a hierarchical multiscale framework for problems with multiscale source terms this paper presents a hierarchical multiscale framework for problems that involve multiscale source terms an assumption on the additive decomposition of the source function results in consistent decoupling of the fully coupled system and constitutes the new method the structure of this decomposition is investigated and its mathematical implications are delineated this method results in variational embedding of fine scale information that is derived from the fine scale equations in the corresponding coarse scale equations it therefore provides a mathematically consistent way of bridging information between disparate spatial scales in the response function that are induced by multiscale forcing functions

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

on the bounded cohomology of semi simple groups s arithmetic groups and products we prove vanishing results for lie groups and algebraic groups over any local field in bounded cohomology the main result is a vanishing below twice the rank for semi simple groups related rigidity results are established for s arithmetic groups and groups over global fields we also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

a measurable group theoretic solution to von neumann s problem we give a positive answer in the measurable group theory context to von neumann s problem of knowing whether a non amenable countable discrete group contains a non cyclic free subgroup we also get an embedding result of the free group von neumann factor into restricted wreath product factors

variational multiscale element free galerkin method for 2d burgers equation a new numerical method which is based on the coupling between variational multiscale method and meshfree methods is developed for 2d burgers equation with various values of re the proposed method takes full advantage of meshfree methods therefore no mesh generation and mesh recreation are involved meanwhile compared with the variational multiscale finite element method a strong assumption that is the fine scale vanishes identically over the element boundaries although non zero within the elements is not needed subsequently two problems which have an available analytical solution of their own are solved to analyze the convergence behaviour of the proposed method finally a 2d burgers equation having large re is solved and the results have also been compared with the ones computed by two other methods the numerical results show that the proposed method can indeed obtain accurate numerical results for 2d burgers equation having large re which does not refer to the choice of a proper stabilization parameter

some prime factorization results for type ii1 factors we prove several unique prime factorization results for tensor products of type ii1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected lie groups of real rank 1 in particular we show that if mathcal r overline otimes mathcal l mathbb f r 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f r m is isomorphic to a subfactor in mathcal r overline otimes mathcal l mathbb f s 1 overline otimes cdot cdot cdot overline otimes mathcal l mathbb f s n for some 2 ri sj then m n

a stabilized mixed finite element method for the incompressible shear rate dependent non newtonian fluids variational multiscale framework and consistent linearization abstract this paper presents a stabilized mixed finite element method for shear rate dependent incompressible fluids the viscosity of the fluid is considered a function of the second invariant of the rate of deformation tensor thus making the shear stress shear strain relation nonlinear the weak form of the generalized navier stokes equations is cast in the variational multiscale vms framework that leads to a two level description of the problem consistent linearization of the fine scale problem with respect to the fine scale velocity field and the use of bubble functions to expand the fine scale trial and test functions lead to an analytical expression for the fine scale velocity along with a definition of the stabilization tensor the ensuing nonlinear stabilized form is presented and the consistent tangent tensor is derived numerical convergence of the proposed method on structured and unstructured meshes that are composed of linear triangles and bilinear quadrilaterals are presented shear thinning and shear thickening effects are investigated via the backward facing step problem and the effects of geometric parameters on the flow characteristics are highlighted time dependent features are investigated via the transient vortex shedding problem and the accuracy and stability properties of the new method are shown

the partition of unity finite element approach with hp refinement for the stationary fokker planck equation abstract in this paper the stationary fokker planck equation fpe is solved for nonlinear dynamical systems using a local numerical technique based on the meshless partition of unity finite element method pufem the method is applied to stationary fpe for 2 3 and 4 d systems and is argued to be an excellent candidate for higher dimensional problems and the transient problem local refinement is applied by introducing higher order polynomials in selected subdomains local p refinement to keep the problem size small while ensuring high approximation accuracy various local approximations are blended using novel pasting functions that provide any desired order of continuity results are compared with existing global and local techniques local p refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of fpe

a multiscale stabilized finite element method for the advection diffusion equation this paper presents a multiscale method that yields a stabilized finite element formulation for the advection diffusion equation the multiscale method arises from a decomposition of the scalar field into coarse resolved scale and fine unresolved scale the resulting stabilized formulation possesses superior properties like that of the supg and the gls methods a significant feature of the present method is that the definition of the stabilization term appears naturally and therefore the formulation is free of any user designed or user defined parameters another important ingredient is that since the method is residual based it satisfies consistency ab initio based on the proposed formulation a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed numerical results show the good performance of the method on uniform skewed as well as composite meshes and confirm convergence at optimal rates 2004 elsevier b v all rights reserved

nonlinear and semilinear spectrum for asymptotically linear or positively homogeneous operators we prove some properties for the nonlinear spectrum f and semilinear spectrum l n when f and n are asymptotically linear or positively homogeneous thus close to a linear operator the results generalize a previous result which required n to be a linear operator and l to be the identity map applying the theorems we prove a result on the existence of a positive eigenvalue and eigenvector for a compact positive operator examples of applications to the study of a three point boundary value problem are also given

statistics of nonlinear stochastic dynamical systems under levy noises by a convolution quadrature approach this paper describes a novel numerical approach to find the statistics of the non stationary response of scalar nonlinear systems excited by levy white noises the proposed numerical procedure relies on the introduction of an integral transform of the wiener hopf type into the equation governing the characteristic function once this equation is rewritten as partial integro differential equation it is then solved by applying the method of convolution quadrature originally proposed by lubich here extended to deal with this particular integral transform the proposed approach is relevant for two reasons 1 statistics of systems with several different drift terms can be handled in an efficient way independently from the kind of white noise 2 the particular form of wiener hopf integral transform and its numerical evaluation both introduced in this study are generalizations of fractional integro differential operators of potential type and grunwald letnikov fractional derivatives respectively

asymptotics of cheeger constants and unitarisability of groups abstract given a group we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of cayley graphs of for increasingly large generating sets the connection hinges on an analytic invariant lit 0 which we call the littlewood exponent finiteness amenability unitarisability and the existence of free subgroups are related respectively to the thresholds 0 1 2 and for lit using graphical small cancellation theory we prove that there exist groups for which 1 lit further applications examples and problems are discussed

some oe and w rigidity results for actions by wreath product groups we use deformation rigidity theory in the von neumann algebra framework to study probability measure preserving actions by wreath product groups in particular we single out large families of wreath product groups satisfying various types of orbit equivalence oe rigidity for instance we show that whenever h k are icc property t groups such that h and k admit stably orbit equivalent action and such that h and k are ergodic then automatically is stably orbit equivalent to and h is stably orbit equivalent to k rigidity results for von neumann algebras arising from certain actions of such groups i e w rigidity results are also obtained

orbit equivalence rigidity and bounded cohomology we establish new results and introduce new methods in the theory of measurable orbit equivalence using bounded cohomology of group representations our rigidity statements hold for a wide uncountable class of groups arising from negative curvature geometry amongst our applications are a measurable mostow type rigidity theorems for products of negatively curved groups b prime factorization results for measure equivalence c superrigidity for orbit equivalence d the first examples of continua of type ii1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

a converse to dye s theorem every non amenable countable group induces orbit inequivalent ergodic equivalence relations on standard borel probability spaces not every free ergodic measure preserving action of f 2 on a standard borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces there is a treeable non hyperfinite borel equivalence relation which is not universal for treeable in the b ordering

on the structural theory of ii1 factors of negatively curved groups ii actions by product groups abstract this paper contains a series of structural results for von neumann algebras arising from measure preserving actions by product groups on probability spaces expanding upon the methods used earlier by the first two authors we obtain new examples of strongly solid factors as well as von neumann algebras with unique or no cartan subalgebra for instance we show that every ii 1 factor associated with a weakly amenable group in the class s of ozawa is strongly solid there is also the following product version of this result any maximal abelian subalgebra of any ii 1 factor associated with a finite product of weakly amenable groups in the class s of ozawa has an amenable normalizing algebra finally pairing some of these results with a cocycle superrigidity result of ioana it follows that compact actions by finite products of lattices in s p n 1 n 2 are virtually w superrigid

variational multiscale element free galerkin method for the water wave problems a new numerical method which is based on the coupling between variational multiscale method and meshfree methods is developed for the water wave problems in which the free surface capturing technique is used to capture the position of the free surface the proposed method takes full advantage of meshfree methods therefore no mesh generation and mesh reconstruction are involved meanwhile due to that the proposed method belongs to meshfree methods thus it is suitable for the highly deformed free surface flow problems finally two water wave problems are solved and the results have also been analyzed the numerical results show that the proposed method can indeed obtain accurate numerical results for the water wave problems which does not refer to the choice of a proper stabilization parameter

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

global bifurcation of the p laplacian and related operators abstract we prove some global bifurcation theorems of rabinowitz type for a large class of operators the results apply in particular to equations involving the p laplacian

product groups acting on manifolds we analyse volume preserving actions of product groups on riemannian manifolds under a natural spectral irreducibility assumption we prove the following dichotomy either the action is measurably isometric in which case there are at most two factors or the action is infinitesimally linear which means that the derivative cocycle arises from unbounded linear representations of all factors as a first application this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group another application is a strong restriction for actions of non linear groups we prove our results by means of a new cocycle superrigidity theorem of independent interest in analogy to zimmer s programme

one cohomology and the uniqueness of the group measure space decomposition of a ii1 factor we provide a unified and self contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a ii1 factor we single out a large class of groups characterized by a one cohomology property and prove that for every free ergodic probability measure preserving action of the associated ii1 factor has a unique group measure space cartan subalgebra up to unitary conjugacy our methods follow closely a recent article of chifan peterson but we replace the usage of peterson s unbounded derivations by thomas sinclair s dilation into a malleable deformation by a one parameter group of automorphisms

measure equivalence rigidity of the mapping class group we show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence if the mapping class group is measure equivalent to a discrete group then they are commensurable up to finite kernels moreover we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group we obtain similar results for finite direct products of mapping class groups

groups of piecewise projective homeomorphisms the group of piecewise projective homeomorphisms of the line provides straightforward torsion free counterexamples to the so called von neumann conjecture the examples are so simple that many additional properties can be established

gromov s measure equivalence and rigidity of higher rank lattices in this paper the notion of measure equivalence me of countable groups is studied me was introduced by gromov as a measure theoretic analog of quasi isometries all lattices in the same locally compact group are measure equivalent this is one of the motivations for this notion the main result of this paper is me rigidity of higher rank lattices any countable group which is me to a lattice in a simple lie group g of higher rank is commensurable to a lattice in g

the dixmier problem lamplighters and burnside groups j dixmier asked in 1950 whether every non amenable group admits uniformly bounded representations that cannot be unitarised we provide such representations upon passing to extensions by abelian groups this gives a new characterisation of amenability furthermore we deduce that certain burnside groups are non unitarisable answering a question raised by g pisier

integrable measure equivalence and rigidity of hyperbolic lattices we study rigidity properties of lattices in operatorname isom mathbf h n simeq mathrm so n 1 mathbb r n 3 and of surface groups in operatorname isom mathbf h 2 simeq mathrm sl 2 mathbb r in the context of integrable measure equivalence the results for lattices in operatorname isom mathbf h n n 3 are generalizations of mostow rigidity they include a cocycle version of strong rigidity and an integrable measure equivalence classification despite the lack of mostow rigidity for n 2 we show that cocompact lattices in operatorname isom mathbf h 2 allow a similar integrable measure equivalence classification

on the similarity problem for locally compact quantum groups abstract a well known theorem of day and dixmier states that any uniformly bounded representation of an amenable locally compact group g on a hilbert space is similar to a unitary representation within the category of locally compact quantum groups the conjectural analogue of the day dixmier theorem is that every completely bounded hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a representation we prove that this conjecture is false for a large class of non kac type compact quantum groups including all q deformations of compact simply connected semisimple lie groups on the other hand within the kac framework we prove that the day dixmier theorem does indeed hold for several new classes of examples including amenable discrete quantum groups of kac type

strong rigidity of generalized bernoulli actions and computations of their symmetry groups abstract we study equivalence relations and ii 1 factors associated with quotients of generalized bernoulli actions of kazhdan groups specific families of these actions are entirely classified up to isomorphism of ii 1 factors this yields explicit computations of outer automorphism and fundamental groups in particular every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic ii 1 factors

unique cartan decomposition for ii1 factors arising from arbitrary actions of free groups we prove that for any free ergodic probability measure preserving action mathbb f n curvearrowright x mu of a free group on n generators mathbb f n 2 leq n leq infty the associated group measure space ii1 factor l infty x rtimes mathbb f n has l x as its unique cartan subalgebra up to unitary conjugacy we deduce that group measure space ii1 factors arising from actions of free groups with different number of generators are never isomorphic we actually prove unique cartan decomposition results for ii1 factors arising from arbitrary actions of a much larger family of groups including all free products of amenable groups and their direct products

ergodic actions of semisimple groups and product relations suppose f is a discrete group and s yt is a finite measure space on which f acts so that i is invariant and ergodic if f is amenable it follows from results of 4 3 that the f action is orbit equivalent to a product action of a product group say f1 x 2 acting on s1 x s2 in other words the equivalence relation on s defined by the j orbits is a product of two other relations in a nontrivial way the point of this paper is to show that equivalence relations defined by actions of semisimple groups or their discrete subgroups or equivalence relations defined by foliations in which the leaves have negative curvature do not have this property more precisely we show the following

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

a stabilized mixed finite element method for darcy flow based on a multiscale decomposition of the solution recently masud and hughes proposed a stabilized mixed finite element formulation for darcy flow an interesting feature of this formulation is that there are no mesh dependent parameters in the present work we provide a derivation of this formulation based on a multiscale decomposition of the solution we also extend the work of masud and hughes to three dimensional problems and show the convergence rates for various three dimensional finite elements we also show that this formulation passes three dimensional constant flow patch tests for distorted element geometries i e elements with non constant jacobian robustness of this formulation is illustrated by performing numerical simulations on complex geometries

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

actions of _ whose ii factors and orbit equivalence relations have prescribed fundamental group we show that given any subgroup f of r which is either countable or belongs to a certain large class of uncountable subgroups there exist continuously many free ergodic measure preserving actions i of the free group with infinitely many generators f1 on probability measure spaces xi i such that their associated group measure space ii1 factors mi l 1 xi o i f1 and orbit equivalence relationsri r f1 y xi have fundamental group equal tof and with mi respectivelyri stably non isomorphic moreover these actions can be taken so thatri has no outer automorphisms and any automorphism of mi is unitary conjugate to an automorphism that acts trivially on the subalgebra l 1 xi of mi

cocycle superrigidity and bounded cohomology for negatively curved spaces we introduce new techniques to extend superrigidity theory beyond the scope of lie or algebraic groups we construct a cohomological invariant which accounts for and generalizes all known superrigidity results for actions on negatively curved spaces together with a new vanishing result and the machinery of bounded cohomology this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature we also prove a cocycle version a la zimmer

measure equivalence rigidity and bi exactness of groups we get three types of results on measurable group theory direct product groups of ozawa s class s groups wreath product groups and amalgamated free products we prove measure equivalence factorization results on direct product groups of ozawa s class s groups as consequences monod shalom type orbit equivalence rigidity theorems follow we prove that if two wreath product groups a g b of non amenable exact direct product groups g with amenable bases a b are measure equivalent then g and are measure equivalent we get bass serre rigidity results on amalgamated free products of non amenable exact direct product groups

revisiting stabilized finite element methods for the advective diffusive equation we give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection diffusion equation this article presents a concise perspective of the developments emanated from the works started in the 1980s through today 2005 elsevier b v all rights reserved pacs 47 11 j

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

on isometry groups and maximal symmetry we study problems of maximal symmetry in banach spaces this is done by providing an analysis of the structure of small subgroups of the general linear group gl x where x is a separable reflexive banach space in particular we provide the first known example of a banach space x without any equivalent maximal norm or equivalently such that gl x contains no maximal bounded subgroup moreover this space x may be chosen to be super reflexive in memory of greg hjorth 1963 2011 and nigel kalton 1946 2010

stochastic elastic plastic finite elements abstract a computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic plastic materials uncertain elastic plastic material properties are modeled as random fields which appear as the coefficient term in the governing partial differential equation of mechanics a spectral stochastic elastic plastic finite element method with fokker planck kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non linear elastic plastic partial differential equation with stochastic coefficient to this end the random field material properties are discretized in both spatial and stochastic dimension into finite numbers of independent basic random variables using karhunen loeve expansion those random variables are then propagated through the elastic plastic constitutive rate equation using fokker planck kolmogorov equation approach to obtain the evolutionary material properties as the material plastifies the unknown displacement solution random field is then assembled using polynomial chaos as a function of known basic random variables and unknown deterministic coefficients which are obtained by minimizing the error of finite representation by galerkin technique

a multiscale finite element method for the incompressible navier stokes equations this paper presents a new multiscale finite element method for the incompressible navier stokes equations the proposed method arises from a decomposition of the velocity field into coarse resolved scales and fine unresolved scales modeling of the unresolved scales corrects the lack of stability of the standard galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind petrov galerkin supg method and the galerkin least squares gls method the multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields specifically the equal order interpolations that are easy to implement but violate the celebrated babuska brezzi condition a significant feature of the present method is that the structure of the stabilization tensor s appears naturally via the solution of the fine scale problem a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed convergence studies for the method on uniform skewed as well as composite meshes are presented numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method 2005 elsevier b v all rights reserved

application of multi scale finite element methods to the solution of the fokker planck equation abstract this paper presents an application of multi scale finite element methods to the solution of the multi dimensional fokker planck equation the fokker planck or forward kolmogorov equation is a degenerate convective diffusion equation arising in markov process theory it governs the evolution of the transition probability density function of the response of a broad class of dynamical systems driven by gaussian noise and completely describes the response process analytical solutions for the fokker planck equation have been developed for only a limited number of low dimensional systems leading to a large body of approximation theory one such approach successfully applied to the solution of these problems in the past is the finite element method though for systems of dimension three or less in this paper a multi scale finite element method is applied to the fokker planck equation in an effort to develop a formulation that can yield higher accuracy on cruder spatial discretizations thus reducing the computational overhead associated with large scale problems that arise in higher dimensions

hyperfinite and ii1 actions for nonamenable groups abstract the actions of certain nonamenable groups on the lebesgue space are studied an example is constructed of a group which has a continuum of weakly nonequivalent actions of type ii 1 it is also proved that if a free group with two generators has hyperfinite action on the lebesgue space then at least one generator acts dissipatively a hyperfinite action is constructed for any nonamenable group

orbit equivalence rigidity consider a countable group acting ergodically by measure p reserving transformations on a probability space x and let r be the corresponding orbit equivalence relation on x the following rigidity phenomenon is shown there exist group actions such that the equivalence relation r on x determines the group and the action x uniquely up to finite groups the natural action of sln z on the n torus r n z n for n 2 is one of such examples the interpretation of these results in the context of von neumann algebras provides some support to the conjecture of connes on rigidity of group algebras for groups with property t our rigidity results also give examples of countable equivalence relations of type ii1 which cannot be generated mod 0 by a free action of any group this gives a negative answer to a long standing problem of feldman and moore

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100