ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

rigidity of coxeter groups let w be a coxeter group acting properly discontinuously and cocompactly on manifolds n and m 9m 0 such that the fixed point sets of finite subgroups are contractible let f n n mx dk m xsk1 be a w homotopy equivalence which restricts to a w homeomorphism on the boundary under an assumption on the three dimensional fixed point sets we show that then f is w homotopic to a w homeomorphism

the periodicity in stable equivariant surgery the classical surgery theory see 5 and 23 computes the structure set sm m rel of manifolds homotopy equivalent to m relative to the boundary siebenmann showed that in topological category the structure set is 4 periodic sm m rel sm 4 m d4 rel up to a copy of see 12 cappell and weinberger gave a geometric interpretation of this periodicity in 8 by using weinberger s stratified surgery theory see 24 we extend this to an equivariant periodicity result for topological manifolds with homotopically stratified actions by compact lie groups with d4 replaced by the unit ball of certain group representations in particular if g is an odd order group acting on a topological manifold m then the equivariant stable structure sets satisfy s m rel s m d 4 g rel up to copies of 1993 john wiley sons inc

on the relationship between monstrous moonshine and the uniqueness of the moonshine module we consider the relationship between the conjectured uniqueness of the moonshine module open image in new window and monstrous moonshine the genus zero property of the modular invariance group for each monster group thompson series we first discuss a family of possiblezn meromorphic orbifold constructions of open image in new window based on automorphisms of the leech lattice compactified bosonic string we reproduce the thompson series for all 51 non fricke classes of the monster groupm together with a new relationship between the centralisers of these classes and 51 corresponding conway group centralisers generalising a well known relationship for 5 such classes assuming that open image in new window is unique we consider meromorphic orbifoldings of open image in new window and show that monstrous moonshine holds if and onlyzr if the only meromorphic orbifoldings of open image in new window are open image in new window itself or the leech theory this constraint on the meromorphic orbifoldings of open image in new window therefore relates monstrous moonshine to the uniqueness of open image in new window in a new way

topological classification of affine operators on unitary and euclidean spaces abstract we study affine operators on a unitary or euclidean space u up to topological conjugacy an affine operator is a map f u u of the form f x a x b in which a u u is a linear operator and b u two affine operators f and g are said to be topologically conjugate if g h 1 fh for some homeomorphism h u u if an affine operator f x a x b has a fixed point then f is topologically conjugate to its linear part a the problem of classifying linear operators up to topological conjugacy was studied by kuiper and robbin topological classification of linear endomorphisms invent math 19 2 1973 83 106 and other authors let f u u be an affine operator without fixed point we prove that f is topologically conjugate to an affine operator g u u such that u is an orthogonal direct sum of g invariant subspaces v and w the restriction g v of g to v is an affine operator that in some orthonormal basis of v has the form x 1 x 2 x n x 1 1 x 2 x n 1 e x n uniquely determined by f where e 1 if u is a unitary space e 1 if u is a euclidean space and n 2 if e 1 and the restriction g w of g to w is a linear operator that in some orthonormal basis of w is given by a nilpotent jordan matrix uniquely determined by f up to permutation of blocks

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

horn silver distributions in d dimensional euclidean spaces a geometric interpretation is found for a class of finite discrete probability distributions that generalizes the distribution found by horn and silver to represent an unusual distribution of neutral versus charged pions in some hadronic production models connections to de finetti s representation theorem are discussed

on the classification of g spheres i equivariant transversality this paper is the first in a series of three stated in geometric terms the papers examine locally linear group actions on spheres for odd order groups g in essentially equivalent homotopy theoretic terms the papers study the homotopy types of the spaces plc v and tope v of equivariant pl homeomorphisms and homeomorphisms of a linear representation v in fact it is the homogeneous spaces f6 v plg v and top6 v pl6 v we study where fc v is the space of proper equivariant homotopy equivalences our results generalize theorems of haefliger kirby siebenmann sullivan and wall and others the link between classification of manifolds and classification of homotopy types is transversality and this is the subject of the present first paper the second paper generalizes wall s classification of fake lens spaces to the classification of g spheres which are equivariantly homotopy equivalent to a given linear action this involves the determination of the equivariant homotopy type of the g space f pl and a complete calculation of the pl equivariant surgery sequence for a linear g sphere as a result we show that the homotopy groups of pl v u plc v vanish in dimensions less than dim v g in the third paper we study g triangulation theory in particular we study the homotopy groups of top6 v u topc v in a range the homotopy groups of the stiefel spaces plc v u plc v and topc v u top6 v are in turn needed for equivariant transversality so the three papers are locked together in an inductive fashion the reader is referred to w 4 below and to the individual parts ii and iii for more information we now give a brief discussion of the content of the present paper let g denote an arbitrary finite group we consider the g transversality question in the locally linear

some consequences of exchangeability in random matrix theory properties of infinite sequences of exchangeable random variables result directly in explicit expressions for calculating asymptotic densities of eigenvalues of any ensemble of random matrices h whose distribution depends only on tr h h where h is the hermitian conjugate of h for real symmetric matrices and for hermitian matrices the densities are constructed by summing up wigner semicircles with varying radii and weights as confirmed by monte carlo simulations extensions to more general matrix ensembles are also considered

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

topological classification of systems of bilinear and sesquilinear forms abstract let a and b be two systems consisting of the same vector spaces c n 1 c n t and bilinear or sesquilinear forms a i b i c n k i c n l i c for i 1 s we prove that a is transformed to b by homeomorphisms within c n 1 c n t if and only if a is transformed to b by linear bijections within c n 1 c n t

a nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes a nine point scheme is presented for discretizing diffusion operators on distorted quadrilateral meshes the advantage of this method is that highly distorted meshes can be used without the numerical results being altered remarkably and it treats material discontinuities rigorously and offers an explicit expression for the face centered flux moreover it has only the cell centered unknowns we prove that the method is stable and has first order convergence on distorted meshes numerical experiments show that the method has second order or nearly second order accuracy on distorted meshes

nonlinear similarity begins in dimension six the real representations pi and p2 of a finite group g are topologically similar written pi t p2 if there is a homeomorphism h v p1 v pa where v pi denotes the vector space of the representation pi such that h p1 g v p2 g h v for v e v p1 and g e g i e the representation spaces are equivariantly homeomorphic de rham dr conjectured that topological similarity implies the linear equivalence of the two representations the first two authors showed that de rham s conjecture is true for representations of dimension less than six cs6 but false for representations of dimension greater than or equal to nine cs2 here we give examples of nonlinear similarity i e topological similarity between linearly inequivalent representations in dimension six which is therefore the minimal dimension in which nonlinear similarity may occur since linear equivalence is detected by cyclic subgroups a minimal counterexample to de rham s conjecture will occur for g cyclic moreover by hp and mr nonlinear similarity may only occur for groups whose order is divisible by four thus we shall restrict attention to g z4q the cyclic group of order 4q we shall construct six dimensional nonlinear similarities of z4q for every q greater than two moreover we shall show in a later paper cssww2 joint with weinberger that the examples here represent all six dimensional nonlinear similarities of cyclic groups actually we shall give the complete topological classification of representations of the form p 6 c where p is free i e z4q acts freely away from the origin in the representation space of p and 6 and e are the nontrivial and trivial one dimensional representations respectively representations of the form p 6 were studied in cs2 suppose that pi 6 c 1t p2 6 e with p1 and p2 free let lp be the orbit space of the unit

efficiency of mixed hybrid finite element and multipoint flux approximation methods on quadrangular grids and highly anisotropic media the mixed hybrid finite element mhfe and the multipoint flux approximation mpfa methods are well suited for anisotropic heterogeneous domains since both are locally conservative and can handle general irregular grids in this work behaviours and performances of mhfe and mpfa methods are studied numerically for different heterogeneities and anisotropy factors on parallelograms and then on a more general quadrilateral grid the superiority of mpfa in terms of accuracy and efficiency is clearly demonstrated for parallelogram grids in the case of more general quadrilateral grids mpfa becomes more central processing unit time consuming than mhfe for high anisotropy factors both methods give results with significant non physical oscillations copyright 2008 john wiley sons ltd

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

the incorporation of the semi implicit linear equations into newton s method to solve radiation transfer equations abstract for large time steps the nonlinear equations of radiation transfer may not be solved adequately by the semi implicit linear approximation to yield physical solutions this deficiency is rectified in three steps the equations of the semi implicit linear method are modified the modified equations are incorporated into newton s method to solve nonlinear equations and the transfer equations are solved by the resulting method the new method also uses the photon free method to search for the solution in a lower dimensional space than the space of the underlying transfer equations two algorithms are developed from the new method they solve the modified semi implicit linear equations by different approaches the first is a physics approach it solves the linear equations approximately by the grey transport approximation the second is a mathematical approach it solves the linear equations exactly by the sherman morrison woodbury formula of linear algebra however both algorithms yield the solution to the nonlinear system derived by the photon free method therefore their solutions are equal to within the specified tolerance of the nonlinear solver moreover both methods can take advantage of the unconditional stability which comes with the implicit differencing of the time derivative the time step which both methods can take is much larger than the time step in which time discretization error is discernible we shall relate the mathematical approach to the photon free method numerical results for three test problems are presented

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

the mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes we study the mimetic finite difference discretization of diffusion type problems on unstructured polyhedral meshes we demonstrate high accuracy of the approximate solutions for general diffusion tensors the second order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes on non matching meshes and even on meshes with irregular shaped polyhedra with flat faces we show that in general the meshes with non flat faces require more than one flux unknown per mesh face to get optimal convergence rates

a piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids we develop a piecewise linear pwl galerkin finite element spatial discretization for the multi dimensional radiation diffusion equation it uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal 2d or polyhedral 3d grids we first demonstrate some analytical properties of the pwl method and perform a simple mode analysis to compare the pwl method with palmer s vertex centered finite volume method and with a bilinear continuous finite element method we then show that this new pwl method gives solutions comparable to those from palmer s however since the pwl method produces a symmetric positive definite coefficient matrix it should be substantially more computationally efficient than palmer s method which produces an asymmetric matrix we conclude that the galerkin pwl method is an attractive option for solving diffusion equations on unstructured grids

symmetric positive definite flux continuous full tensor finite volume schemes on unstructured cell centered triangular grids novel cell centered full tensor finite volume methods are presented for general unstructured grids in two spatial dimensions the numerical schemes are flux continuous and based on computing the transmissibilities in a local subcell transform space ensuring that local flux matrices are symmetric as a result the global discretization matrix is shown to be symmetric positive definite for any grid type a symmetric physical space method is also introduced and the symmetric methods are shown to be closely related discrete ellipticity conditions are derived for positive definiteness of the physical space and subcell space schemes computational examples are presented for unstructured triangular grids demonstrating good performance of the scheme the schemes are compared with the so called multipoint flux approximation mpfa o method i aavatsmark t barkve o boe and t mannseth siam j sci comput 19 1998 pp 1700 1716 good agreement between the methods is obtained but the new scheme shows improved behavior in challenging cases

regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings abstract by kronecker s theorem each matrix pencil a b over a field f is strictly equivalent to its regularizing decomposition i e a direct sum i r d m 1 n 1 m t n t where d is an r r nonsingular matrix and each m i n i is of the form i k j k 0 j k 0 i k l k r k or l k t r k t in which l k and r k are obtained from i k by deleting its last or respectively first row and j k 0 is a singular jordan block we give a method for constructing a regularizing decomposition of an m n matrix pencil a b which is formulated in terms of the linear mappings a b f n f m two m n pencils a b and a b over f r or c are said to be topologically equivalent if the pairs of linear mappings a b f n f m and a b f n f m coincide up to homeomorphisms of the spaces f n and f m we prove that two pencils are topologically equivalent if and only if their regularizing decompositions coincide up to permutation of summands and replacement of d by a nonsingular matrix d such that the linear operators d d f r f r coincide up to a homeomorphism of f r

the repair paradigm new algorithms and applications to compressible flow the repair paradigm leads to several algorithms for redistributing mass momentum and energy while adhering to local maximum principles as an adjunct to the remapping step in certain compressible flow codes that use remapping such as arbitrary lagrangian eulerian codes or for just redistributing mass in advection codes in the case of advection of a concentration repair keeps the newly computed concentration in a cell between the maximum and minimum concentrations in neighboring old cells thus guaranteeing at least that the new concentration is between zero and one for compressible flow density velocity and internal energy are similarly constrained while maintaining conservation of mass momentum and total energy in this way positive density and internal energy are achieved as a side effect we propose a new algorithm combining both local and global repair that maintains causality and is efficient in a parallel computational setting the local global algorithm is independent of the order in which the distribution is performed and it maintains 1d symmetry this is applied to advection in two dimensions and to among others the leblanc problem the sedov problem and an interacting 2d blast wave problem the latter is done with a lagrangian code for which rezoning remapping and repair are essential

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

symbolic implicit monte carlo radiation transport in the difference formulation a piecewise constant discretization the equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit one source of this difficulty is that in thick absorbing media thermal emission and absorption are almost completely balanced a new formulation for thermal radiation transport called the difference formulation was recently introduced in order to remove the stiff balance between emission and absorption in the new formulation thermal emission is replaced by derivative terms that become small in thick media it was proposed that the difficulties of solving the transport equation in thick media would be ameliorated by the difference formulation while preserving full rigor and accuracy of the transport solution in the streaming limit in this paper the transport equation is solved by the symbolic implicit monte carlo method and comparisons are made between the standard formulation and the difference formulation the method is easily adapted to the derivative source terms of the difference formulation and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media

a discrete duality finite volume approach to hodge decomposition and div curl problems on almost arbitrary two dimensional meshes we define discrete differential operators such as gradient divergence and curl on general two dimensional nonorthogonal meshes these discrete operators verify discrete analogues of usual continuous theorems discrete green formulae discrete hodge decomposition of vector fields and vector curls have a vanishing divergence and gradients have a vanishing curl we apply these ideas to discretize div curl systems we give error estimates based on the reformulation of these systems into equivalent equations for the potentials numerical results illustrate the use of the method on several types of meshes some of which are degenerating triangular meshes and nonconforming locally refined meshes

rigidity and crystallographic groups i this is the first in a series of articles about rigidity theorems for crystallographic groups the main theorem is 1 1 it is an h cobordism rigidity statement it implies see 1 6 that any crystallographic manifold with odd order holonomy which is h cobordant and simply homotopically equivalent to the standard one is actually homeomorphic to it in the second part of this work 9 we will exploit this result to prove that one can remove the word h cobordant from the previous sentence the main theorem concerns an involution on the group wh p mr this is the group of g h cobordisms of the flat torus mr of a crystallographic group f with holonomy group g definitions of these terms appear in 1 4 we prove a vanishing result of the tate cohomology

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

approximation by equivariant homeomorphisms i locally linear locally smoothable actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other for these actions an equivariant o approximation theorem is proved using engulfing techniques as corollaries are obtained equivariant fibrations are bundles and controlled h cobordism theorems as well as an equivariant version of edwards cell like mapping theorem and the vanishing of the set of transfer invariant g homotopy topological structures rel bound ary on tn x dp when tn is the n torus with trivial g action and dp is a representation disc here we consider locally linear locally smooth br pl and topological actions of a finite group g on n manifolds in addition we require that a g manifold m have gaps of codimension 3 i e for h c g if mh is a component of the h fixed point set mh and m h c mh is the subspace of points x for which the isotropy subgroup gx strictly contains h then either m h mh or m h has codimension 3 in mh we shall use the term not necessarily locally linear g manifold when we wish to drop the local linearity hypothesis but retain the property that if mh c m but mh m then mh is a locally flat inequivariantly submanifold of mk of codimension at least three this while self contained is the second in a series of papers sw2 8 s in which we analyze the extent of failure of the topological invariance of equivari ant whitehead torsion and the consequent failure of subgroups of the equivariant whitehead group whgl m of illman ill cf rothenberg r to classify pl or smooth g h cobordisms up to topological equivalence under our gap hypotheses the arguments of browder and quinn bq and rothenberg r show that g h cobordisms are classified up to pl or smooth equivalence by such subgroups we eventually conclude in sw7 s cf sw2 that the topological equivalence classes of g h cobordisms on a g manifold m that are products over the union of fixed point components of m having dimension 5 are classised by a group whg p m to which whgl m maps homomorphically the image in whg p m of the per tinent subgroup of whgl m classifies up to equivariant homeomorphism rel m those g h cobordisms on m that admit equivariant handlebody decompositions relative to m this homomorphism is not generally surjective and there are g h cobordisms that do not admit such handle decompositions cf sw5 s in order

an improved monotone finite volume scheme for diffusion equation on polygonal meshes we construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes the new scheme uses the cell edge unknowns instead of cell vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes specially numerical results show that the new scheme is robust and more accurate than the existing monotone scheme on some kinds of meshes

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

a family of linearity preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids a family of cell centered finite volume schemes are proposed for anisotropic diffusion problems on arbitrary polyhedral grids with planar facets the derivation of the schemes is done under a general framework through a certain linearity preserving approach the key ingredient of our algorithm is to employ solely the so called harmonic averaging points located at the cell interfaces to define the auxiliary unknowns which not only makes the interpolation procedure for auxiliary unknowns simple and positivity preserving but also reduces the stencil of the schemes the final schemes are cell centered with a small stencil of 25 point on the structured hexahedral grids moreover the schemes satisfy the local conservation condition treat discontinuity exactly and allow for a simple stability analysis a second order accuracy in the l2 norm and a first order accuracy in the h1 norm are observed numerically on general distorted meshes in case that the diffusion tensor is anisotropic and discontinuous

a finite volume method for approximating 3d diffusion operators on general meshes a finite volume method is presented for discretizing 3d diffusion operators with variable full tensor coefficients this method handles anisotropic non symmetric or discontinuous variable tensor coefficients while distorted non matching or non convex n faced polyhedron meshes can be used for meshes of polyhedra whose faces have not more than four edges the associated matrix is positive definite and symmetric if the diffusion tensor is symmetric a second order resp first order accuracy is numerically observed for the solution resp gradient of the solution

a subcell remapping method on staggered polygonal grids for arbitrary lagrangian eulerian methods we describe a new remapping algorithm for use in arbitrary lagrangian eulerian ale simulations the new features of this remapper are designed to complement a staggered mesh lagrangian phase in which the cells may be general polygons in two dimensions and which uses subcell discretizations to control unphysical mesh distortion and hourglassing our new remapping algorithm consists of three stages a gathering stage in which we interpolate momentum internal energy and kinetic energy to the subcells in a conservative way a subcell remapping stage in which we conservatively remap mass momentum internal and kinetic energy from the subcells of the lagrangian mesh to the subcells of the new rezoned mesh a scattering stage in which we conservatively recover the primary variables subcell density nodal velocity and cell centered specific internal energy on the new rezoned mesh we prove that our new remapping algorithm is conservative reversible and satisfies the debar consistency condition we also demonstrate computationally that our new remapping method is robust and accurate for a series of test problems in one and two dimensions

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

semi implicit time integration for pn thermal radiative transfer implicit time integration involving the solution of large systems of equations is the current paradigm for time dependent radiative transfer in this paper we present a semi implicit linear discontinuous galerkin method for the spherical harmonics p n equations for thermal radiative transfer in planar geometry our method is novel in that the material coupling terms are treated implicitly via linearizing the emission source and the streaming operator is treated explicitly using a second order accurate runge kutta method the benefit of this approach is that each time step only involves the solution of equations that are local to each cell this benefit comes at the cost of having the time step limited by a cfl condition based on the speed of light to guarantee positivity and avoid artificial oscillations we use a slope limiting technique we present analysis and numerical results that show the method is robust in the diffusion limit when the photon mean free path is not resolved by the spatial mesh also in the diffusion limit the time step restriction relaxes to a less restrictive explicit diffusion cfl condition we demonstrate with numerical results that away from the diffusion limit our method demonstrates second order error convergence as the spatial mesh is refined with a fixed cfl number

asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo we perform an asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo imc a monte carlo technique for simulating nonlinear radiative transfer specifically we examine the approximation of absorption and re emission by a spatially continuous artificial scattering process and either a piecewise constant or piecewise linear emission source within each spatial cell we consider three asymptotic scalings representing i a time step that resolves the mean free time ii a courant limit on the time step size and iii a fixed time step that does not depend on any asymptotic scaling for the piecewise constant approximation we show that only the third scaling results in a valid discretization of the proper diffusion equation which implies that imc may generate inaccurate solutions with optically large spatial cells if time steps are refined however we also demonstrate that for a certain class of problems the piecewise linear approximation yields an appropriate discretized diffusion equation under all three scalings we therefore expect imc to produce accurate solutions for a wider range of time step sizes when the piecewise linear instead of piecewise constant discretization is employed we demonstrate the validity of our analysis with a set of numerical examples

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

a decoupled and positivity preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes abstract we suggest a new positivity preserving discrete duality finite volume ddfv scheme for anisotropic diffusion problems on general polygonal meshes like existing ddfv schemes this scheme is built on the primary and dual meshes and has finite volume fv equations for both vertex and cell centered unknowns the transpose of the coefficient matrix for the fv equations of the cell centered unknowns is an m matrix while that for the vertex unknowns is not an m matrix but a symmetric and positive definite matrix by employing a certain truncation technique for the vertex unknowns the positivity preserving property for both categories of unknowns is guaranteed local conservation is strictly maintained for the cell centered unknowns and conditionally maintained for the vertex unknowns since the fv equations of the vertex unknowns can be solved independently the two sets of fv equations are decoupled in contrast to existing nonlinear positivity preserving schemes the new scheme requires no nonlinear iterations for linear problems for nonlinear problems the positivity preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted moreover the positivity of the discrete solution is proved and the well posedness is analyzed rigorously for linear problems in numerical experiments the new scheme is examined extensively and compared with two positivity preserving cell centered schemes and a nonlinear ddfv scheme numerical results show that the present ddfv scheme achieves second order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations

circumspheres of sets of n 1 random points in the d dimensional euclidean unit ball 1 n d in the d dimensional euclidean space any set of n 1 independent random points uniformly distributed in the interior of a unit ball of center o determines almost surely a circumsphere of center c and radius 1 n d and an n flat 1 n d 1 the orthogonal projection of o onto this flat is called o while designates the distance o c the classical problem of the distance between two random points in a unit ball corresponds to n 1 the focus is set on the family of circumspheres which are contained in this unit ball for any d 2 and 1 n d 1 the joint probability density function of the distance o c and circumradius has a simple closed form expression the marginal probability density functions of and are both products of powers and a gauss hypergeometric function stochastic representations of the latter random variables are described in terms of geometric means of two independent beta random variables for n d 1 and have a joint dirichlet distribution with parameters d d2

recognition of linear actions on spheres let g be a finite group acting smoothly on a homotopy sphere ym we wish to establish necessary and sufficient conditions for the given g action on i to be topologically equivalent to a linear action that is we want to be able to decide whether or not there exists a g homeomorphism y i sm p where sm p c r i p denotes the unit sphere in an orthogonal representation space r p for g in order for a g action on i to be topologically equivalent to a linear action it is clearly necessary that i for each subgroup h of g the fixed point set h is homeomorphic to a sphere or empty ii for any subgroups h and h c h i s i s k of g the pair h u i1lh is homeomorphic to a standard pair s uk is where each sr i s i s k is a standard n subsphere of sn in this paper we consider the case where the fixed point set eg is nonempty and all other fixed point sets have dimension at least 5 in giving efficient sufficient conditions we do not need the full strength of condition ii we only need ii for any subgroups h and h c h i s i s p of g such that dim ih dim h 2 the pair ih uft1 h is homeomorphic to a standard pair s n up sin 2 where each s0 2 1 s i s p is a standard n 2 subsphere of n our main results are then that in the case when g is abelian conditions i and ii are riecessary and sufficient for a given g action on i to be topologically equivalent to a linear action and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension i and 2 fixed point situation is simple our results generalize for actions of finite groups a well known theorem of connell montgomery and yang and are the first to also cover the case where codimension 2 fixed point situations occur let g be a finite group acting smoothly on a homotopy sphere 2m a homotopy sphere 2m is a smooth manifold which is homeomorphic to sm we wish to establish necessary and sufficient conditions for the given g action on l to be topologically equivalent to a linear action that is we want to be able to decide whether or not there exists a g homeomorphism y l sm p where sm p c rn l p denotes the unit sphere in an orthogonal representation space rn l p for g in order for a given g action on l to be topologically equivalent to a linear action it is clearly necessary that the following two conditions hold i for each subgroup h of g the fixed point set 2h is homeomorphic to a sphere or empty received by the editors january 7 1981 1980 mathematics subject classification primary 57s17 57s25 secondary 57q30

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

computation of canonical matrices for chains and cycles of linear mappings van dooren linear algebra appl 27 1979 103 constructed an algorithm for the computation of all irregular summands in kronecker s canonical form of a matrix pencil the algorithm is numerically stable since it uses only unitary transformations we construct a unitary algorithm for computation of the canonical form of the matrices of a chain of linear mappings v1 v2 vt andextend van dooren s algorithm to the matrices of a cycle of linear mappings whereall vi are complex vector spaces and each line denotes or

topological classification of sesquilinear forms reduction to the nonsingular case abstract two sesquilinear forms c m c m c and c n c n c are called topologically equivalent if there exists a homeomorphism c m c n i e a continuous bijection whose inverse is also a continuous bijection such that x y x y for all x y c m r a horn and v v sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix a that is a direct sum s a s r j n 1 j n p in which s and r are nonsingular and each j n i is the n i by n i singular jordan block in this paper we prove that and are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands j n i and replacement of r c r r by a nonsingular matrix r c r r such that r and r are the matrices of topologically equivalent forms c r c r c analogous results for bilinear forms over c and over r are also obtained

approximating second order vector differential operators on distorted meshes in two space dimensions a new finite volume method is presented for approximating second order vector differential operators in two space dimensions this method allows distorted triangle or quadrilateral meshes to be used without the numerical results being too much altered the matrices that need to be inverted are symmetric positive definite therefore the most powerful linear solvers can be applied the method has been tested on a few second order vector partial differential equations coming from elasticity and fluids mechanics areas these numerical experiments show that it is second order accurate and locking free copyright 2008 john wiley sons ltd

high order multi material ale hydrodynamics we present a new approach for multi material arbitrary lagrangian eulerian ale hydrodynamics simulations based on high order finite elements posed on high order curvilinear meshes the method builds on and extends our previous work in the lagrangian v a dobrev t v kolev and r n rieben siam j sci comput 34 2012 pp b606 b641 and remap r w anderson et al internat j numer methods fluids 77 2015 pp 249 273 phases of ale and depends critically on a functional perspective that enables subzonal physics and material modeling v a dobrev et al internat j numer methods fluids 82 2016 pp 689 706 curvilinear mesh relaxation is based on node movement which is determined through the solution of an elliptic equation the remap phase is posed in terms of advecting state variables between two meshes over a fictitious time interval the resulting advection equation is solved by a discontinuous galerkin dg formulation combined with a customized flux corrected transpor

generalized binomial distributions the definition properties and calculational advantages of the binomial transform representation of a class of finitely enumerated discrete probability distributions that generalize the standard binomial distribution are investigated discrete distributions of this type appear in various models of hadron multiparticle production among other physical applications the compact representations of the generating functions for factorial moments of these discrete distributions obtained using the binomial transform are used to relate the behavior of the complex zeros of the generating functions to different models for the distribution of charged and neutral particles in multihadron production

topological classification of chains of linear mappings abstract we consider systems of linear mappings a 1 a t 1 of the form a u 1 a 1 2 a 2 u 3 a 3 a t 1 u t in which u 1 u t are unitary or euclidean spaces and each line is either the arrow or the arrow let a be transformed to b v 1 b 1 v 2 b 2 v 3 b 3 b t 1 v t by a system i u i v i i 1 t of bijections we say that a and b are linearly isomorphic if all i are linear considering all u i and v i as metric spaces we say that a and b are topologically isomorphic if all i and i 1 are continuous we prove that a and b are topologically isomorphic if and only if they are linearly isomorphic

a deterministic photon free method to solve radiation transfer equations a new method to solve radiation transfer equations is presented in the absence of scattering material motion and heat conduction the photon variables can be eliminated from the fully implicit multi group discrete ordinate finite difference finite element equations of continuum radiation transfer to yield a smaller set of equations which depends only on temperature the solution to this smaller set of equations is used to generate the solution to the original set of equations from which the reduced set is derived the reduced system simplifies to a nonlinear heat equation in the regime of strong absorption and strong emission we solve the reduced set of equations by the newton gmres method in which the jacobian update is preconditioned by a linearization of this nonlinear heat equation the performances of this new method and of the semi implicit linear method which is preconditioned by grey transport acceleration combined with diffusion synthetic acceleration are compared on two test problems the test results indicate that the new method can take larger time steps requires less memory is more accurate and is competitive in speed with the semi implicit linear method

analytic and topological torsion for manifolds with boundary and symmetry letg be a finite group acting on a riemannian manifoldm by isometries we introduce analytic torsion an m m1 v r z repr g pl torsion pl m m1 v k1 rg poincare torsion pd m m1 v k1 rg and euler characteristic m m1 v repr g for m the disjoint union of m1 and m2 and v an equivariant coefficient system the analytic torsionis defined in terms of the spectrum of the laplace operator the pl torsionis based on the cellular chain complex and poincare torsionmeasures the failure of equivariant poincare duality in the pl setting which does hold in the analytic context denote by repr g the subgroup of repr g generated by the irreducible representations of real or complex type we define an isomorphism

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

a subzone reconstruction algorithm for efficient staggered compatible remapping staggered grid lagrangian hydrodynamics algorithms frequently make use of subzonal discretization of state variables for the purposes of improved numerical accuracy generality to unstructured meshes and exact conservation of mass momentum and energy for arbitrary lagrangian eulerian ale methods using a geometric overlay it is difficult to remap subzonal variables in an accurate and efficient manner due to the number of subzone subzone intersections that must be computed this becomes prohibitive in the case of 3d unstructured polyhedral meshes a new procedure is outlined in this paper to avoid direct subzonal remapping the new algorithm reconstructs the spatial profile of a subzonal variable using remapped zonal and nodal representations of the data the reconstruction procedure is cast as an under constrained optimization problem enforcing conservation at each zone and node on the remapped mesh provides the set of equality constraints the objective function corresponds to a quadratic variation per subzone between the values to be reconstructed and a set of target reference values numerical results for various pure remapping and hydrodynamics tests are provided ideas for extending the algorithm to staggered grid radiation hydrodynamics are discussed as well as ideas for generalizing the algorithm to include inequality constraints

group actions and equivariant lipschitz analysis the idea that one could prove by analytic methods the topological invariance of characteristic classes was suggested in an influential lecture by singer the concrete realization of this in the case of rational pontrjagin classes was the result of works by sullivan who showed that topological manifolds have essentially unique lipschitz structures and teleman who showed that lipschitz manifolds are the type of object for which one can analyze the signature operator a refinement of the same method proves the ko l 2 orientability of topological bundles much of this can be done for manifolds with group actions the main implication is that for certain sorts of topological group actions including all smooth and pl actions one can construct an equivariant signature operator whose class in equivariant if homology is a topological invariant this class satisfies the g signature theorem the topological method has implications for various precise classification problems e g semifree topological actions on the sphere but here we shall present only the consequences of these lipschitzian ideas a topologically locally linear g manifolds for odd order g have kog 1 2 orientations madsen rothenberg for all finite g there are lipschitz structures but the signature class is a zero divisor for even order groups the same method produces upon application of the localization theorem atiyah singer classes for some actions with nonmanifold fixed set weinberger extending earlier joint work with cappell b for odd order groups topological and linear conjugacy of linear representations are equivalent hsiang pardon madsen rothenberg this is a consequence of a alternatively one can return to the original argument of atiyah bott milnor for semifree representations a stronger result is true the atiyah bott local numbers associated to representations agree for elements that have discrete fixed sets in topologically equivalent representations this distinguishes many pairs of representations of even order groups c r top g r lip g for all finite g where r cat is the grothendieck group of representations up to cat conjugacy weinberger after inverting 2 this implies that there is no analytic definition of reidemeister torsion that works well in the lipschitz setting the reason is that following

topological equivalence of linear representations for cyclic groups i in the two parts of this paper we prove that the reidemeister torsion invariants determine topological equivalence of g representations for g a finite cyclic group

delaunay s mesh of a convex polyhedron in dimension d application to arbitrary polyhedra this paper presents a method for creating a delaunay triangulation connected to a set of specified points the theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach valid for dimensions 2 and 3 which is simple robust and well adapted to computation convex polyhedral and arbitrary polyhedral situations are introduced

nonlinear analogs of linear group actions on spheres introduction this article is based upon a principle which is so standard that it is almost a cliche the first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation to be more specific we shall describe some applications of this to the symmetry questions of topological transformation groups given a topological space x let homeo denote the set of self homeomorphisms of x this is a group under composition of mappings if g is an arbitrary group then a group action of g on i is a homomorphism homeo x frequently we wish to impose some weak assumptions on p for example if g is a topological group then we might want cp to have suitable continuity properties the usual assumption is that the map