constructing exact symmetric informationally complete measurements from numerical solutions recently several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements sic povms or sics and algebraic number theory these conjectures relate the sics to their minimal defining algebraic number field testing or sharpening these conjectures requires that the sics are expressed exactly rather than as numerical approximations while many exact solutions of sics have been constructed previously using grobner bases this method has probably been taken as far as is possible with current computer technology except in special cases where there are additional symmetries here we describe a method for converting high precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the galois symmetries of an sic using this method we have calculated 69 new exact solutions including nine new dimensions where previously only numerical solutions were known which more than triples the number of known exact solutions in some cases the solutions require number fields with degrees as high as 12 288 we use these solutions to confirm that they obey the number theoretic conjectures and address two questions suggested by the previous work

a two stage two level finite difference scheme for moving boundary problems abstract a two stage two level finite difference scheme is devised which after applying a coordinate transformation requires only a single iteration of a modified newton method to produce second order approximations to the solution of nonlinear parabolic moving boundary problems numerical evidence of unconditional stability and second order convergence in space and time to both the solution and the moving boundary is presented for two particular problems the proposed scheme is expected to be competitive with most current front tracking methods

on the robust solution of an isogeometric discretization of bilaplacian equation by using multigrid methods abstract in this paper we propose a multigrid method for the solution of the biharmonic problem isogeometric analysis iga is considered in order to easily obtain h 2 conforming discretizations of the bilaplacian equation which are difficult to get by means of standard finite element methods typically the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task here we achieve such robustness by using multiplicative schwarz methods as smoothers within the multigrid algorithm the design of the proposed solver is also supported by a local fourier analysis lfa which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization the robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one and two dimensional cases

symmetric informationally complete positive operator valued measures a new computer study we report on a new computer study of the existence of d2 equiangular lines in d complex dimensions such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory we provide numerical solutions in all dimensions d 67 and moreover a putatively complete list of weyl heisenberg covariant solutions for d 50 a symmetry analysis of this list leads to new algebraic solutions in dimensions d 24 35 and 48 which are given together with algebraic solutions for d 4 15 and 19

complex hadamard matrices and equiangular tight frames abstract in this paper we give a new construction of parametric families of complex hadamard matrices of square orders and connect them to equiangular tight frames the results presented here generalize some of the recent ideas of bodmann et al 3 and extend the list of known equiangular tight frames in particular a 36 21 frame coming from a nontrivial cube root signature matrix is obtained for the first time

the lie algebraic significance of symmetric informationally complete measurements examples of symmetric informationally complete positive operator valued measures sic povms have been constructed in every dimension 67 however it remains an open question whether they exist in all finite dimensions a sic povm is usually thought of as a highly symmetric structure in quantum state space however its elements can equally well be regarded as a basis for the lie algebra gl d c in this paper we examine the resulting structure constants which are calculated from the traces of the triple products of the sic povm elements and which it turns out characterize the sic povm up to unitary equivalence we show that the structure constants have numerous remarkable properties in particular we show that the existence of a sic povm in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl d c we hope that transforming the problem in this way from a question about quantum state space to a question about lie algebras may help to make the existence problem tractable

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

the stability and response of a flexible rod in a quick return mechanism a quick return mechanism is analyzed for deflection and stability when the rod is considered an euler bernoulli beam the crank is assumed rigid and to be rotating at a constant angular velocity the equations of motion and natural boundary conditions are obtained using hamilton s principle spatial dependence is suppressed using galerkin s method with time dependent pinned pinned overhanging beam modes using a small crank length approximation zones of parametric resonance are found using hsu s method the accuracy of these is verified using a monodromy matrix technique the technique is also used to explore the possibility of resonances not covered by hsu s first order method a particular solution instability is found to exist using hsu s method and is verified by direct numerical integration for a somewhat flexible configuration all instabilities were found to lie outside the range of normal operating speeds for stiffer configurations likely to be found in practice this conclusion can be asserted even stronger at least for small cranks

efficient multigrid methods for solving upwind biased discretizations of the convection equation this paper studies a novel multigrid approach to the solution for a second order upwind biased discretization of the convection equation in two dimensions this approach is based on semi coarsening and well balanced explicit correction terms added to coarse grid operators to maintain on coarse grids the same cross characteristic interaction as on the target fine grid multicolor relaxation schemes are used on all the levels allowing a very efficient parallel implementation the results of the smoothing analysis and numerical tests can be summarized as follows 1 the residual asymptotic convergence rate is about 3 per cycle which far surpasses the theoretical limit 4 3 predicted for standard multigrid algorithms with full coarsening 2 the full multigrid fmg algorithm with two v 0 2 cycles on the target grid and just one v 0 2 cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error 3 a novel algorithm for deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed the computational complexity of this algorithm is nearly optimal comparable with the complexity of the fmg algorithm applied to solve the problem on the optimally spaced target grid

equiangular tight frames and unistochastic matrices we demonstrate that a complex equiangular tight frame composed of n vectors in dimension d denoted etf d n exists if and only if a certain bistochastic matrix univocally determined by n and d belongs to a special class of unistochastic matrices this connection allows us to find new complex etfs in infinitely many dimensions and to derive a method to introduce non trivial free parameters in etfs we present an explicit six parametric family of complex etf 6 16 which defines a family of symmetric povms minimal and maximal possible average entanglement of the vectors within this qubit qutrit family are described furthermore we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix which we apply to find all existing classes of complex etfs containing up to 20 vectors

efficient geometric multigrid implementation for triangular grids this paper deals with a stencil based implementation of a geometric multigrid method on semi structured triangular grids triangulations obtained by regular refinement of an irregular coarse triangulation for linear finite element methods an efficient and elegant procedure to construct these stencils using a reference stencil associated to a canonical hexagon is proposed local fourier analysis lfa is applied to obtain asymptotic convergence estimates numerical experiments are presented to illustrate the efficiency of this geometric multigrid algorithm which is based on a three color smoother

modelling and estimation for a moving boundary problem the displacement of a slightly compressible liquid by another in a porous medium has been modelled this problem which involves a moving boundary has been numerically solved for the one dimensional case by using the galerkin method state and parameter estimation have been carried out using the extended kalman filter

on three grid fourier analysis for multigrid in this paper we present three grid fourier analysis for multigrid methods due to the recursive structure of a multigrid iteration this analysis can be deduced from the well known two grid fourier analysis the coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three grid analysis we apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation the main focus lies on possible improvements by carefully chosen galerkin operators and or by an additional acceleration with restarted gmres gmres m numerical test calculations validate the theoretical predictions

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

hp multigrid as smoother algorithm for higher order discontinuous galerkin discretizations of advection dominated flows part i multilevel analysis the hp multigrid as smoother algorithm hp mgs for the solution of higher order accurate space time discontinuous galerkin discretizations of advection dominated flows is presented this algorithm combines p multigrid with h multigrid at all p levels where the h multigrid acts as smoother in the p multigrid the performance of the hp mgs algorithm is further improved using semi coarsening in combination with a new semi implicit runge kutta method as smoother a detailed multilevel analysis of the hp mgs algorithm is presented to obtain more insight into the theoretical performance of the algorithm as model problem a fourth order accurate space time discontinuous galerkin discretization of the advection diffusion equation is considered the multilevel analysis shows that the hp mgs algorithm has excellent convergence rates both for steady state and time dependent problems and low and high cell reynolds numbers including highly stretched meshes

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

fibonacci lucas sic povms we present a conjectured family of symmetric informationally complete positive operator valued measures which have an additional symmetry group whose size is growing with the dimension the symmetry group is related to fibonacci numbers while the dimension is related to lucas numbers the conjecture is supported by exact solutions for dimensions d 4 8 19 48 124 and 323 as well as a numerical solution for dimension d 844

acceleration of multigrid flow computations through dynamic adaptation of the smoothing procedure abstract the paper presents the development and investigation of an adaptive smoothing as procedure in conjunction with the full multigrid full approximation storage method the latter has been previously implemented by the authors 1 for the incompressible navier stokes equations in conjunction with the artificial compressibility method and forms the basis for investigating the current as approach the principle of adaptive smoothing is to exploit the nonuniform convergence behavior of the numerical solution during the iterations to reduce the size of the computational domain and subsequently to reduce the total computing time the implementation of the as approach is investigated in conjunction with three different adaptivity criteria for two and three dimensional incompressible flows furthermore a dynamic procedure henceforth labeled dynamic adaptivity for defining variably the as parameters during the computation is also proposed and its performance is investigated in contrast to as with constant parameters henceforth labeled static adaptivity both static and dynamic adaptivity can provide similar acceleration but the latter additionally provides more stable numerical solutions and also requires less experimentation for optimizing the performance of the method numerical experiments are presented for attached and separated flows around airfoils as well as for three dimensional flow in a curved channel for external flows the as performs better when it is applied in all grid levels of the multigrid method while for internal flows the best performance is achieved when as is applied in the finest grid only overall the results show that substantial acceleration of multigrid computations can be achieved by using adaptive smoothing

direct multi grid methods for linear systems with harmonic aliasing patterns multi level numerical methods that obtain the exact solution of a linear system are presented the methods are devised by combining ideas from the full multi grid algorithm and perfect reconstruction filters the problem is stated as whether a direct solver is possible in a full multi grid scheme by avoiding smoothing iterations and using different coarse grids at each step the coarse grids must form a partition of the fine grid and thus establishes a strong connection with domain decomposition methods an important analogy is established between the conditions for direct solution in multi grid solvers and perfect reconstruction in filter banks furthermore simple solutions of these conditions for direct multi grid solvers are found by using mirror filters as a result different configurations of direct multi grid solvers are obtained and studied

bccb complex hadamard matrices of order 9 and mubs abstract a new type of complex hadamard matrices of order 9 are constructed the studied matrices are symmetric block circulant with circulant blocks b c c b and form an until now unknown non reducible and non affine two parameter orbit several suborbits are identified including a one parameter intersection with the fourier orbit f 9 4 the defect of this new type of hadamard matrices is observed to vary from a generic value 2 to the anomalous values 4 and 10 for some sub orbits and to 12 and 16 for some single matrices the latter matrices are shown to be related to complete sets of mubs in dimension 9

dynamic analyses of a flexible quick return mechanism by the fixed and variable finite difference grids the finite difference method fdm with fixed and variable grids is proposed to approximate the numerical solutions of a flexible quick return mechanism in the dynamic analysis and simulation the flexible rod is divided into two regions each region with time dependent length is modeled by euler beam theory sufficient stability and convergence conditions are established for these finite difference schemes it is found that for the fixed grid method numerical divergence occurs when the moving boundary moves across any of the neighboring nodes the possibility of break down can be avoided via the variable grid method in which a coordinate transformation is employed to fix the moving boundary numerical results are discussed and provided to justify the stability and convergence

high order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems high mixed order finite difference discretization of optimality systems arising from elliptic nonlinear constrained optimal control problems are discussed for the solution of these systems an efficient and robust multigrid algorithm is presented theoretical and experimental results show the advantages of higher order discretization and demonstrate that the proposed multigrid scheme is able to solve efficiently constrained optimal control problems also in the limit case of bang bang control

fourier mode analysis of multigrid methods for partial differential equations with random coefficients partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models various approaches exist for computing the stochastic characteristics of the solution of such a differential equation in this paper we consider the spectral expansion approach this method transforms the continuous model into a large discrete algebraic system we study the convergence properties of iterative methods for solving this discretized system we consider one level and multi level methods the classical fourier mode analysis technique is extended towards the stochastic case this is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem we show how the convergence properties depend on the particulars of the algorithm on the discretization parameters and on the stochastic characteristics of the model numerical results are added to illustrate some of our theoretical findings

algebraic multigrid for stokes equations a method is investigated for solving stationary or time dependent discrete stokes equations it uses one of the standard flavors of algebraic multigrid for coupled partial differential equations which however is not applied directly to the linear system stemming from discretization but to an equivalent system obtained with a simple algebraic transformation which may be seen as a form of preconditioning in the literal sense a two grid analysis is provided showing that the eigenvalues of the preconditioned matrix are within a region of the complex plane that is both bounded and away from the origin independently of the mesh or grid size as well as of other main problem parameters on the other hand whereas the approach can in principle be combined with any type of algebraic multigrid scheme an investigation of the properties of the coarse grid matrices reveals that plain aggregation has to be preferred to maintain nice two grid convergence at coarser levels eventually numerical experiments are

on the existence of equiangular tight frames abstract an equiangular tight frame etf is a d n matrix that has unit norm columns and orthogonal rows of norm n d its key property is that the absolute inner products between pairs of columns are i identical and ii as small as possible etfs have applications in communications coding theory and sparse approximation numerical experiments indicate that etfs arise for very few pairs d n and it is an important challenge to develop restrictions on the pairs for which they can exist this article uses field theory to provide detailed conditions on real and complex etfs in particular it describes restrictions on harmonic etfs a specific type of complex etf that appears in applications finally the article offers empirical evidence that these conditions are sharp or nearly sharp especially in the real case

multigrid solvers for nonaligned sonic flows we investigate an approach to the solution of nonelliptic equations on a rectangular grid the multigrid algorithms presented here demonstrate the textbook multigrid efficiency even in the case that the equation characteristics do not align with the grid to serve as a model problem the two dimensional 2d and three dimensional 3d linearized sonic flow equations have been chosen efficient full multigrid fmg solvers for the problems are demonstrated

analysis of a multigrid strokes solver model problem analysis or local mode analysis is used to investigate the two level convergence rates for multigrid strokes solvers the solvers are based on the discretization on the staggered grid and the distributive relaxation of the continuity equation the influence of the grid geometry is analyzed by also considering the poisson equation on the semistaggered grid various grid transfer operators are compared for an isotropic h x h y discretization simple tranfer operators are sufficient to obtain optimal rates but not in the anisotropic case the theoretical results are compared with observed multigrid convergence rates

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

general planar dynamics of a sliding flexible link abstract a finite element dynamic model of a sliding link through a prismatic joint where the prismatic joint hub is executing general planar motion is formulated in contrast to previously reported formulations a finite element mesh with a fixed number of elements is used where the element length is constant the time dependent boundary conditions manifested by the prismatic joint constraints are considered a transition element with variable stiffness is introduced at the interface with the joint hub in this finite element formulation all the inertia coupling terms between the beam reference motions and the local elastic deformations are considered in addition the model accounts for the dynamics of the end mass as well as the associated coupling effects numerical simulations and comparisons with results obtained by other methods are presented to demonstrate the validity and accuracy of the model

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

automated transformations for pde systems with application to multigrid solvers an approach for transforming systems of partial differential equations in order to obtain new formulations which are more accessible to numerical solution is studied an algorithm is developed for generating such transformations automatically using symbolic computations employing grobner bases the algorithm is implemented using freely available software this approach along with planned developments will potentially provide a powerful set of tools for handling large systems of partial differential equations

convergence analysis of iterative methods by pseudodifference operators abstract the convergence of iterative methods to solve linear partial differential equations numerically is analyzed by the theory of pseudodifference operators approximate inverses are determined from the symbol of the iterative operator so that implicit and preconditioned methods are covered the behavior of the symbol for higher wave numbers describes the convergence rate for the corresponding error modes the results are applied to krylov subspace methods stationary iterations of gauss seidel type the multigrid algorithm and time stepping methods for flow problems

geometry of the welch bounds abstract a geometric perspective involving grammian and frame operators is used to derive the entire family of welch bounds this perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k tensors tight frames homogeneous polynomials and t designs in particular a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k tensors it is also shown that tightness of the bounds requires tight frames the lack of tight frames of symmetric k tensors in many cases however leads to consideration of sets that come as close as possible to attaining the bounds the geometric derivation is then extended in the setting of generalized or continuous frames the welch bounds for finite sets and countably infinite sets become special cases of this general setting

dimension towers of sics i aligned sics and embedded tight frames algebraic number theory relates sic povms in dimension d 3 to those in dimension d d 2 we define a sic in dimension d d 2 to be aligned to a sic in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d d 2 in a specified way we give 19 mostly numerical examples of aligned sics we conjecture that given any sic in dimension d there exists an aligned sic in dimension d d 2 in all our examples the aligned sic has lower dimensional equiangular tight frames embedded in it if d is odd so that a natural tensor product structure exists we prove that the individual vectors in the aligned sic have a very special entanglement structure and the existence of the embedded tight frames follows as a theorem if d 2 is an odd prime number we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned sic to this dimension

on quantitative analysis methods for multigrid solutions limitations of the classical local mode fourier lmf analysis in application to multigrid solutions of variable coefficient problems are well known this paper introduces alternative very general quantitative analysis methods for multigrid solutions of partial differential equations the methods are applied to available nonperfect multigrid solvers that deal with practical problems the analysis methods considered in this paper focus on the main complementary parts of a multigrid cycle relaxation and coarse grid correction tasks for these parts e g error smoothing for relaxation and reduction of smooth errors for coarse grid correction are assumed to be assigned in advance idealized relaxation ir and idealized coarse grid icg iterations are introduced in these iterations one part of the cycle relaxation for ir iterations and coarse grid correction for icg iterations is replaced with an idealized imitation known to be efficient for the assigned task its complementary part is the actual part of a two grid cycle the analysis compares performances of the actual cycle and idealized iterations the ir and icg iterations are very general and can be directly applied in the most complicated simulations including highly variable or nonlinear coefficients complex geometries and unstructured grids the results of this analysis are not single number estimates they are rather convergence patterns of the iterations that may either confirm or refute expectations indicating what part of the actual solver is inefficient in carrying out the assigned task the generality of the analysis makes it a valuable tool for analyzing complicated large scale computational problems where no other analysis methods are currently available in this paper the ir and icg iterations are applied to model problems of progressive complexity the analysis proved sensitive to very delicate details of the actual multigrid cycle pointing clearly to each instance when a part of the algorithm was inefficient

a multigrid method for constrained optimal control problems we consider the fast and efficient numerical solution of linear quadratic optimal control problems with additional constraints on the control discretization of the first order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints the complementarity conditions are treated by a primal dual active set strategy that serves as outer iteration at each iteration step a kkt system has to be solved here we develop a multigrid method for its fast solution to this end we use a smoother which is based on an inexact constraint preconditioner we present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying elliptic pde problem furthermore when combined with a nested iteration the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the pde solution

a sharpening of the welch bounds and the existence of real and complex spherical t designs the welch bounds for a finite set of unit vectors are a family of inequalities indexed by t 1 2 ldots which describe how evenly spread the vectors are they have important applications in signal analysis where sequences giving equality in the first welch bound are known as welch bound equality sequences or as unit norm tight frames here we consider sequences of vectors giving equality in the higher order welch bounds these are seen to correspond to tight frames for the complex symmetric t tensors which we prove always exist we show that for t 1 the welch bounds can be sharpened for real vectors and again vectors giving equality always exist we give a unified treatment of various conditions for equality in both the real and complex cases in particular we give an explicit description of the corresponding cubature rules t designs our results set up a framework for the construction and classification several configurations of vectors of recent interest these include mutually unbiased bases complex equiangular lines spherical half designs projective t designs and minimisers of the higher order frame potential one interesting consequence is a construction of sets of complex equiangular lines which were previously unknown

local mode analysis of multicolor and composite relaxation schemes abstract we develop local mode analysis of multicolor or more generally multicoupling numerical processes and provide practical procedures for computing their symbols and predicting their convergence and smoothing properties in the context of multigrid methods we apply our analysis including simple numerical examples to multicolor point relaxation especially in the context of the novel composite relaxation a brandt barriers to achieving textbook multigrid efficiency tme in cfd tech report gauss center report wi gc 10 weizmann institute december 1998 o e livne multiscale eigenbasis algorithms ph d thesis the weizmann institute of science rehovot israel 2000 for products of discrete operators such as the biharmonic

fourier analysis for multigrid methods on triangular grids in this paper a local fourier analysis technique for multigrid methods on triangular grids is presented the analysis is based on an expression of the fourier transform in new coordinate systems both in space variables and in frequency variables associated with reciprocal bases this tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids different smoothers for the discrete laplace operator obtained with linear finite elements are analyzed a new three color smoother has been studied and has proven to be the best choice for near equilateral triangles it is also shown that the block line smoothers are more appropriate for irregular triangles numerical test calculations validate the theoretical predictions

rigorous quantitative analysis of multigrid i constant coefficients two level cycle with l 2 norm exact numerical convergence factors for any multigrid cycle can be predicted by local mode fourier analysis for general linear elliptic partial differential equation pde systems with piecewise smooth coefficients in general domains discretized by uniform grids it is proved that in the limit of small meshsizes these predicted factors are indeed obtained provided the cycle is supplemented with a proper processing at and near the boundaries that processing it is proved costs negligible extra computer work apart from mode analysis a coarse grid approximation cga condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly the present part studies the l 2 convergence in one cycle for equations with constant coefficients in the sequel brandt rigorous quantitative analysis of multigrid ii extensions and practical implications manuscript extensions are discussed to many cycles asymptotic convergence to more levels with arbitrary cycle ty

on multigrid solution of high reynolds incompressible entering flows an approach is presented for effectively separating the solution process of the elliptic component of high reynolds incompressible steady entering flow for which classical multigrid techniques are well suited from that of the non elliptic part for which other methods are more effective it is shown by analysis and numerical calculations that such an approach is very effective in terms of asymptotic convergence as well as reduction of errors well below discretization level in a 1 fmg algorithm

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina