authors we also compare the asymptotic optimal design concepts of sacks and ylvisaker 1966 1968 and bickel and herzberg 1979 and point out some inconsistencies of the latter finally we combine the best features of both concepts to develop a new approach for the design of experiments for correlated observations and it is demonstrated that the resulting design problems are related to the logarithmic potential theory