isogeometric variational multiscale modeling of wall bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes abstract in this work we combine i nurbs based isogeometric analysis ii residual driven turbulence modeling and iii weak imposition of no slip and no penetration dirichlet boundary conditions on unstretched meshes to compute wall bounded turbulent flows while the first two ingredients were shown to be successful for turbulence computations at medium to high reynolds number i akkerman y bazilevs v m calo t j r hughes s hulshoff the role of continuity in residual based variational multiscale modeling of turbulence comput mech 41 2008 371 378 y bazilevs v m calo j a cottrell t j r hughes a reali g scovazzi variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows comput methods appl mech engrg 197 2007 173 201 it is the weak imposition of no slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher reynolds number y bazilevs t j r hughes weak imposition of dirichlet boundary conditions in fluid mechanics comput fluids 36 2007 12 26 y bazilevs c michler v m calo t j r hughes weak dirichlet boundary conditions for wall bounded turbulent flows comput methods appl mech engrg 196 2007 4853 4862 these three ingredients form a basis of a possible practical strategy for computing engineering flows somewhere between rans and les in complexity we demonstrate this by solving two challenging incompressible turbulent benchmark problems channel flow at friction velocity reynolds number 2003 and flow in a planar asymmetric diffuser we observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data this lends some credence to the proposed approach which we believe may become a viable engineering tool

isogeometric boundary element analysis using unstructured t splines we couple collocated isogeometric boundary element methods and unstructured analysis suitable t spline surfaces for linear elastostatic problems we extend the definition of analysis suitable t splines to encompass unstructured control grids unstructured meshes and develop basis functions which are smooth rational polynomials defined in terms of the bezier extraction framework and which pass standard patch tests we then develop a collocation procedure which correctly accounts for sharp edges and corners extraordinary points and t junctions this approach is applied to several three dimensional problems including a real world t spline model of a propeller we believe this work clearly illustrates the power of combining new analysis suitable computer aided design technologies with established analysis methodologies in this case the boundary element method

the hanna neumann conjecture for demushkin groups abstract we confirm the hanna neumann conjecture for topologically finitely generated closed subgroups u and w of a nonsolvable demushkin group g namely we show that g u g w d u g w g 1 d u d w where d k max d k 1 0 and d k is the least cardinality of a topological generating set for the group k

hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

finite group extensions and the atiyah conjecture the atiyah conjecture for a discrete group g states that the l 2 betti numbers of a finite cw complex with fundamental group g are integers if g is torsion free and in general that they are rational numbers with denominators determined by the finite subgroups of g here we establish conditions under which the atiyah conjecture for a torsion free group g implies the atiyah conjecture for every finite extension of g the most important requirement is that h g z p is isomorphic to the cohomology of the p adic completion of g for every prime p an additional assumption is necessary e g that the quotients of the lower central series or of the derived series are torsion free we prove that these conditions are fulfilled for a certain class of groups which contains in particular artin s pure braid groups and more generally fundamental groups of fiber type arrangements free groups fundamental groups of orientable compact surfaces certain knot and link groups primitive positive one relator groups and products of these therefore every finite in fact every elementary amenable extension of these groups satisfies the atiyah conjecture as a consequence if such an extension h is torsion free then the group ring ch contains no non trivial zero divisors i e h fulfills the zero divisor conjecture in the course of the proof we prove that if these extensions are torsionfree then they have plenty of non trivial torsion free quotients which are virtually nilpotent all of this applies in particular to artin s full braid group therefore answering question b6 on www grouptheory org our methods also apply to the baum connes conjecture this is discussed in 47 where for example the baum connes conjecture is proved for the full braid groups msc 55n25 homology with local coecients 16s34 group rings laurent rings 57m25 knots and links

sofic groups and diophantine approximation we prove the algebraic eigenvalue conjecture of j dodziuk p linnell v mathai t schick and s yates see 2 for sofic groups moreover we give restrictions on the spectral measure of elements in the integral group ring finally we define integer operators and prove a quantization of the operator norm below 2 to the knowledge of the author there is no group known that is not sofic

submultiplicativity and the hanna neumann conjecture in this article we dene submultiplicativity of 2 numbers in the category of complexes over a given complex x which generalizes the statement of the strengthened hanna neumann conjecture shnc in the case when is a left orderable group and x is a free complex we prove submultiplicativity for the subcategory consisting of ordered leafages over x with an additional analytic assumption called the deep fall property we show that the deep fall property is satised for graphs this implies shnc

the strong atiyah and luck approximation conjectures for one relator groups it is shown that the strong atiyah conjecture and the luck approximation conjecture in the space of marked groups hold for locally indicable groups in particular this implies that one relator groups satisfy both conjectures we also show that the center conjecture the independence conjecture and the strong eigenvalue conjecture hold for these groups as a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a hughes free epic division algebra and in particular it is embedded in a division algebra

isogeometric analysis of the advective cahn hilliard equation spinodal decomposition under shear flow we present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective cahn hilliard equation a stiff nonlinear parabolic equation characterized by the presence of fourth order spatial derivatives our numerical solution procedure is based on isogeometric analysis an approximation technique for which basis functions of high order continuity are employed these basis functions allow us to directly discretize the advective cahn hilliard equation without resorting to a mixed formulation we present steady state solutions for rectangular domains in two dimensions and for the first time in three dimensions we also present steady state solutions for the two dimensional taylor couette cell to enforce periodic boundary conditions in this curved domain we derive and utilize a new periodic bezier extraction operator we present an extensive numerical study showing the effects of shear rate surface tension and the geometry of the domain on the phase evolution of the binary fluid theoretical and experimental results are compared with our simulations

on the multi level splitting of finite element spaces in this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases we show that the condition number of such a stiffness matrix behaves like o log 2 where is the condition number of the stiffness matrix with respect to a nodal basis in the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like o left left log frac 1 h right 2 right instead of o left left frac 1 h right 2 right for a nodal basis the proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization especially we do not need the quasiuniformity of the employed triangulations as the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis our results mean that the method of conjugate gradients needs onlyo log n steps ando n log n computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures heren denotes the dimension of the finite element space and of the discrete linear problem to be solved

group cocycles and the ring of affiliated operators in this article we study cocycles of discrete countable groups with values in 2 g and the ring of affiliated operators ug we clarify properties of the first cohomology of a group g with coefficients in 2 g and answer several questions from de cornulier et al transform groups 13 1 125 147 2008 moreover we obtain strong results about the existence of free subgroups and the subgroup structure provided the group has a positive first 2 betti num ber we give numerous applications and examples of groups which satisfy our assumptions

a short proof of the approximation conjecture for amenable groups abstract we give a short proof of the approximation conjecture with complex coefficients for amenable groups g elek the strong approximation conjecture holds for amenable groups j funct anal 239 1 2006 345 355

approximating spectral invariants of harper operators on graphs abstract we study harper operators and the closely related discrete magnetic laplacians dml on a graph with a free action of a discrete group as defined by sunada sun a main result in this paper is that the spectral density function of dmls associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the dmls on a regular exhaustion with either dirichlet or neumann boundary conditions this then gives a criterion for the existence of gaps in the spectrum of the dml as well as other interesting spectral properties of such dmls the technique used incorporates some results of algebraic number theory

the parity of the cochran harvey invariants of 3 manifolds given a finitely presented group and an epimorphism cochran and harvey defined a sequence of invariants which can be viewed as the degrees of higher order alexander polynomials cochran and harvey showed that up to a minor modification this is a never decreasing sequence of numbers if is the fundamental group of a 3 manifold with empty or toroidal boundary furthermore they showed that these invariants give lower bounds on the thurston norm using a certain cohn localization and the duality of reidemeister torsion we show that for a fundamental group of a 3 manifold any jump in the sequence is necessarily even this answers in particular a question of cochran furthermore using results of turaev we show that under a mild extra hypothesis the parity of the cochran harvey invariant agrees with the parity of the thurston norm

soficity amenability and dynamical entropy in a previous paper the authors developed an operator algebraic approach to lewis bowen s sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure preserving transformations on a standard probability space we show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable

superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis abstract in this article we address adaptive methods for isogeometric analysis based on local refinement guided by recovery based a posteriori error estimates isogeometric analysis was introduced a decade ago and an impressive progress has been made related to many aspects of numerical methods and advanced applications however related to adaptive mesh refinement guided by a posteriori error estimators rather few attempts are pursued besides the use of classical residual based error estimators in this article we explore a feature common for isogeometric analysis iga namely the use of structured tensorial meshes that facilitates superconvergence behavior of the gradient in the galerkin discretization by utilizing the concept of structured mesh refinement using lr b splines our aim is to facilitate superconvergence behavior for locally refined meshes as well superconvergence behavior matches well with the use of recovery based a posteriori estimator in the superconvergent patch recovery spr procedure however to our knowledge so far the spr procedure has not been exploited in the iga community we start out by addressing the existence of derivative superconvergent points in the computed finite element solution based on b splines and lr b splines for an elliptic model problem 1d and 2d poisson then we present some recovery procedures for improving the derivatives or gradient of the isogeometric finite element solution where the spr procedure will be the main focus in particular we show that our spr procedure for the improvement of derivatives fulfills the desired consistency criteria at the end we develop a posteriori error estimator where the improved gradient obtained from the proposed recovery procedures is used numerical results are presented to illustrate the efficiency of using spr procedure for the improvement of derivatives or gradient of computed solution in isogeometric analysis then the proposed a posteriori error estimator based adaptive refinement methodology is tested to solve smooth and non smooth elliptic benchmark problems the focus is put on whether optimal convergence rates are obtained in the computed solution or not as well as the effectivity index of the proposed error estimators

isogeometric analysis of nearly incompressible large strain plasticity abstract we study the behavior of nurbs based isogeometric analysis on problems of large deformation plasticity we evaluate the performance of standard nurbs elements and elements based on the f formulation of elguedj et al 2008 1 our experience shows that standard measures of evaluation employed in the literature namely displacements at selected locations and graphs of reaction forces versus displacements are often misleading metrics in problems of plasticity on the other hand stress distributions in the form of contour plots are in our experience revealing measure of element performance we also determine that the concept of mesh locking which has dominated investigations of low order elements is not a relevant issue for higher order nurbs elements for problems of large deformation plasticity however standard higher order nurbs elements of type qk of continuity class c k 1 k 2 typically exhibit spurious stress oscillations whereas the f elements of type qk qk 1 produce good results in all cases

a hierarchic family of isogeometric shell finite elements a hierarchic family of isogeometric shell finite elements based on nurbs shape functions is presented in contrast to classical shell finite element formulations inter element continuity of at least c1 enables a unique and continuous representation of the surface normal within one nurbs patch this does not only facilitate formulation of kirchhoff love type shell models for which the standard galerkin weak form has a variational index of 2 but it also offers significant advantages for shear deformable reissner mindlin type shells and higher order shell models for a 5 parameter shell formulation with reissner mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated kirchhoff love type director of the deformed configuration this split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration assumed natural strains or mixed finite element formulations the third member of the hierarchy is a 7 parameter model including thickness change and allowing for application of unmodified three dimensional constitutive laws the phenomenon of curvature thickness locking coming along with this kinematic extension again is automatically avoided by the hierarchic difference vector concept without any further treatment membrane locking and in plane shear locking are removed by two different approaches firstly elimination via the discrete strain gap dsg method and secondly removal of parasitic membrane strains using a hybrid mixed method based on the hellinger reissner variational principle the hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach

weakly enforced essential boundary conditions for nurbs embedded and trimmed nurbs geometries on the basis of the finite cell method summary enforcing essential boundary conditions plays a central role in immersed boundary methods nitsche s idea has proven to be a reliable concept to satisfy weakly boundary and interface constraints we formulate an extension of nitsche s method for elasticity problems in the framework of higher order and higher continuity approximation schemes such as the b spline and non uniform rational basis spline version of the finite cell method or isogeometric analysis on trimmed geometries furthermore we illustrate a significant improvement of the flexibility and applicability of this extension in the modeling process of complex 3d geometries with several benchmark problems we demonstrate the overall good convergence behavior of the proposed method and its good accuracy we provide extensive studies on the stability of the method its influence parameters and numerical properties and a rearrangement of the numerical integration concept that in many cases reduces the numerical effort by a factor two a newly composed boundary integration concept further enhances the modeling process and allows a flexible discretization independent introduction of boundary conditions finally we present our strategy in the framework of the modeling and isogeometric analysis process of trimmed non uniform rational basis spline geometries copyright 2013 john wiley sons ltd

sectional curvature of polygonal complexes with planar substructures in this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature these complexes can be viewed as generalizations of 2 dimensional euclidean and hyperbolic buildings we focus on the case of non positive and negative combinatorial curvature as geometric results we obtain a hadamard cartan type theorem thinness of bigons gromov hyperbolicity and estimates for the cheeger constant we employ the latter to get spectral estimates show discreteness of the spectrum in the sense of a donnelly li type theorem and present corresponding eigenvalue asymptotics moreover we prove a unique continuation theorem for eigenfunctions and the solvability of the dirichlet problem at infinity

spectral analysis of percolation hamiltonians we study the family of hamiltonians which corresponds to the adjacency operators on a percolation graph we characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions this set of energies is a dense subset of the algebraic integers the integrated density of states has discontinuities precisely at this set of energies we show that the convergence of the integrated densities of states of finite box hamiltonians to the one on the whole space holds even at the points of discontinuity for this we use an equicontinuity from the right argument the same statements hold for the restriction of the hamiltonian to the infinite cluster in this case we prove that the integrated density of states can be constructed using local data only finally we study some mixed anderson quantum percolation models and establish results in the spirit of wegner and delyon and souillard

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

integrality of l 2 betti numbers the atiyah conjecture predicts that the l 2 betti numbers of a finite cw complex with torsion free fundamental group are integers we establish the atiyah conjecture under the condition that it holds for g and that h lhd g is a normal subgroup for amalgamated free products g h h rtimes f here f is a free group and h rtimes f is an arbitrary semi direct product this includes free products g f and semi direct products g rtimes f we also show that the atiyah conjecture holds with an additional technical condition for direct and inverse limits of groups for which it is true as a corollary it holds for positive 1 relator groups with torsion free abelianization putting everything together we establish a new bigger class of groups for which the atiyah conjecture holds which contains all free groups and in particular is closed under taking subgroups direct sums free products extensions with torsion free elementary amenable quotient or with free quotient and under certain direct and inverse limits

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

approximating l2 invariants and the atiyah conjecture let g be a torsion free discrete group and let denote the field of algebraic numbers in we prove that g fulfills the atiyah conjecture if g lies in a certain class of groups d which contains in particular all groups that are residually torsion free elementary amenable or are residually free this result implies that there are no nontrivial zero divisors in g the statement relies on new approximation results for l2 betti numbers over g which are the core of the work done in this paper another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial laplacian on l2 cochains on any normal covering space of a finite cw complex we establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the linnell class we also establish the absence of eigenvalues that are liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 2003 wiley periodicals inc

isogeometric shape optimization of photonic crystals via coons patches abstract in this paper we present an approach that extends isogeometric shape optimization from optimization of rectangular like nurbs patches to the optimization of topologically complex geometries we have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps salient features of this approach include the following 1 multi patch coons representation of design geometry the design geometry is represented as a collection of coons patches where the four boundaries of each patch are represented as nurbs curves the use of multiple patches is motivated by the need for representing topologically complex geometries the coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization 2 exact boundary conversion to the analysis geometry with guaranteed mesh injectivity the analysis geometry is a collection of nurbs patches that are converted from the multi patch coons representation with geometric exactness in patch boundaries the internal nurbs control points are embedded in the parametric domain of the coons patches with a built in mesh rectifier to ensure the injectivity of the resulting b spline geometry i e every point in the physical domain is mapped to one point in the parametric domain 3 analytical sensitivities sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities the nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding coons patches

the strong approximation conjecture holds for amenable groups abstract let g be a finitely generated group and g g 1 g 2 be normal subgroups such that k 1 g k 1 let a mat d d c g and a k mat d d c g g k be the images of a under the maps induced by the epimorphisms g g g k according to the strong form of the approximation conjecture of luck w luck l 2 invariants theory and applications to geometry and k theory ergeb math grenzgeb 3 vol 44 springer verlag berlin 2002 dim g ker a lim k dim g g k ker a k where dim g denotes the von neumann dimension in j dodziuk p linnell v mathai t schick s yates approximating l 2 invariants and the atiyah conjecture comm pure appl math 56 7 2003 839 873 dodziuk et al proved the conjecture for torsion free elementary amenable groups in this paper we extend their result for all amenable groups using the quasi tilings of ornstein and weiss d s ornstein b weiss entropy and isomorphism theorems for actions of amenable groups j anal math 48 1987 1 141

approximatingl2invariants of amenable covering spaces a combinatorial approach abstract in this paper we prove that the l 2 betti numbers of an amenable covering space can be approximated by the average betti numbers of a regular exhaustion proving a conjecture in dm we also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

on the asymptotic convergence of spline collocation methods for partial differential equations we examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions we prove optimal rates of convergence in l 2 for partial derivatives of the approximate solution which are of order at least two in one variable while the solution itself and its gradient converge in l 2 at rates less than the optimal approximation theoretic results

linear wegner estimate for alloy type schrodinger operators on metric graphs we study spectra of alloy type random schrodinger operators on metric graphs for finite edge subsets we prove a wegner estimate which is linear in the volume i e the total length of the edges and the length of the energy interval the single site potential needs to have fixed sign the metric graph does not need to have a periodic structure a further result is the existence of the integrated density of states for ergodic random hamiltonians on metric graphs with a z structure for certain models the two above results together imply the lipschitz continuity of the integrated density of states

maximum principle and convergence analysis for the meshfree point collocation method the discrete laplacian operator is considered in the sense of the meshfree point collocation method which will be called the strong meshfree laplacian operator to define the strong meshfree laplacian operator we use the fast version of the generalized moving least square approximation which can calculate the approximated derivatives of shape functions some types of the locally layered node distribution are defined in this paper and two specific domains are constructed onto which we can distribute locally layered nodes at such nodes the discrete maximum principle can be shown to hold through the representation formula for the strong meshfree laplacian operator the discrete maximum principle together with the reproducing property of the meshfree approximations results in an a priori estimate for the strong meshfree laplacian operator in the nodal solution space furthermore the a priori estimate we have obtained guarantees the existence and the uniqueness of the numerical solution and plays a central role in achieving converged results for the poisson problem with dirichlet boundary conditions in the nodal solution space the order of convergence of the nodal solutions can be raised up to o h2 at the proposed type of nodes in specific domains for generally shaped domains immersed in the previously mentioned domains we can obtain the first order convergence result of o h

applications of variational inequalities to the existence theorem on quadrature domains in this paper we shall study quadrature domains for the class of subharmonic functions by using the theory of variational inequalities we shall give a new proof of the existence and uniqueness theorem as an application we deal with hele shaw flows with a free boundary and show that their two weak solutions one of which was defined by the author using quadrature domains and the other was defined by gustafsson 3 using variational inequalities are identical with each other introduction in a previous paper 7 the author has defined the quadrature domains of positive measures for the class of subharmonic functions and studied their applications to complex function theory let v be a finite positive measure on the two dimensional euclidean space r2 let sl1 q be the class of subharmonic functions in an open set q which are integrable with respect to the two dimensional lebesgue measure m a nonempty open set q is called a quadrature domain of v for class sl1 if qi v is concentrated in q namely v qc 0 where qc denotes the complement of q qii jus s dv 1 a e in a connected open set w with finite area f 0 a e in wc and jf dm m w then q fm sl1 0 and there exists a minimum domain w in q fm sl1 namely q e q jm sl1 if and only if w c q and m q w o received by the editors october 5 1981 and in revised form february 23 1982 1980 mathematics subject classification primary 3 1aos 3 1bos secondary 35a15

local variational principle concerning entropy of a sofic group action abstract recently lewis bowen introduced a notion of entropy for measure preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy and then david kerr and hanfeng li developed an operator algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space and established the global variational principle concerning measure theoretic and topological entropy in this sofic context by localizing these two kinds of entropy in this paper we prove a local version of the global variational principle for any finite open cover of the space and show that these local measure theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group

bezier projection a unified approach for local projection and quadrature free refinement and coarsening of nurbs and t splines with particular application to isogeometric design and analysis abstract we introduce bezier projection as an element based local projection methodology for b splines nurbs and t splines this new approach relies on the concept of bezier extraction and an associated operation introduced here spline reconstruction enabling the use of bezier projection in standard finite element codes bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global l 2 projection bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging degree elevation and reduction basis roughening and smoothing and spline reparameterization in fact bezier projection provides a quadrature free approach to refinement and coarsening of splines in this sense bezier projection provides the fundamental building block for h p k r adaptivity in isogeometric analysis

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

the hp d adaptive finite cell method for geometrically nonlinear problems of solid mechanics summary the finite cell method fcm combines the fictitious domain approach with the p version of the finite element method and adaptive integration for problems of linear elasticity it offers high convergence rates and simple mesh generation irrespective of the geometric complexity involved this article presents the integration of the fcm into the framework of nonlinear finite element technology however the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map as a consequence of the weak penalization nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries leading to a low algebraic rate of convergence therefore the fcm approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp d method which synergetically uses the h adaptivity of the integration scheme numerical experiments show that the hp d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map the hp d adaptive fcm is thus able to restore high convergence rates for the geometrically nonlinear case while preserving the easy meshing property of the original fcm accuracy and performance of the present scheme are demonstrated by several benchmark problems in one two and three dimensions and the nonlinear simulation of a complex foam sample copyright 2011 john wiley sons ltd

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

uniform existence of the integrated density of states for random schrodinger operators on metric graphs over zd abstract we consider ergodic random schrodinger operators on the metric graph z d with random potentials and random boundary conditions taking values in a finite set we show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable this limit the integrated density of states can be expressed by a closed shubin pastur type trace formula it supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions among other examples we discuss random magnetic fields and percolation models

imposing nonlocal boundary conditions in galerkin type methods based on non interpolatory functions abstract the imposition of inhomogeneous dirichlet essential boundary conditions is a fundamental challenge in the application of galerkin type methods based on non interpolatory functions i e functions which do not possess the kronecker delta property such functions typically are used in various meshfree methods as well as methods based on the isogeometric paradigm the present paper analyses a model problem consisting of the poisson equation subject to non standard boundary conditions namely instead of classical boundary conditions the model problem involves dirichlet and neumann type nonlocal boundary conditions variational formulations with strongly and weakly imposed inhomogeneous dirichlet type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non uniform rational b splines nurbs the attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods

two dimensional mesh embedding for b spline methods many factors motivate consideration of b splines as basis functions for solving partial differential equations these are arbitrary orders of accuracy and high resolving powers similar to those of compact schemes furthermore if one uses a galerkin scheme one gets in addition to conservation of the discretized quantities conservation of quadratic invariants such as energy this work develops another property namely the ability to treat semi structured embedded or zonal meshes for two dimensional geometries this can drastically reduce the number of grid points in many applications an algorithm is presented for constructing a global spline basis that automatically hasd 1 continuous derivatives at mesh block boundaries as everywhere else heredis the polynomial degree the basis functions are simply suitable products of one dimensional b splines both integer and noninteger refinement ratios are allowed across mesh blocks finally test cases for linear scalar equations such as the poisson and advection equation are presented

convergence of an efficient local least squares fitting method for bases with compact support abstract the least squares projection procedure appears frequently in mathematics science and engineering it possesses the well known property that a least squares approximation formed via orthogonal projection to a given data set provides an optimal fit in the chosen norm the orthogonal projection of the data onto a finite basis is typically approached by the inversion of a gram matrix involving the inner products of the basis functions even if the basis functions have compact support so that the gram matrix is sparse its inverse will be dense thus computing the orthogonal projection is expensive an efficient local least squares algorithm for non orthogonal projection onto smooth piecewise polynomial basis functions is analyzed the algorithm runs in optimal time and delivers the same order of accuracy as the standard orthogonal projection numerical results indicate that in many computational situations the new algorithm offers an effective alternative to global least squares approximation

spline approximation of advection diffusion problems using upwind type collocation nodes abstract a spline collocation method for linear advection diffusion equations is proposed the method is based on an operator dependent collocation grid and provides stabilized approximated solutions with respect to the coefficient of the diffusive term when problems are advection dominated

l2 spectral invariants and convergent sequences of finite graphs abstract using the spectral theory of weakly convergent sequences of finite graphs we prove the uniform existence of the integrated density of states for a large class of infinite graphs

characters l 2 betti numbers and an equivariant approximation theorem let g be a group with a finite subgroup h we define the l 2 multiplicity of an irreducible representation of h in the l 2 homology of a proper g cw complex these invariants generalize the l 2 betti numbers our main results are approximation theorems for l 2 multiplicities which extend the approximation theorems for l 2 betti numbers of luck farber and elek szabo respectively the main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups we discuss applications to the cohomology of arithmetic groups

arithmetic properties of eigenvalues of generalized harper operators on graphs let open image in new window denote the field of algebraic numbers in open image in new window a discrete group g is said to have the multiplier algebraic eigenvalue property if for every matrix a m d open image in new window g regarded as an operator on l 2 g d the eigenvalues of a are algebraic numbers where z 2 g open image in new window is an algebraic multiplier and open image in new window denotes the unitary elements of open image in new window such operators include the harper operator and the discrete magnetic laplacian that occur in solid state physics we prove that any finitely generated amenable free or surface group has this property for any algebraic multiplier in the special case when is rational n 1 for some positive integer n this property holds for a larger class of groups open image in new window containing free groups and amenable groups and closed under taking directed unions and extensions with amenable quotients included in the paper are proofs of other spectral properties of such operators

group ring elements with large spectral density given delta 0 we construct a group g and a group ring element s in mathbb z g such that the spectral measure mu of s fulfils mu 0 varepsilon frac c log varepsilon 1 delta for small varepsilon in particular the novikov shubin invariant of any such s is 0 the constructed examples show that the best known upper bounds on mu 0 varepsilon are not far from being optimal

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

multigrid methods for isogeometric discretization abstract we present geometric multigrid methods for isogeometric discretization of scalar second order elliptic problems the smoothing property of the relaxation method and the approximation property of the intergrid transfer operators are analyzed these properties when used in the framework of classical multigrid theory imply uniform convergence of two grid and multigrid methods supporting numerical results are provided for the smoothing property the approximation property convergence factor and iterations count for v w and f cycles and the linear dependence of v cycle convergence on the smoothing steps for two dimensions numerical results include the problems with variable coefficients simple multi patch geometry a quarter annulus and the dependence of convergence behavior on refinement levels l whereas for three dimensions only the constant coefficient problem in a unit cube is considered the numerical results are complete up to polynomial order p 4 and for c 0 and c p 1 smoothness

analysis of an upstream weighted collocation approximation to the transport equation the numerical behavior of a modified orthogonal collocation method as applied to the transport equations can be examined through the use of a fourier series analysis the necessity of such a study becomes apparent in the analysis of several techniques which emulate classical upstream weighting schemes these techniques are employed in orthogonal collocation and other numerical methods as a means of handling parabolic partial differential equations with significant first order terms divergent behavior can be shown to exist in one upstream weighting method applied to orthogonal collocation

on turing dynamical systems and the atiyah problem main theorems of the article concern the problem of atiyah on possible values of l 2 betti numbers it is shown that all non negative real numbers are l 2 betti numbers and that many for example all non negative algebraic real numbers are l 2 betti numbers of simply connected manifolds with respect to a free cocompact action also an explicit example is constructed which leads to a simply connected manifold with a transcendental l 2 betti number with respect to an action of the threefold direct product of the lamplighter group mathbb z 2 mathbb z wr mathbb z the main new idea is embedding turing machines into integral group rings the main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

hyperlinearity essentially free actions and l2 invariants the sofic property we prove that connes embedding conjecture holds for the von neumann algebras of sofic groups that is sofic groups are hyperlinear hence we provide some new examples of hyperlinearity we also show that the determinant conjecture holds for sofic groups as well we introduce the notion of essentially free actions and amenable actions and study their properties

a high order b spline collocation method for linear boundary value problems abstract collocation methods are investigated because of their simplicity and inherent efficiency for applications to linear boundary value problems on a b the objective of the present research is obtaining numerical solution of the boundary value problems for d th order linear boundary value problem by a b spline collocation method using b splines of order k and their index of regularity is m d 1 m k 2 the collocation points in our method form a strictly increasing sequence of points in a b each interior j th collocation point belongs to the interior of the compact support of corresponding j th b spline basis element and the number of b spline basis elements equals the number of collocation points the order of accuracy of the proposed method is shown to be optimal the mathematical properties of this collocation method are less well established primarily because the order of accuracy depends on the regularity and order of b spline basis and location of the collocation points the error analysis is through the green s function approach than the matrix approach we compare the efficiency and accuracy of our method to nodal and orthogonal collocation methods as applied to linear ordinary differential equations with boundary conditions our collocation method like greville collocation method is more convenient than nodal or orthogonal collocation because exactly the correct number of collocation points is available the greville and botella collocation methods are special cases of our collocation method

application of variational inequalities to the moving boundary problem in a fluid model for biofilm growth vi betraktar ett problem med rorlig rand motsvarande ett gransfall i en fluid modell for biofilmstillvaxt som foreslagits av j dockery och i klapper siam j appl math 62 3 2001

a unified approach for embedded boundary conditions for fourth order elliptic problems summary an efficient procedure for embedding kinematic boundary conditions in the biharmonic equation for problems such as the pure streamfunction formulation of the navier stokes equations and thin plate bending is based on a stabilized variational formulation obtained by nitsche s approach for enforcing boundary constraints the absence of kinematic admissibility constraints allows the use of non conforming meshes with non interpolatory approximations thereby providing added flexibility in addressing the higher continuity requirements typical of these problems variationally conjugate pairs weakly enforce kinematic boundary conditions the use of a scaling factor leads to a formulation with a single stabilization parameter for plates the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners the single stabilization parameter is determined from a local generalized eigenvalue problem guaranteeing coercivity of the discrete bilinear form the accuracy of the approach is verified by representative computations with bicubic b splines providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter copyright 2014 john wiley sons ltd

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

geometric modeling isogeometric analysis and the finite cell method the advent of isogeometric analysis iga using the same basis functions for design and analysis constitutes a milestone in the unification of geometric modeling and numerical simulation however an important class of geometric models based on the csg constructive solid geometry concept such as trimmed nurbs surfaces do not fully support the isogeometric paradigm since basis functions do not explicitly represent the boundary the finite cell method fcm is a high order fictitious domain method which offers simple meshing of potentially complex domains into a structured grid of cuboid cells while still achieving exponential rates of convergence for smooth problems in the present paper we first discuss the possibility to directly couple the finite cell method to csg without any necessity for meshing the three dimensional domain and then explore a combination of the best of the two approaches iga and fcm closely following ideas of the recently introduced shell fcm the resulting finite cell extension to isogeometric analysis achieves a truly straightforward transfer of a trimmed nurbs surface into an analysis suitable nurbs basis while benefiting from the favorable properties of the high order and high continuity basis functions accuracy and efficiency of the new approach are demonstrated by a numerical benchmark and its versatility is outlined by the analysis of different trimmed design variants of a brake disk

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

erratum to integrality of l2 betti numbers a gap has been found in the proof of proposition 3 of 3 consequently this result and the consequences drawn in the paper have to be considered as conjectures in particular it is not proved that the atiyah conjecture is stable under extensions with free quotient groups and under free products with free groups

efficient isogeometric nurbs based solid shell elements mixed formulation and b method abstract in this work solid shell nurbs elements are developed in order to address static problems of slender structures under small perturbations a single layer of elements is considered through the thickness of the shell and the degree of approximation in that direction is chosen to be equal to two a full 3d constitutive relation is assumed the objective is to obtain highly accurate low degree elements to be used in coarse meshes in order to do that we propose a mixed method from which we derive a b projection to deal with locking the main idea is to modify the interpolation of the average stresses and strains through the thickness more precisely we develop two finite elements the first element which is based on the mixed method or equivalently on the b projection is extremely accurate but leads to a fully populated global stiffness matrix to improve the efficiency the second element uses a local least squares type procedure to define a new b projection leading to a sparse global stiffness matrix the quality and efficiency of the methods are assessed through several usual test cases and by comparison with other published techniques