wavelet solvers for hp fem discretizations in 3d using hexahedral elements in this paper we investigate the discretization of an elliptic boundary value problem in 3d by means of the hp version of the finite element method using a mesh of hexahedrons the corresponding linear system is solved by a preconditioned conjugate gradient method the construction of the preconditioner is based on an inexact additive overlapping schwarz method the remaining subproblems are treated by a tensor product based preconditioner this preconditioner uses a basis transformation into a basis which is stable in l 2 and h 1 several numerical examples show the efficiency of the proposed method

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

extension operators on tensor product structures in two and three dimensions in this paper a uniformly elliptic second order boundary value problem in two dimensions is discretized by the p version of the finite element method an inexact dirichlet dirichlet domain decomposition preconditioner for the system of linear algebraic equations is investigated the ingredients of such a preconditioner are a preconditioner for the schur complement a preconditioner for the subdomains and an extension operator operating from the edges of the elements into their interior using methods of multiresolution analysis we propose a new method to compute the extension efficiently we prove that this type of extension is optimal i e the h 1 omega norm of the extended function is bounded by the h 0 5 partial omega norm of the given function numerical experiments show the optimal performance of the described extension

impact of element level static condensation on iterative solver performance this paper provides theoretical estimates that quantify and clarify the savings associated to the use of element level static condensation as a first step of an iterative solver these estimates are verified numerically the numerical evidence shows that static condensation at the element level is beneficial for higher order methods for lower order methods or when the number of iterations required for convergence is low the setup cost of the elimination as well as its implementation may offset the benefits obtained during the iteration process however as the iteration count e g above 50 or the polynomial order e g above cubics grows the benefits of element level static condensation are significant

bddc and feti dp preconditioners for spectral element discretizations abstract two of the most recent and important nonoverlapping domain decomposition methods the bddc method balancing domain decomposition by constraints and the feti dp method dual primal finite element tearing and interconnecting are here extended to spectral element discretizations of second order elliptic problems in spite of the more severe ill conditioning of the spectral element discrete systems compared with low order finite elements and finite differences these methods retain their good properties of scalability quasi optimality and independence on the discontinuities of the elliptic operator coefficients across subdomain interfaces

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

equivariance brst symmetry and superspace the structure of equivariant cohomology in non abelian localization formulas and topological field theories is discussed equivariance is formulated in terms of a nilpotent becchi rouet stora tyutin brst symmetry and another nilpotent operator which restricts the brst cohomology onto the equivariant or basic sector a superfield formulation is presented and connections to reducible batalin fradkin vilkovisky bfv quantization of topological yang mills theory are discussed

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

a preconditioner for the h p version of the finite element method in two dimensions a preconditioner based on a two level mesh and a two level orthogonalization is proposed for the h p version of the finite element method for two dimensional elliptic problems in polygonal domains its implementation is in parallel on the subdomain level for the linear or bilinear nodal modes and in parallel on the element level for the high order side and internal modes the condition number of the preconditioned linear system is of order max limits i 1 ln textstyle h ip i over textstyle h i 2 where h i is the diameter of the i th subdomain h i and p i are the diameter of elements and the maximum polynomial degree used in the subdomain this result reduces to well known results for the h version i e p i 1 and the p version i e h i h i as the special cases of the h p version

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

inexact solvers on element interfaces for the p and h p finite element method parallel and iterative solvers for large scale linear system arising from finite element discretization based on domain decomposition decompose the approximation spaces into a number of subspaces the corresponding subproblems are solved in these subspaces and the solutions are corrected in an iterative process the subproblems are much smaller in size and much easier to be solved one of these subspaces is spanned by high order element side modes in two dimensions and high order element face modes in three dimensions solving the subproblems on the element interfaces is a major cost in each step of iteration to reduce computational cost various inexact solvers on element interfaces are proposed in this paper with theoretical analysis and numerical illustrations

the p version of the finite element method compared to an adaptive h version for the deformation theory of plasticity a p version of the finite element method is applied to the deformation theory of plasticity and the results are compared to a state of the art adaptive h version it is demonstrated that even for nonlinear elliptic problems the p version is a very efficient discretization strategy

nonexistence of abelian difference sets lander s conjecture for prime power orders in 1963 ryser conjectured that there are no circulant hadamard matrices of order 4 and no cyclic difference sets whose order is not coprime to the group order these conjectures are special cases of lander s conjecture which asserts that there is no abelian group with a cyclic sylow p subgroup containing a difference set of order divisible by p we verify lander s conjecture for all difference sets whose order is a power of a prime greater than 3

planar functions relative difference sets and character theory planar functions from2nto3nare studied in this paper by investigating the character values of the corresponding relative difference sets we obtain some nonexistence results of planar functions in particular we show that there is no planar functions fromzpqto4pq wherepandqare any primes we also show that except for four undecided cases there is no planar function from5nto6nifnis not a prime andn 50 000

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

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

new bounds and optimal binary signature sets part ii aperiodic total squared correlation we derive new bounds on the aperiodic total squared correlation atsc of binary antipodal signature sets for any number of signatures k and any signature length l we then present optimal designs that achieve the new bounds for several k l cases as interesting arguably side results we show that individual maximal merit factor sequences for example barker sequences are single user atsc optimal while neither the familiar gold nor the kasami set designs are atsc optimal in general the atsc optimal signature set designs provided in this work are in this sense better suited for asynchronous and or multipath code division multiplexing applications

cyclotomic integers and finite geometry the most powerful method for the study of finite geometries with regular or quasiregular automorphism groups g is to translate their definition into an equation over the integral group ring 2 g and to investigate this equation by applying complex representations of g for the definitions and the basic facts see section 2 if g is abelian this approach boils down to proving or disproving the existence of elements of 2 g with 0 1 coefficients whose character values are cyclotomic integers of certain prescribed absolute values up to now there have been two general methods to tackle this problem namely hall s multiplier concept and turyn s selfconjugacy approach however both methods need severe technical assumptions and thus are not applicable to many classes of problems despite many efforts over a period of more than 30 years no general method has been found to overcome these difficulties in this paper we present a new approach to the study of combinatorial structures via group ring equations which works without any restrictive assumptions in order to understand the method of the present paper it will be instructive to briefly discuss the self conjugacy concept first turyn 51 demonstrated that the character method for the study of group ring equations works very nicely under the so called self conjugacy condition an integer n is called self conjugate modulo m if all prime ideals above n in the mth cyclotomic field q m are invariant under complex conjugation under this condition it is possible to find all cyclotomic integers in q m of absolute value nt 2 for any positive integer t it is the complete knowledge of the cyclotomic integers of prescribed absolute value which makes the character method work so well under the self conjugacy condition since turyn s fundamental work 51 there have been dozens of papers extending and refining his approach however all these results are restricted to the case of self conjugacy and that is a very severe restriction indeed namely the probability that n is selfconjugate modulo m decreases exponentially fast in the number of distinct prime divisors of n and m see remark 2 2 this means that the self conjugacy method fails in almost all cases one may ask if it is possible to extend turyn s method in order to get rid of the self conjugacy assumption it turns out that in general this is impossible at least with present day methods the required complete knowledge of the cyclotomic integers of prescribed absolute value would yield an

preconditioning on element interfaces for the p version finite element method and spectral element method in the preconditioned iterative method based on nonoverlapping domain decompositions the preconditioner is usually of block diagonal type to calculate its action on a residual vector it is required to solve a linear system of algebraic equations with the coefficient matrix being the schur complement block corresponding to the interface to reduce the computational cost the schur complement blocks in the preconditioners can be replaced with other simpler matrices which are called the interface preconditioners in this paper we discuss various interface preconditioners for the p version finite element method and spectral element method in r2 and r3 emphases are placed on 1 the lower and upper bounds of the condition numbers for the preconditioned schur complement blocks using the interface preconditioners which characterize the enlargement of the overall condition numbers and 2 the algorithms to efficiently implement the interface preconditioners numerical results are presented for the condition numbers using various interface preconditioners

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

multilevel additive schwarz method for the h p version of the galerkin boundary element method we study a multilevel additive schwarz method for the h p version of the galerkin boundary element method with geometrically graded meshes both hypersingular and weakly singular integral equations of the first kind are considered as it is well known the h p version with geometric meshes converges exponentially fast in the energy norm however the condition number of the galerkin matrix in this case blows up exponentially in the number of unknowns m we prove that the condition number p of the multilevel additive schwarz operator behaves like o mlog 2 m as a direct consequence of this we also give the results for the 2 level preconditioner and also for the h p version with quasi uniform meshes numerical results supporting our theory are presented

identification and estimation of dynamic errors in variables models abstract we consider here the factor analysis approach to the analysis of linear dynamic errors in variables models identification of the most general models and the question of nonuniqueness are discussed we show how these models are generalizations of the total least squares model and how the grassmann manifold provides a natural setting for their analysis under certain assumptions we analyze the problem of maximum likelihood estimation and indicate how this involves maximization of a functional on a grassmann manifold

an iterative and parallel solver based on domain decomposition for the h p version of the finite element method in this paper we propose a new interative and parallel solver based on domain decomposition for the h p version of the finite element method in two dimensions it improves our previous work in two aspects 1 a subdomain may contain several super elements of the coarse mesh thus can be of arbitrary shape and size this makes the solver more efficient and more flexible in computational practice 2 the p version components i e the high order side and internal modes in every element are treated separately which results in better parallelism

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

the p version of the finite element method for three dimensional curved thin walled structures in this paper we present an implementation of a three dimensional p version for structural problems of solids with almost arbitrarily curved surfaces applying the blending function method complex structures can often be modelled by a few p elements being the basis for a higher order approximation numerical examples will demonstrate that the p version with anisotropic ansatz spaces allows to predict the structural behaviour of three dimensional plates and shells with approximately the same amount of degrees of freedom as in the two dimensional case yet significantly more accurate due to the three dimensional model furthermore it is advantageous to compute complex structures exclusively with three dimensional discretizations as no special elements are needed to model the transition from dimensionally reduced formulations like plates or shells to fully three dimensional solid elements using the p version with anisotropic ansatz spaces the whole structure can be efficiently discretized with solid elements even if the aspect ratio of the elements becomes very large copyright 2001 john wiley sons ltd

domain decomposition method for the h p version finite element method domain decomposition method for the h p version of the finite element method in two and three dimensions are discussed using the framework of additive schwarz method various iterative methods are described with their condition numbers estimated further to reduce the cost for solving the sub problems on element interfaces different inexact interface solvers are proposed the effects on the overall condition number as well as their efficient implementation are analyzed

domain decomposition preconditioners for p and hp finite element approximation of stokes equations abstract domain decomposition preconditioning techniques are developed in the context of hp finite element approximation of the stokes problem two basic types of preconditioner are considered a block diagonal scheme based on decoupling the velocity and pressure components and a scheme based on an indefinite system similar to the original stokes system for each type of scheme theoretical estimates are obtained for the location of the eigenvalues of the preconditioned operators in terms of the polynomial degree the mesh sizes on the coarse and fine grids and the inf sup constant for the method theoretical estimates show that the growth of the bounds is modest as the mesh is refined and the polynomial order is increased the preconditioners are shown to be applicable to various iterative schemes for the stokes systems the theoretical bounds are compared with actual quantities obtained in practical computations for several representative problems

multilevel preconditioned qmr methods for unstructured mesh computation we present a variant of the quasi minimal residual qmr algorithm of freund and nachtigal with a preconditioner based on the algebraic multilevel amli algorithm of axelsson and vassilevski this combination provides an effective solution method for indefinite algebraic systems and is tested by application to finite element discretizations of the helmholtz equation the implementation is applicable with both h and p refinements and easily extends to a parallel environment

a preconditioner based on domain decomposition for h p finite element approximation on quasi uniform meshes the problem of solving the algebraic systems arising from the discretization of a symmetric elliptic boundary value problem using h p finite element methods in two dimensions is addressed a preconditioning technique similar to those of bramble pasciak and schatz math comp 47 1986 pp 103 135 based on domain decomposition also known as substructuring is developed the method is applicable to problems on general domains involving differential operators with quite general coefficients the algorithm reduces to that of bramble pasciak and schatz if linear elements are used it is shown that the condition number of the preconditioned system which determines the rate of convergence of the iterative method grows at most as 1 log 2 p 1 log 2 hp h where p is the polynomial degree h is the size of the elements and h is the size of the subdomains

bohr sommerfeld conditions for integrable systems with critical manifolds of focus focus type we present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators ph ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity typical examples include the quantum spherical pendulum or the quantum champagne bottle in the spirit of colin de verdiere and parisse we show that such systems have a universal behavior described by singular quantization conditions of bohr sommerfeld type these conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears thanks to numerical computations done by ms child for the case of the champagne bottle we are able to accurately illustrate our statements

a completely integrable hamiltonian system associated with line fitting in complex vector spaces on presente et on integre explicitement un systeme hamiltonien completement integrable qui apparait dans le probleme de l ajustement de courbe a un ensemble de donnees dans un espace vectoriel complexe

total linear least squares and the algebraic riccati equation abstract it is shown that the solution to a total linear least squares problem satisfies a quadratic matrix equation which turns into an algebraic riccati equation when the matrix of unknowns is square if there is an additional symmetry constraint on the solution the optimal solution is given by the anti stabilizing solution of this riccati equation

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

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

fully discrete hp finite elements fast quadrature a fully discrete hp finite element method fem is presented it combines the features of the standard hp fem conforming galerkin formulation variable order quadrature schemes geometric meshes static condensation and of the spectral element method special shape functions and spectral quadrature techniques the speed up relative to standard hp elements is analyzed in detail both theoretically and computationally

non hamiltonian actions with isolated fixed points we construct a non hamiltonian symplectic circle action on a closed connected six dimensional symplectic manifold with exactly 32 fixed points

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

fast domain decomposition algorithm discretizations of 3 d elliptic equations by spectral elements the main obstacle for obtaining fast domain decomposition solvers for the spectral element discretizations of the 2 nd order elliptic equations was the lack of fast solvers for local internal problems on subdomains of decomposition and their faces as was recently shown by korneev rytov such solvers can be derived on the basis of the specific interrelation between the stiness matrices of the spectral and hierarchical p reference elements the coordinate polynomials of the latter are tensor products of the integrated legendre s polynomials this interrelation allows to apply to the spectral element discretizations fast solvers which in some basic features are quite similar to those developed for the discretizations by the hierarchical elements using these facts and the preceding results on the wire basket preconditioners we present an almost optimal in total arithmetical cost domain decomposition preconditioner solver for the spectral element discretizations of the 2 nd order elliptic equations in 3 d domains 2000 mathematics subject classification 65n22 65m30 65n55

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

estimation principal components and hamiltonian systems in this paper we show how symplectic geometry can be used to analyse a fundamental problem in estimation theory the fitting of lines and planes to a data set in n dimensions symplectic methods yield information not only on the critical point structure of the maximum likelihood estimation mle function see byrnes and willems 7 but also on the asymmetries in the distribution of the data points in n space

structure preserving model reduction for mechanical systems this paper focuses on methods of constructing of reduced order models of mechanical systems which preserve the lagrangian structure of the original system these methods may be used in combination with standard spatial decomposition methods such as the karhunen loeve expansion balancing and wavelet decompositions the model reduction procedure is implemented for three dimensional finite element models of elasticity and we show that using the standard newmark implicit integrator significant savings are obtained in the computational costs of simulation in particular simulation of the reduced model scales linearly in the number of degrees of freedom and parallelizes well

a numerical study on neumann neumann and feti methods for hp approximations on geometrically refined boundary layer meshes in two dimensions abstract we present extensive numerical tests showing the performance and robustness of certain balancing neumann neumann and one level feti methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions the numerical results confirm the theoretical results derived in a toselli x vasseur technical report 02 15 seminar fur angewandte mathematik eth zurich september 2002 the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients in addition they only grow polylogarithmically with the polynomial degree as in the case of p approximations on shape regular meshes

preconditioning of the p version of the finite element method the p version finite element method for linear second order elliptic equations in an arbitrary sufficiently smooth incl polygonal bounded domain is studied in the framework of the domain decomposition dd method two types of square reference elements are used with coordinate functions given by the products of the integrated legendre polynomials estimates for the condition numbers and some useful inequalities are given we consider preconditioning of the problems arising on subdomains and of the schur complement as well as the derivation and analysis of the dd preconditioner for the entire system this is done for a class of curvilinear finite elements we obtain several dd preconditioners for which the generalized condition numbers vary from o log p 3 to o 1 this paper is based on 19 21 27 we have omitted most of the proofs in order to shorten it and have described instead what could be done as well as outlined some additional ideas the full proofs omitted can in most cases be found in 19 20 27

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

on the construction of certain 6 dimensional symplectic manifolds with hamiltonian circle actions let m w be a connected compact 6 dimensional symplectic manifold equipped with a semi free hamiltonian s 1 action such that the fixed point set consists of isolated points or surfaces assume dim h 2 m 3 in an earlier paper we defined a certain invariant of such spaces which consists of fixed point data and twist type and we divided the possible values of these invariants into six types in this paper we construct such manifolds with these types as a consequence we have a precise list of the values of these invariants

an infinite dimensional hamiltonian system on projective hilbert space we consider here the explicit integration of a hamiltonian system on infinite dimensional complex projective space the hamiltonian which is the restriction of a linear functional to this projective space arises in the problem of line fitting in complex hilbert space or equivalently the problem of functional approximation or as the expectation value of a model quantum mechanical system we formulate the system here as a lax system with parameter showing how this leads to an infinite set of conserved integrals associated with the problem and to an explicit formulation of the flow in action angle form via an extension of some work of j moser in addition we find the algebraic curve naturally associated with the system introduction in recent years there has been a great deal of interest in integrable hamiltonian systems in both finite and infinite dimensions see for example the work of adler and van moerbeke 2 griffiths 11 mckean 16 moser 18 and mumford 19 most classical hamiltonian systems of interest arise in the modelling of mechanical systems in contrast our hamiltonian here arises as the maximum likelihood estimation function for fitting a line to a set of data in a complex hilbert space or equivalently as the distance between an l2 function and its first fourier component alternatively the hamiltonian may be viewed as the expectation value of a model quantum mechanical system where the hamiltonian operator is compact this expectation then generates the heisenberg equations of motion as discussed in 8 the work discussed here is a generalization to infinite dimensions and detailed discussion of the work announced in 4 an announcement of the infinite dimensional case was made in 7 for the most part the finite dimensional case may be subsumed in the discussion of the infinite dimensional case one aspect that does not appear to carry over in a rigorous sense is the moment map argument for complete integrability see 4 5 although a convincing heuristic argument may be made as remarked in 4 the finite dimensional system may be viewed as a system of the type dubbed by adler and van moerbeke 2 to be spinning top or ellipsoid it is interesting that the analysis of the finite dimensional system carries over so naturally to the infinite dimensional context in particular aspects of the work of moser on the integration of finite dimensional integrable systems extend to this infinite dimensional setting 1 the hamiltonian a method of arriving at our hamiltonian is to consider the problem of finding and minimizing the total perpendicular distance of a set received by the editors august 18 1986 1980 mathematics subject classification 1985 revision primary 34a05 70h05 53c57 81c25 partially supported by a horace a rockham faculty grant at the university of michigan r 1987 american mathematical society 787 0002 9947 87 1 00 25 per page this content downloaded from 207 46 13 113 on thu 06 oct 2016 04 24 47 utc all use subject to http about jstor org terms

new shape functions for triangular p fem using integrated jacobi polynomials in this paper the second order boundary value problem x y u f is discretized by the finite element method using piecewise polynomial functions of degree p on a triangular mesh on the reference element we define integrated jacobi polynomials as interior ansatz functions if is a constant function on each triangle and each triangle has straight edges we prove that the element stiffness matrix has not more than nonzero matrix entries an application for preconditioning is given numerical examples show the advantages of the proposed basis

domain decomposition techniques for reaction diffusion equations in two dimensional regions with re entrant corners a system of two non linear reaction diffusion equations is solved numerically by means of linearized q methods and both overlapping and non overlapping domain decomposition techniques in two dimensional regions with re entrant corners two numerical methods based on either approximate factorization af or the bi conjugate gradient stabilized bicgstab technique are employed a study of the effects of the number of overlapping grid lines on both the accuracy and numerical efficiency is presented for non overlapping domain decomposition techniques the unknown values at the common interface between adjacent subdomains have been updated by means of dirichlet neumann and robin couplings and combinations thereof it is shown that non overlapping domain techniques are less accurate than overlapping ones for domains with re entrant corners because the interfaces between adjacent subdomains are evaluated by imposing continuity of the unknowns and their normal derivatives there and therefore the partial differential equations are not solved at the interfaces between adjacent subdomains nevertheless the accuracy of these techniques increases as the grid spacing is decreased although they still exhibit large errors near the re entrant corners

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

bps preconditioners for a weak galerkin finite element method for 2d diffusion problems with strongly discontinuous coefficients abstract we analyze two bps preconditioners for a weak galerkin wg finite element method for 2d diffusion equations with strongly discontinuous coefficients the first preconditioner uses nonconforming linear elements on the coarse mesh and the corresponding condition number is bounded from above by c 1 ln h h 3 with c independent of the coefficients the second one uses h 1 conforming linear elements on the coarse mesh and the corresponding condition number is bounded from above by c 1 ln h h 2 with c depending on the coefficients in addition we construct and analyze a preconditioner for the sub problems encountered in the procedure of applying the two preconditioners the condition number of the preconditioned system is bounded from above by c 1 ln h h 2 with c independent of the coefficients finally we carry out some numerical experiments to verify our theoretical results

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

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

sufficient conditions for a conjecture of ryser about hadamard circulant matrices let h be a hadamard circulant matrix of order n 4h2 where h 1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial n t are bounded by a constant independent of n then for all the n n th primitive roots w of 1 p w n is not an algebraic integer in the cyclotomic field k q w where p t is the representer polynomial of h and is the euler function this implies that p w is not a real number

multiplicities formula for geometric quantization part ii 1 introduction let p be a compact manifold let h be a compact lie group acting on the right on p we assume that the stabilizer of each element y e p is a finite subgroup of h the space m p h is an orbifold and every orbifold can be presented this way if h acts freely then m is a manifold if ae graded space of h invariant solutions of the h horizontal dirac operator on p twisted by the line bundle let m o 9 be the moment map for the g action assume 0 is a regular value of let mred be the reduced orbifold of m that is mred o g consider the reduced orbifold line bundle rd lu lto g on m oa in the ease where both g andh are torus we prove here the formula

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter

ginzburg weinstein via gelfand zeitlin let u n be the unitary group and u n the dual of its lie al gebra equipped with the kirillov poisson structure in their 1983 paper guillemin sternberg introduced a densely defined hamil tonian action of a torus of dimension n 1 n 2 on u n with mo ment map given by the gelfand zeitlin coordinates a few years later flaschka ratiu described a similar multiplicative gelfand zeitlin system for the poisson lie group u n by the ginzburg weinstein theorem u n is isomorphic to u n as a poisson manifold flaschka ratiu conjectured that one can choose the ginzburg weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear gelfand zeitlin systems our main result gives a proof of this conjecture and produces a canonical ginzburg weinstein diffeomorphism

a p version finite element approach for two and three dimensional problems of the j2 flow theory with non linear isotropic hardening in this paper an implementation of a two and three dimensional p version approach to the j 2 flow theory with non linear isotropic hardening for small displacements and small strains is presented based on higher order quadrilateral and hexahedral element formulations a newton raphson iteration scheme combined with a radial return algorithm is applied to find approximate solutions for the underlying physically non linear model problem curved boundaries are taken care of with the blending function method allowing an accurate representation of geometry with only a few p elements numerical examples demonstrate that the p version supplies efficient and accurate approximations to this class of physically non linear problems