the measurable kesten theorem we give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d regular graphs it follows that the a finite d regular ramanujan graph g contains a negligible number of cycles of size less than c log log g we prove that infinite d regular ramanujan unimodular random graphs are trees through benjamini schramm convergence this leads to the following rigidity result if most eigenvalues of a d regular finite graph g fall in the alon boppana region then the eigenvalue distribution of g is close to the spectral measure of the d regular tree in particular g contains few short cycles in contrast we show that d regular unimodular random graphs with maximal growth are not necessarily trees institute of mathematical statistics 2016

asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

toric complexes and artin kernels abstract a simplicial complex l on n vertices determines a subcomplex t l of the n torus with fundamental group the right angled artin group g l given an epimorphism g l z let t l be the corresponding cover with fundamental group the artin kernel n we compute the cohomology jumping loci of the toric complex t l as well as the homology groups of t l with coefficients in a field k viewed as modules over the group algebra k z we give combinatorial conditions for h r t l k to have trivial z action allowing us to compute the truncated cohomology ring h r t l k we also determine several lie algebras associated to artin kernels under certain triviality assumptions on the monodromy z action and establish the 1 formality of these not necessarily finitely presentable groups

a partition of unity enriched dual boundary element method for accurate computations in fracture mechanics we introduce a novel enriched boundary element method bem and dual boundary element method dbem approach for accurate evaluation of stress intensity factors sifs in crack problems the formulation makes use of the partition of unity method pum such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation an enrichment strategy is described in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns in addition an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals finally results are shown for mixed mode crack problems these illustrate that the introduction of pum enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched dbem

a meshless method with enriched weight functions for fatigue crack growth the meshless method is particularly appropriate to solve crack propagation problems in this paper the fatigue growth of cracks in two dimensional bodies is considered the analysis is based upon paris equation new enriched weight functions are introduced in the meshless method formulation to capture the singularity at the crack tip simple problems show the accuracy and efficiency of this method then it is applied to fatigue analysis of single and multi cracked bodies under mixed mode conditions copyright 2004 john wiley sons ltd

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

asymptotic mean density of sub unitary ensembles the asymptotic behaviour of the mean spectral density for the product of n n random unitary matrices distributed according to the haar measure and a fixed diagonal matrix is investigated the large n limit of the exact expression derived by wei and fyodorov 2008 j phys a math theor 41 502001 is calculated by a modification of the saddle point method it is shown that in the bulk the result coincides with the one obtained within the free probability theory by haagerup and larsen 2000 j funct anal 176 331 close to the edge points of the bulk asymptotics the mean density for large n is described by a universal function depending on a certain scaling variable the large deviation formulae valid outside the bulk are also derived obtained formulae agree well with the results of direct numerical calculations

cogrowth and amenability of discrete groups abstract let g be a group and g1 gt a set of generators there are approximately 2t 1 n reduced words in g1 gt of length n let gg n be the number of those which represent 1g we show that lim n gg n 1 n exists clearly 1 2t 1 log log 2t 1 is the cogrowth 0 1 in fact 0 1 2 1 the entropic dimension of g is shown to be 1 it is then proved that d g 1 if and only if g is free on g1 gt and d g 0 if and only if g is amenable

element free galerkin method for wave propagation and dynamic fracture abstract element free galerkin method efg is extended to dynamic problems efg method which is based on moving least square interpolants mls requires only nodal data no element connectivity is needed this makes the method particularly attractive for moving dynamic crack problems since remeshing can be avoided in contrast to the earlier formulation for static problems by authors the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions with this formulation the stiffness matrix is symmetric and positive semi definite and hence the consistency conergence and stability analyses of time integration remain the same as those in finite element method numerical examples are presented to illustrate the performance of this method the relationship between the element free galerkin method and the smooth particle hydrodynamics sph method is also discussed in this paper results are presented for some one dimensional problems and two dimensional problems with static and moving cracks

a meshless method with enriched weight functions for three dimensional crack propagation the propagation of cracks in three dimensions is analysed by a meshless method the cracks are modelled by a set of triangles that are added when the propagation occurs since the method is meshless no remeshing of the domain is necessary during the propagation to avoid using a large number of degrees of freedom the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions this enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper several algorithms for efficiently implementing the meshless method in three dimensions are detailed the accuracy of the enrichment is first assessed on simple examples and some results of non planar propagation of multiple cracks are then presented copyright 2005 john wiley sons ltd

an element free method for in plane notch problems with anisotropic materials summary this study developed an element free galerkin method efgm to simulate notched anisotropic plates containing stress singularities at the notch tip two dimensional theoretical complex displacement functions are first deduced into the moving least squares interpolation the interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the galerkin method in the numerical validation an interface layer of the efgm is used to combine the mesh between the traditional finite elements and the proposed singular notch efgm the h integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method the comparisons indicate that the proposed method obtains more accurate results for the displacement stress and energy fields than those determined from the standard finite element method copyright 2013 john wiley sons ltd

an incremental formulation for the prediction of two dimensional fatigue crack growth with curved paths this paper presents a new incremental formulation for predicting the curved growth paths of two dimensional fatigue cracks the displacement and traction boundary integral equations bies are employed to calculate responses of a linear elastic cracked body the paris law and the principle of local symmetry are adopted for defining the growth rate and direction of a fatigue crack respectively the three governing equations i e the bies the paris law and the local symmetry condition are non linear with respect to the crack growth path and unknowns on the boundary iterative forms of three governing equations are derived to solve problems of the fatigue crack growth by the newton raphson method the incremental crack path is modelled as a parabola defined by the crack tip position and the trapezoidal rule is employed to integrate the paris law the validity of the proposed method is demonstrated by two numerical examples of plates with an edge crack

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

the spectral sequence of an equivariant chain complex and homology with local coefficients we study the spectral sequence associated to the filtration by powers of the augmentation ideal on the twisted equivariant chain complex of the universal cover of a connected cw complex x in the process we identify the d 1 differential in terms of the coalgebra structure of h x k and the k 1 x module structure on the twisting coefficients in particular this recovers in dual form a result of reznikov on the mod p cohomology of cyclic p covers of aspherical complexes this approach provides information on the homology of all galois covers of x it also yields computable upper bounds on the ranks of the cohomology groups of x with coefficients in a prime power order rank one local system when x admits a minimal cell decomposition we relate the linearization of the equivariant cochain complex of the universal abelian cover to the aomoto complex arising from the cup product structure of h x k thereby generalizing a result of cohen and orlik

efficient algorithm for edge cracked geometries the stress field in a finite edge cracked specimen under load is computed using algorithms based on two slightly different integral equations of the second kind these integral equations are obtained through left regularizations of a first kind integral equation in numerical experiments it is demonstrated that the stress field can be accurately computed highly accurate stress intensity factors and t stresses are presented for several setups and extensive comparisons with results from the literature are made for simple geometries the algorithms presented here achieve relative errors of less than 10 10 it is also shown that the present algorithms can accurately handle both geometries with arbitrarily shaped edge cracks and geometries containing several hundred edge cracks all computations were performed on an ordinary workstation copyright c 2005 john wiley sons ltd less

element free galerkin methods for static and dynamic fracture abstract element free galerkin efg methods are presented and applied to static and dynamic fracture problems efg methods which are based on moving least square mls interpolants require only nodal data no element connectivity is needed the description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks this makes the method particularly attractive for growing crack problems since only minimal remeshing is needed to follow crack growth in moving least square interpolants the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

dynamic fracture using element free galerkin methods the element free galerkin method for dynamic crack propagation is described and applied to several problems this method is a gridless method which facilitates the modelling of growing crack problems because it does not require remeshing the growth of the crack is modelled by extending its surfaces the essential feature of the method is the use of moving least squares interpolants for the trial and test functions in these interpolants the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point the discrete equations are obtained by a galerkin method the procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described

accurate fracture modelling using meshless methods the visibility criterion and level sets formulation and 2d modelling fracture modelling using numerical methods is well advanced in 2d using techniques such as the extended finite element method xfem the use of meshless methods for these problems lags somewhat behind but the potential benefits of no meshing particularly in 3d prompt continued research into their development in methods where the crack face is not explicitly modelled as the edge of an element for instance two procedures are instead used to associate the displacement jump with the crack surface the visibility criterion and the diffraction method the visibility criterion is simple to implement and efficient to compute especially with the help of level set coordinates however spurious discontinuities have been reported around crack tips using the visibility criterion whereas implementing the diffraction method in 3d is much more complicated than the visibility criterion in this paper a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved the formulation is based on the use of level set coordinates and the element free galerkin method and is generally applicable for single or multiple crack problems in 2d or 3d the paper explains the formulation and provides verification of the method against a number of 2d crack problems copyright 2011 john wiley sons ltd

chordal loewner families and univalent cauchy transforms we study chordal loewner families in the upper half plane and show that they have a parametric representation we show one that to every chordal loewner family there corresponds a unique measurable family of probability measures on the real line and two that to every measurable family of probability measures on the real line there corresponds a unique chordal loewner family in both cases the correspondence is being given by solving the chordal loewner equation we use this to show that any probability measure on the real line with finite variance and mean zero has univalent cauchy transform if and only if it belongs to some chordal loewner family if the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the cauchy transform the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal cauchy transform and the second in terms of moment inequalities corresponding to the grunsky inequalities

a variational approach for evaluation of stress intensity factors using the element free galerkin method a variational meshfree method has been developed to evaluate the stress intensity factors of mixed mode crack problems the stiffness is evaluated by regular domain integrals and shape functions are determined by both the radial basis function rbf interpolation and the moving least square mls method the stress intensity factors are obtained by two boundary integrals with variation of crack length applications of the proposed technique to two dimensional fracture mechanics have been presented and comparisons are made with benchmark solutions finally the application of the proposed method to modelling fatigue crack growth is presented

ito formula for free stochastic integrals abstract the objects under investigation are the stochastic integrals with respect to free levy processes we define such integrals for square integrable integrands as well as for a certain general class of bounded integrands using the product form of the ito formula we prove the full functional ito formula in this context

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

multiple crack weight for solution of multiple interacting cracks by meshless numerical methods summary 7 we devise a multiple crack weight mcw method for the accurate and effective solution of strongly interacting cracks by meshless numerical methods the mcw method constructs weight functions 9 around cracks so that they simultaneously characterize all the cracks present in the single nodal domain of influence this approach reduces the number of nodes necessary to achieve sufficient 11 accuracy and consequently it decreases the computational effort numerical examples demonstrate that the method allows an accurate solution of multiple cracks problems convergence of the method is 13 analysed and discussed copyright 2006 john wiley sons ltd

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

a study of the representation of cracks with level sets the level set method has been used for a few years to represent cracks in fracture mechanics simulations instead of an explicit description of the cracks faces geometry this paper studies in detail the shape of the level set functions around a crack in two dimensions that is propagating with a sharp kink obtained both with level set update methods found in the literature and with several innovative update methods developed by the author a criterion based on the computation of a j integral of a virtual displacement field obtained with the values of the level set functions is proposed in order to assess the quality of these update methods with the help of this criterion two optimal approaches are identified which predict an accurate evolution of the crack with smooth and consistent level set functions these methods are then applied in three dimensions to the case of an initially penny shaped crack that propagates out of its plane copyright 2006 john wiley sons ltd

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

free stein kernels and an improvement of the free logarithmic sobolev inequality we introduce a free version of the stein kernel relative to a semicircular law we use it to obtain a free counterpart of the hsi inequality of ledoux peccatti and nourdin which is an improvement of the free logarithmic sobolev inequality of biane and speicher as well as a rate of convergence in the multivariate entropic free central limit theorem we also compute the free stein kernels for several relevant families of self adjoint operators

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

the dual boundary element method effective implementation for crack problems summary the present paper is concerned with the effective numerical implementation of the two dimensional dual boundary element method for linear elastic crack problems the dual equations of the method are the displacement and the traction boundary integral equations when the displacement equation is applied on one of the crack surfaces and the traction equation on the other general mixed mode crack problems can be solved with a single region formulation both crack surfaces are discretized with discontinuous quadratic boundary elements this strategy not only automatically satisfies the necessary conditions for the existence of the finite part integrals which occur naturally but also circumvents the problem of collocation at crack tips crack kinks and crack edge corners examples of geometries with edge and embedded crack are analysed with the present method highly accurate results are obtained when the stress intensity factor is evaluated with the j integral technique the accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed mode conditions

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

arbitrary lagrangian eulerian formulation for element free galerkin method abstract arbitrary lagrangian eulerian ale formulation of the element free galerkin efg method is presented efg is a meshless method for solving partial differentia equations in which the trial and test functions employed in the discretization process result from moving least square interpolants the most significant advantage of the method is that it requires only nodes and a description of internal and external boundaries and interfaces such as cracks of the model no element connectivity is needed however as for any discretization method acceptable solutions can only be obtained for a sufficiently refined discretization in dynamic fracture problems where the crack path can be arbitrary and is thus a priori unknown this necessitates a refined discretization in large parts of the computational domain which can lead to prohibitive computation costs ale formulation allows to continuously relocate nodes on the computational domain by combining efg with ale it is thus possible in a crack propagation problem to refine locally the spatial discretization in the neighborhood of a propagating crack tip results are presented for a wave propagation problem as well as for 2 d dynamic crack propagation problems

a new weight function enrichment in meshless methods for multiple cracks in linear elasticity summary a new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions the main novelty consists in modifying the weight function with an intrinsic enrichment which is discontinuous over the finite length of the crack represented by a segment but continuous all around the crack tips an analytical function is used to introduce discontinuities that are incorporated in the kernel in a simple multiplicative manner the resulting method allows a more straightforward implementation and simulation of the presence of multiple cracks crack branching and crack propagation in a meshless framework without using any of the existing algorithms such as visibility transparency and diffraction and without using additional unknowns and additional equations for the evolution of the level sets as in extrinsic partition of unity based methods stress intensity factors calculated using the j integral demonstrate excellent agreement with analytical solutions for classical fracture mechanics benchmarks copyright 2011 john wiley sons ltd

the linearization of the central limit operator in free probability theory we interpret the central limit theorem as a fixed point theorem for a certain operator and consider the problem of linearizing this operator in classical as well as in free probability theory vdn92 we consider two methods giving such a linearization and interpret the result as a weak form of the clt in the classical case the analysis involves dilation operators in the free case more general composition operators appear

smoothing and accelerated computations in the element free galerkin method two topics in the formulation and implementation of meshless methods are considered the smoothing of the approximating functions at concave boundaries and the speedup of the calculation of the approximating functions and their derivatives these techniques are described in the context of the element free galerkin method but they are applicable to other meshless methods results are presented for some elastostatic problems which show a moderate improvement in the accuracy of the smoothed interpolant the speedup in calculating the shape functions is about a factor of two

an adaptive singular es fem for mechanics problems with singular field of arbitrary order this paper presents a singular edge based smoothed finite element method ses fem for mechanics problems with singular stress fields of arbitrary order the ses fem uses a basic mesh of three noded linear triangular t3 elements and a special layer of five noded singular triangular elements st5 connected to the singular point of the stress field the st5 element has an additional node on each of the two edges connected to the singular point it allows us to represent simple and efficient enrichment with desired terms for the displacement field near the singular point with the satisfaction of partition of unity property the stiffness matrix of the discretized system is then obtained using the assumed displacement values not the derivatives over smoothing domains associated with the edges of elements an adaptive procedure for the ses fem is proposed to enhance the quality of the solution with minimized number of nodes several numerical examples are provided to validate the reliability of the present ses fem method c 2012 elsevier b v all rights reserved

derivative recovery and a posteriori error estimate for extended finite elements this paper is the first attempt at error estimation for extended finite elements the goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method xfem in each element the local estimator is the l2 norm of the difference between the raw xfem strain field and an enhanced strain field computed by extended moving least squares xmls derivative recovery obtained from the raw nodal xfem displacements the xmls construction is tailored to the nature of the solution the technique is applied to linear elastic fracture mechanics in which near tip asymptotic functions are added to the mls basis the xmls shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity the result is a very smooth enhanced strain solution including the singularity at the crack tip results are shown for two and three dimensional linear elastic fracture mechanics problems in mode i and mixed mode the effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near tip problem it is also shown that for the linear elastic fracture mechanics problems treated the proposed estimator outperforms one of the superconvergent patch recovery technique of zienkiewicz and zhu which is only c0 parametric studies of the general performance of the estimator are also carried out

dual lukacs regressions for non commutative variables abstract dual lukacs type characterizations of random variables in free probability are studied here first we develop a freeness property satisfied by lukacs type transformations of free poisson and free binomial non commutative variables which are free second we give a characterization of non commutative free poisson and free binomial variables by properties of first two conditional moments which mimic lukacs type assumptions known from classical probability more precisely our result is a non commutative version of the following result known in classical probability if u v are independent real random variables such that e v 1 u u v and e v 2 1 u 2 u v are non random then v has a gamma distribution and u has a beta distribution

higher homotopy groups of complements of complex hyperplane arrangements abstract we generalize results of hattori on the topology of complements of hyperplane arrangements from the class of generic arrangements to the much broader class of hypersolvable arrangements we show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range and we give an explicit z 1 module presentation of p the first non vanishing higher homotopy group we also give a combinatorial formula for the 1 coinvariants of p for affine line arrangements whose cones are hypersolvable we provide a minimal resolution of 2 and study some of the properties of this module for graphic arrangements associated to graphs with no 3 cycles the algorithm for computing 2 is purely combinatorial the fitting varieties associated to 2 may distinguish the homotopy 2 types of arrangement complements with the same 1 and the same betti numbers in low degrees

enrichment of the method of finite spheres using geometry independent localized scalable bubbles summary in this paper we report the development of two new enrichment techniques for the method of finite spheres a truly meshfree method developed for the solution of boundary value problems on geometrically complex domains in the first method the enrichment functions are multiplied by a weight function with compact support while in the second one a floating enrichment node is introduced the scalability of the enrichment bubbles offers flexibility in localizing the spatial extent to which the enrichment field is applied the bubbles are independent of the underlying geometric discretization and therefore provide a means of achieving convergence without excessive refinement several numerical examples involving problems with singular stress fields are provided demonstrating the effectiveness of the enrichment schemes and contrasting them to traditional geometry dependent enrichment strategies in which one or more nodes associated with the geometric discretization of the domain are enriched an additional contribution of this paper is the use of a meshfree numerical integration technique for computing the j integral using the domain integral method copyright 2006 john wiley sons ltd

kesten s theorem for invariant random subgroups an invariant random subgroup of the countable group is a ran dom subgroup of whose distribution is invariant under conjuga tion by all elements of we prove that for a nonamenable invariant random subgroup h the spectral radius of every finitely supported random walk on i s strictly less than the spectral radius of the corresponding random walk on h this generalizes a result of kesten who proved this for normal subgroups as a byproduct we show that for a cayley graph g of a linear group with no amenable normal subgroups any sequence of finite quotients of g that spectrally approximates g converges to g in benjamini schramm convergence in particular this implies that infinite sequences of finite d regular ramanujan schreier graphs have essentially large girth

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

convolution operations arising from vandermonde matrices different types of convolution operations involving large vandermonde matrices are considered the convolutions parallel those of large gaussian matrices and additive and multiplicative free convolution and include additive and multiplicative convolution of vandermonde matrices and deterministic diagonal matrices and cases where two independent vandermonde matrices are involved it is also shown that the convergence of any combination of vandermonde matrices is almost sure the convolutions are divided into two types those which depend on the phase distribution of the vandermonde matrices and those which depend only on the spectra of the matrices a general criterion is presented to find which type applies for any given convolution a simulation is presented verifying the results implementations of the presented convolutions are provided and discussed the implementation is based on the technique of fourier motzkin elimination and is quite general as it can be applied to virtually any combination of vandermonde matrices connections with related matrices such as toeplitz and hankel matrices are also discussed

a mixed augmented lagrangian extended finite element method for modelling elastic plastic fatigue crack growth with unilateral contact the complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods a new technique for the finite element modelling of elastic plastic fatigue crack growth with unilateral contact on the crack faces is presented the extended finite element method x fem is used to discretize the equations allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh this paper presents an augmented lagrangian formulation in the x fem framework that is able to deal with elastic plastic crack growth with treatment of contact an original formulation which takes advantages of two powerful numerical methods is presented next the numerical issues such as contact treatment and numerical integration are addressed and finally numerical examples are shown to validate the method copyright 2007 john wiley sons ltd

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools