exponential decay of eigenfunctions in many body type scattering with second order perturbations abstract we show the exponential decay of eigenfunctions of second order geometric many body type hamiltonians at non threshold energies moreover in the case of first order and small second order perturbations we show that there are no eigenfunctions with positive energy

integral operators on the oshima compactification of a riemannian symmetric space abstract consider a riemannian symmetric space x g k of non compact type where g is a connected real semisimple lie group and k a maximal compact subgroup let x be its oshima compactification and c x the left regular representation of g on x in this paper we examine the convolution operators f for rapidly decaying functions f on g and characterize them within the framework of totally characteristic pseudodifferential operators describing the singular nature of their kernels as a consequence we obtain asymptotics for heat and resolvent kernels associated to strongly elliptic operators on x as a further application a regularized trace for the operators f can be defined yielding a distribution on g which can be interpreted as a global character of and is given by a fixed point formula analogous to the atiyah bott character formula for an induced representation of g

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

analytic continuation of the resolvent of the laplacian on symmetric spaces of noncompact type let m g be a globally symmetric space of noncompact type of arbitrary rank and its laplacian we introduce a new method to analyze and the resolvent 1 this has origins in quantum n body scattering but is independent of the classical theory of spherical functions and is analytically much more robust we expect that suitably modified it will generalize to locally symmetric spaces of arbitrary rank as an illustration of this method we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a riemann surface multiply covering the plane we also show how this continuation may be deduced using the theory of spherical functions in summary this paper establishes a long suspected connection between the analysis on symmetric spaces and n body scattering

analytic continuation of the resolvent of the laplacian on sl 3 so 3 in this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the laplacian on symmetric spaces of rank greater than one and their geometric perturbations in our previous work we described the resolvent and specifically the asymptotic behavior of the green s function on sl 3 so 3 using methods from three particle scattering here we extend the technique of complex scaling to symmetric spaces to show that the resolvent continues analytically across the spectrum

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

scattering theory for the laplacian on manifolds with bounded curvature abstract in this paper we study the behaviour of the continuous spectrum of the laplacian on a complete riemannian manifold of bounded curvature under perturbations of the metric the perturbations that we consider are such that its covariant derivatives up to some order decay with some rate in the geodesic distance from a fixed point especially we impose no conditions on the injectivity radius one of the main results are conditions on the rate of decay depending on geometric properties of the underlying manifold that guarantee the existence and completeness of the wave operators

a fast algebraic multigrid preconditioned conjugate gradient solver abstract this work presents a new approach for selecting the coarse grids allowing a faster algebraic multigrid amg preconditioned conjugate gradient solver this approach is based on an appropriate choice of the parameter considering the matrix density during the coarsening process which implies in a significant reduction in the matrix dimension at all amg levels

a parallel multiselection greedy method for the radial basis function based mesh deformation summary greedy algorithm has been widely adopted for the point selection procedure of radial basis function based mesh deformation however in large deformation simulations with thousands of points selected the greedy point selection will be too expensive and thus become a performance bottleneck to improve the efficiency of the point selection procedure a parallel multi selection greedy method has been developed in this paper multiple points are selected at each step to accelerate the convergence speed of the greedy algorithm in addition two strategies are presented to determine the specific selecting number the parallelization of the greedy point selection is realized based on a master slave model and a hybrid decomposition algorithm is proposed to address the load imbalance problem numerical benchmarks show that both our multi selection method and the parallelization could obviously improve the point selection efficiency specifically total speedups of 20 and 55 are separately obtained for the 3d undulating fish with 106 cell mesh and the 3d rotating hydrofoil with 11 million cell mesh this article is protected by copyright all rights reserved

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

algebraic multigrid based on element interpolation amge we introduce amge an algebraic multigrid method for solving the discrete equations that arise in ritz type finite element methods for partial differential equations assuming access to the element stiffness matrices we have that amge is based on the use of two local measures which are derived from global measures that appear in existing multigrid theory these new measures are used to determine local representations of algebraically smooth error components that provide the basis for constructing effective interpolation and hence the coarsening process for amg here we focus on the interpolation process choice of the coarse grids based on these measures is the subject of current research we develop a theoretical foundation for amge and present numerical results that demonstrate the efficacy of the method

parallel algebraic multigrid methods on distributed memory computers algebraic multigrid methods are well suited as preconditioners for iterative solvers we consider linear systems of equations which are sparse and symmetric positive definite and which stem from a finite element discretization of a second order self adjoint elliptic partial differential equation or a system of them since preconditioners based on algebraic multigrid are very efficient additional speedup can only be achieved by parallelization in this paper we propose a general parallel algebraic multigrid algorithm for finite element discretizations based on domain decomposition ideas which is well suited for distributed memory computers this paper pays special attention to the coarsening strategy which has to be adapted in the parallel case moreover a general framework of data distribution gives rise to a construction scheme for the prolongation operators results of numerical studies on parallel computers with distributed memory are presented which show the high efficiency of the approach

improved point selection method for hybrid unstructured mesh deformation using radial basis functions in this paper an efficient mesh deformation technique for hybrid unstructured grids based on radial basis functions has been developed the principle procedure adopted for this scheme can be divided into two steps first a series of radial basis functions are constructed by an interpolation method according to the displacements of moving boundary points later the displacements of all points in the computation domain are determined by the aforementioned radial basis function series to improve the efficiency data reduction should be introduced into the interpolation process consequently a multilevel subspace radial basis function interpolation method based on a double edged greedy algorithm is presented to create an approximate interpolation for all moving boundary points this method is computationally efficient preserves orthogonality and has no dependency on the type of flow solver typical deformation problems of a lann wing dlr f6 wing body nacelle pylon configuration and dlr f11 high li

an extension problem for the cr fractional laplacian we show that the conformally invariant fractional powers of the sub laplacian on the heisenberg group are given in terms of the scattering operator for an extension problem to the siegel upper halfspace remarkably this extension problem is different from the one studied among others by caffarelli and silvestre we also prove an energy identity that yields a sharp trace sobolev embedding

efficient mesh motion using radial basis functions with data reduction algorithms mesh motion using radial basis functions has been demonstrated previously by the authors to produce high quality meshes suitable for use within unsteady and aeroelastic cfd codes in the aeroelastic case the structural mesh may be used as the set of control points governing the deformation which is efficient since the structural mesh is usually small however as a stand alone mesh motion tool where the surface mesh points control the motion radial basis functions may be restricted by the size of the surface mesh as an update of a single volume point depends on all surface points in this paper a method is presented that allows an arbitrary deformation to be represented to within a desired tolerance by using a significantly reduced set of surface points intelligently identified in a fashion that minimises the error in the interpolated surface this method may be used on much larger cases and is successfully demonstrated here for a 10 6 cell mesh where the initial solve phase cost reduces by a factor of eight with the new scheme and the mesh update by a factor of 55 it has also been shown that the number of surface points required to represent the surface is only geometry dependent i e grid size independent and so this reduction factor actually increases for larger meshes

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

parallel algebraic multigrids for structural mechanics this paper presents the results of a comparison of three parallel algebraic multigrid amg preconditioners for structural mechanics applications in particular we are interested in investigating both the scalability and robustness of the preconditioners numerical results are given for a range of structural mechanics problems with various degrees of difficulty and they show that algebraic multigrid methods can be applied successfully even to difficult problems in structural mechanics

on open scattering channels for manifolds with ends abstract in the framework of time dependent geometric scattering theory we study the existence and completeness of the wave operators for perturbations of the riemannian metric for the laplacian on a complete manifold of dimension n the smallness condition for the perturbation is expressed intrinsically and coordinate free in purely geometric terms using the harmonic radius therefore the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the ricci curvature as an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the riemannian metric this stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations

resonances for the laplacian on products of two rank one riemannian symmetric spaces let x x 1 times x 2 be a direct product of two rank one riemannian symmetric spaces of the noncompact type we show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension the resolvent of the laplacian of x can be lifted to a holomorphic function on a riemann surface which is a branched covering of mathbb c in all other cases the resolvent of the laplacian of x admits a singular meromorphic lift the poles of this function are called the resonances of the laplacian we determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances the ranges of these operators are finite dimensional and explicitly realized as direct sums of finite dimensional irreducible spherical representations of the group of the isometries of x

parallel efficient mesh motion using radial basis functions with application to multi bladed rotors radial basis functions are used to provide a solution to the problem of mesh motion for unsteady aerodynamic simulation the method is independent of connectivity and produces high quality meshes but is expensive for large meshes in its full form hence the efficiency of the technique has been greatly improved here by reducing the number of surface points used to define deformations of the surface and the minor error in position that this implies at other surface points is corrected with a simple decaying perturbation thus splitting the method into a primary basis function method and a secondary local correction method this means that the exact surface is retained but the mesh motion is significantly faster while splitting the motion into two stages allows both the methods to work on appropriate problems given their relative strengths an example deformation for a 5 x10 6 cell helicopter rotor mesh with an exaggerated cyclic pitch motion shows excellent mesh quality thus validating a scheme that is also simple robust and readily parallelized

adaptive radial basis function mesh deformation using data reduction radial basis function rbf mesh deformation is one of the most robust mesh deformation methods available using the greedy data reduction method in combination with an explicit boundary correction results in an efficient method as shown in literature however to ensure the method remains robust two issues are addressed 1 how to ensure that the set of control points remains an accurate representation of the geometry in time and 2 how to use automate the explicit boundary correction while ensuring a high mesh quality in this paper we propose an adaptive rbf mesh deformation method which ensures the set of control points always represents the geometry displacement up to a certain user specified criteria by keeping track of the boundary error throughout the simulation and re selecting when needed opposed to the unit displacement and prescribed displacement selection methods the adaptive method is more robust user independent and efficient for the cases considered secondly the analysis of a single high aspect ratio cell is used to formulate an equation for the correction radius needed depending on the characteristics of the correction function used maximum aspect ratio minimum first cell height and boundary error based on the analysis two new radial basis correction functions are derived and proposed this proposed automated procedure is verified while varying the correction function reynolds number and thus first cell height and aspect ratio and boundary error finally the parallel efficiency is studied for the two adaptive methods unit displacement and prescribed displacement for both the cpu as well as the memory formulation with a 2d oscillating and translating airfoil with oscillating flap a 3d flexible locally deforming tube and deforming wind turbine blade generally the memory formulation requires less work due to the large amount of work required for evaluating rbf s but the parallel efficiency reduces due to the limited bandwidth available between cpu and memory in terms of parallel efficiency scaling the different studied methods perform similarly with the greedy algorithm being the bottleneck in terms of absolute computational work the adaptive methods are better for the cases studied due to their more efficient selection of the control points by automating most of the rbf mesh deformation a robust efficient and almost user independent mesh deformation method is presented

on the c chevalley s theorem the group of orthogonal matrices k so n acts by conjugation on the vector space p of all n x ii symmetric matrices and the spectral theorem implies that every orbit intersects the subspace a c p of the diagonal matrices in fact every orbit k x is orthogonal to a under the invariant inner product x y tr xy and the intersection is the orbit of the permutation group on n letters w s which permutes the eigenvalues of x if we let

efficient mesh deformation based on cartesian background mesh abstract moving mesh is widely used in the simulation of aerodynamic shape optimization multibody relative motion aircraft icing and aeroelasticity the efficient and high quality mesh deformation is the key technology of moving mesh this paper presented a new mesh deformation method based on cartesian background mesh mdcbm first the cartesian background mesh is deformed with radial basis functions rbf second the displacement of cartesian background mesh is algebraically interpolated onto all meshes in the computing domain since the background mesh is coarse the background mesh deformation can be finished fast because the background mesh of mdcbm is regular the mapping relationship between background mesh and the computing mesh is simple so the time spent on mapping search is substantially reduced the examples including naca0012 airfoil multi element airfoil with structured unstructured mesh and dlr f4 wing body show the good performance of mdcbm we highlight the advantages of mdcbm with respect to its computational efficiency and high quality of deformed mesh

influence of mesh sensitivities on computational fluid dynamics based derivative calculation to clarify the influence of mesh sensitivities on the computational fluid dynamics based derivatives a theoretical deduction and treatment of the mesh sensitivities are carried out yielding an explicit mathematical formulation for issues of concern numerical tests focusing on aerodynamic shape derivatives within the context of a transonic viscous flow passing through an rae2822 airfoil are conducted to validate the theoretical analysis the automatic differentiation the adjoint and tangent modes as well as the finite difference are applied to resolve the gradient information various mesh deformation strategies including the finite element analog the spring analog the original radial basis function interpolation and a new modified radial basis function method are applied to provide different mesh sensitivity solutions the numerical results confirm that mesh sensitivities are essential for the geometry related numerical derivatives and the reliance is independent of the perturbation magnitu

numerical mathematics of the subtraction method for the modeling of a current dipole in eeg source reconstruction using finite element head models in electroencephalography eeg source analysis a dipole is widely used as the model of the current source the dipole introduces a singularity on the right hand side of the governing poisson type differential equation that has to be treated specifically when solving the equation toward the electric potential in this paper we give a proof for existence and uniqueness of the weak solution in the function space of zero mean potential functions using a subtraction approach the method divides the total potential into a singularity and a correction potential the singularity potential is due to a dipole in an infinite region of homogeneous conductivity we then state convergence properties of the finite element fe method for the numerical solution to the correction potential we validate our approach using tetrahedra and regular and geometry conforming node shifted hexahedra elements in an isotropic three layer sphere model and a model with anisotropic middle compartment validation is carried out using sophisticated visualization techniques correlation coefficient cc and magnification factor mag for a comparison of the numerical results with analytical series expansion formulas at the surface and within the volume conductor for the subtraction approach with regard to the accuracy in the anisotropic three layer sphere model cc of 0 998 or better and mag of 4 3 or better over the whole range of realistic eccentricities and to the computational complexity 2mm node shifted hexahedra achieve the best results a relative fe solver accuracy of 10 4 is sufficient for the used algebraic multigrid preconditioned conjugate gradient approach finally we visualize the computed potentials of the subtraction method in realistically shaped fe head volume conductor models with anisotropic skull compartments

analytic continuation of resolvent kernels on noncompact symmetric spaces let x g k be a symmetric space of noncompact type and let be the laplacian associated with a g invariant metric on x we show that the resolvent kernel of admits a holomorphic extension to a riemann surface depending on the rank of the symmetric space this riemann surface is a branched cover of the complex plane with a certain part of the real axis removed it has a branching point at the bottom of the spectrum of it is further shown that this branching point is quadratic if the rank of x is odd and is logarithmic otherwise in case g has only one conjugacy class of cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of with the only branching point being the bottom of the spectrum

related aspects of positivity in riemannian geometry pour une variete de riemann on etudie le lien entre le point 0 du l 2 spectre le plus proche de zero pour l extension de friedrich d un operateur semi defini symetrique et le taux exponentiel de transition d un operateur de markoff conservant la positivite

domain element method for aerodynamic shape optimization applied to a modern transport wing generic wraparound aerodynamic shape optimization technology is presented and applied to a modern commercial aircraft wing in transonic cruise the wing geometry is parameterized by a novel domain element method which uses efficient global interpolation functions to deform both the surface geometry and corresponding computational fluid dynamics volume mesh the technique also provides a method that allows geometries to be parameterized at various levels ranging from global three dimensional planform alterations to detailed local surface changes combining all levels of parameterization allows for free form design control with very few design variables the method provides an efficient combined shape parameterization and high quality mesh deformation technique that is totally independent of mesh type structured or unstructured optimization independence from the flow solver is achieved by obtaining sensitivity information for an advanced gradient based optimizer by finite differences the entire optimization suite has also been parallelized to allow optimization with highly flexible parameterization in practical times results are presented for highly constrained optimizations of the modern aircraft wing in transonic cruise using three levels of parameterization number of design variables to assess the effect of parameterization level on the optimization the highest level optimization results in a totally shock free geometry with an associated substantial reduction in drag

efficient mesh motion using radial basis functions with volume grid points reduction algorithm abstract as one of the most robust mesh deformation technique available the radial basis function rbf mesh deformation has been accepted widely however for volume mesh deformation driven by surface motion the rbf system may become impractical for large meshes due to the large number of both surface control points and volume points surface points selection procedure based on the greedy algorithm results in an efficient implementation of the rbf based mesh deformation procedure the greedy algorithm could reduce the number of surface points involved in the rbf interpolation while acquire an acceptable accuracy as shown in literature to improve the efficiency of the rbf method furthermore an issue that how to reduce the number of the volume points needed to be moved is addressed in this paper we propose an algorithm for volume points reduction based on a wall distance based restricting function which is added to the formulation of the rbf based interpolation this restricting function is firstly introduced by the current article to support large deformation a multi level subspace interpolation is essentially needed although this technique was used to improve the efficiency of the surface points selection procedure in the existed literature the key point of this technique is setting the error of previous interpolation step as the object of current step and restricting interpolation region gradually because the tolerance of the error is decreased hierarchically the number of the surface points is increased but the number of the volume points needed to be moved is reduced gradually therefore the cpu cost of updating the mesh motion could be reduced eventually since it scales with the product of these two numbers the computational requirement of the proposed procedure is reduced evidently compared with the standard procedure as proved by some examples

a full subtraction approach for finite element method based source analysis using constrained delaunay tetrahedralisation a mathematical dipole is widely used as a model for the primary current source in electroencephalography eeg source analysis in the governing poisson type differential equation the dipole leads to a singularity on the right hand side which has to be treated specifically in this paper we will present a full subtraction approach where the total potential is divided into a singularity and a correction potential the singularity potential is due to a dipole in an infinite region of homogeneous conductivity the correction potential is computed using the finite element fe method special care is taken in order to evaluate the right hand side integral appropriately with the objective of achieving highest possible convergence order for linear basis functions our new approach allows the construction of transfer matrices for fast computation of the inverse problem for anisotropic volume conductors a constrained delaunay tetrahedralisation cdt approach is used for the generation of high quality fe meshes we validate the new approach in a four layer sphere model with a highly conductive cerebrospinal fluid csf and an anisotropic skull compartment for radial and tangential sources with eccentricities up to 1 mm below the csf compartment we achieve a maximal relative error of 0 71 in a cdt fe model with 360 k nodes which is not locally refined around the source singularity and therefore useful for arbitrary dipole locations the combination of the full subtraction approach with the high quality cdt meshes leads to accuracies that to the best of the author s knowledge have not yet been presented before

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

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

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

pseudodifferential operator calculus for generalized q rank 1 locally symmetric spaces i abstract this paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on q rank 1 locally symmetric spaces and riemannian manifolds generalizing these this generalization is the interior of a manifold with boundary where the boundary has the structure of a tower of fibre bundles the class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary in directions given by the tower of fibrations as well as q rank 1 locally symmetric spaces examples include ricci flat metrics on the complement of a divisor in a smooth variety constructed by tian and yau in this first part of the calculus construction parametrices are found for fully elliptic differential a operators which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity in the second part we will consider operators that do not satisfy this condition