factoriality of boejko speicher von neumann algebras we study the von neumann algebra generated by q deformed gaussian elements li l i where operators li fulfill the q deformed canonical commutation relations lil j ql jli ij for 1 q 1 we show that if the number of generators is finite greater than some constant depending on q it is a ii1 factor which does not have the property our technique can be used for proving factoriality of many examples of von neumann algebras arising from some generalized brownian motions both for the type ii1 and type iii case

free monotone transport without a trace we adapt the free monotone transport results of guionnet and shlyakhtenko to the type iii case as a direct application we obtain that the q deformed araki woods algebras are isomorphic for sufficiently small q

symmetric hilbert spaces arising from species of structures symmetric hilbert spaces such as the bosonic and the fermionic fock spaces over some lsquo one particle space mathcal k are formed by certain symmetrization procedures performed on the full fock space we investigate alternative ways of symmetrization by building on joyal s notion of a combinatorial species any such species f gives rise to an endofunctor gamma f of the category of hilbert spaces with contractions mapping a hilbert space mathcal k to a symmetric hilbert space gamma f mathcal k with the same symmetry as the species f a general framework for annihilation and creation operators on these spaces is developed and compared to the generalised brownian motions of r speicher and m bozejko as a corollary we find that the commutation relation a ia j a j a i f n delta ij with na i a i n a i admits a realization on a symmetric hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients

the bi poisson process a quadratic harness this paper is a continuation of our previous research on quadratic harnesses that is processes with linear regressions and quadratic conditional variances our main result is a construction of a markov process from given orthogonal and martingale polynomials the construction uses a two parameter extension of the al salam chihara polynomials and a relation between these polynomials for different values of parameters

a multigrid method for eigenvalue problem a multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem in this scheme solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space besides the multigrid scheme all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems the total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem therefore this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving some numerical experiments are presented to validate the efficiency of the method

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

stochastic integration with respect to q brownian motion we develop a stochastic integration with respect to a q brownian motion for i e a non commutative process where the operator at and its adjoint fulfill the q commutation relation under the vacuum state expectation we show that this process enjoys a predictable representation type property

quadratic harnesses q commutations and orthogonal martingale polynomials we introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a commutation relation this implies that quadratic harnesses are essentially determined uniquely by five numerical constants explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest

dynamic adaptive partitioning for nonlinear time series summary we propose a dynamic adaptive partitioning scheme for nonparametric analysis of stationary nonlinear time series it yields estimates of the whole probability distribution of the underlying process we use information from past values to construct adaptive partitioning in a dynamic fashion which is then different from the more common static schemes in the regression set up the idea of dynamic partitioning is new we make it constructive by an approach based on quantisation of the data and adaptively modelling partition cells with a parsimonious markov chain the methodology is formulated in terms of a new model class the so called quantised variable length markov chains it is a new extension of finite valued variable length markov chains to processes with values in rd we discuss estimation explore asymptotic properties of the new method and give some numerical results which reflect the finite sample behaviour

wick product for commutation relations connected with yang baxter operators and new constructions of factors we analyze a certain class of von neumann algebras generated by selfadjoint elements for satisfying the general commutation relations such algebras can be continuously embedded into some closure of the set of finite linear combinations of vectors where is an orthonormal basis of a hilbert space the operator which represents the vector is denoted by and called the wick product of the operators we describe explicitly the form of this product also we estimate the operator norm of for finally we apply these two results and prove that under the assumption all the von neumann algebras considered are ii 1 factors

noncommutative levy processes for generalized particularly anyon statistics let t mathbb r d let a function qt 2 to mathbb c satisfy q s t overline q t s and q s t 1 a generalized statistics is described by creation operators partial t dagger and annihilation operators t t in t which satisfy the q commutation relations partial s partial dagger t q s t partial dagger t partial s delta s t partial s partial t q t s partial t partial s partial dagger s partial dagger t q t s partial dagger t partial dagger s from the point of view of physics the most important case of a generalized statistics is the anyon statistics for which q s t is equal to q if s t here q in mathbb c q 1 we start the paper with a detailed discussion of a q fock space and operators partial t dagger partial t t in t in it which satisfy the q commutation relations next we consider a noncommutative stochastic process white noise omega t partial t dagger partial t lambda partial t dagger partial t t in t here lambda in mathbb r is a fixed parameter the case 0 corresponds to a q analog of brownian motion while 0 corresponds to a centered q poisson process we study q hermite q charlier respectively polynomials of infinitely many noncommutatative variables omega t t in t the main aim of the paper is to explain the notion of independence for a generalized statistics and to derive corresponding levy processes to this end we recursively define q cumulants of a field xi t t in t this allows us to define a q levy process as a field xi t t in t whose values at different points of t are q independent and which possesses a stationarity of increments in a certain sense we present an explicit construction of a q levy process and derive a nualart schoutens type chaotic decomposition for such a process

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

an optimal two level overlapping domain decomposition method for elliptic problems in two and three dimensions the solution of linear systems of algebraic equations that arise from elliptic finite element problems is considered a two level overlapping domain decomposition method that can be viewed as a combination of the additive and multiplicative schwarz methods is studied this method combines the advantages of the two methods it converges faster than the additive schwarz algorithm and is more parallelizable than the multiplicative schwarz algorithm and works for general not necessarily self adjoint linear second order elliptic equations the gmres method is used to solve the resulting preconditioned linear system of equations and it is shown that the algorithm is optimal in the sense that the rate of convergence is independent of the mesh size and the number of subregions in both r 2 and r 3 a numerical comparison with the additive and multiplicative schwarz preconditioned gmres is reported

infinitesimal non crossing cumulants and free probability of type b free probabilistic considerations of type b first appeared in the paper of biane goodman and nica p biane f goodman a nica non crossing cumulants of type b trans amer math soc 355 2003 2263 2303 recently connections between type b and infinitesimal free probability were put into evidence by belinschi and shlyakhtenko s t belinschi d shlyakhtenko free probability of type b analytic aspects and applications preprint 2009 available online at www arxiv org under reference arxiv 0903 2721 the interplay between type b and infinitesimal is also the object of the present paper we study infinitesimal freeness for a family of unital subalgebras a1 ak in an infinitesimal noncommutative probability space a and we introduce a concept of infinitesimal non crossing cumulant functionals for a obtained by taking a formal derivative in the formula for usual non crossing cumulants we prove that the infinitesimal freeness of a1 ak is equivalent to a vanishing condition for mixed cumulants this gives the infinitesimal counterpart for a theorem of speicher from usual free probability we show that the lattices nc b n of non crossing partitions of type b appear in the combinatorial study of a in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables as an application of alternating free products we observe the infinitesimal analogue for the well known fact that freeness is preserved under compression with a free projection as another application we observe the infinitesimal analogue for a well known procedure used to construct free families of free poisson elements finally we discuss situations when the freeness of a1 ak in a can be naturally upgraded to infinitesimal freeness in a for a suitable choice of a companion functional a c

generalised brownian motion and second quantisation a new approach to the generalised brownian motion introduced by m bouzejko and r speicher is described based on symmetry rather than deformation the symmetrisation principle is provided by joyal s notions of tensorial and combinatorial species any such species v gives rise to an endofunctor fv of the category of hilbert spaces with contractions a generalised brownian motion is an algebra of creation and annihilation operators acting on fv h for arbitrary hilbert spaces h and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions the positivity is encoded by a semigroup of broken pair partitions whose representation space with respect to t is v the existence of the second quantisation as functor t from hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t for a certain one parameter interpolation between the fermionic and the free brownian motion it is shown that the field algebras k are type ii1 factors when k is infinite dimensional

the normal distribution is infinitely divisible abstract we prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution we study the voiculescu transform first by giving a survey of its combinatorial implications and then analytically including a proof of free infinite divisibility in fact we prove that a sub family of askey wimp kerov distributions are freely infinitely divisible of which the normal distribution is a special case at the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution the others being the cauchy distribution and the free 1 2 stable distribution

free monotone transport by solving a free analog of the monge ampere equation we prove a non commutative analog of brenier s monotone transport theorem if an n tuple of self adjoint non commutative random variables z 1 z n satisfies a regularity condition its conjugate variables 1 n should be analytic in z 1 z n and j should be close to z j in a certain analytic norm then there exist invertible non commutative functions f j of an n tuple of semicircular variables s 1 s n so that z j f j s 1 s n moreover f j can be chosen to be monotone in the sense that and g is a non commutative function with a positive definite hessian in particular we can deduce that c z 1 z n c s 1 s n and w z 1 dots z n cong l mathbb f n thus our condition is a useful way to recognize when an n tuple of operators generate a free group factor we obtain as a consequence that the q deformed free group factors vargamma q mathbb r n are isomorphic for sufficiently small q with bound depending on n to free group factors we also partially prove a conjecture of voiculescu by showing that free gibbs states which are small perturbations of a semicircle law generate free group factors lastly we show that entrywise monotone transport maps for certain gibbs measure on matrices are well approximated by the matricial transport maps given by free monotone transport

domain decomposition methods for eigenvalue problems abstract this paper proposes several domain decomposition methods to compute the smallest eigenvalue of linear self adjoint partial differential operators let us be given a partial differential operator on a domain which consists of two nonoverlapping subdomains suppose a fast eigenvalue solver is available for each of these subdomains but not for the union of them the first proposed method is a scheme which determines an appropriate boundary condition at the interface separating the two regions this boundary condition can be derived from the zero of a nonlinear operator which plays the same role for the eigenvalue problem as the steklov poincare operator for the linear equation an iterative method can be used to solve this nonlinear equation yielding the exact boundary condition at the interface this enables the determination of the eigenpair on the whole domain the same concept can be applied to a schwarz alternating method for the eigenvalue problem in case the subdomains overlap a nonoverlapping scheme in the spirit of schur complement will also be discussed these ideas are also applicable to a domain imbedding algorithm

domain decomposition algorithms for fourth order nonlinear elliptic eigenvalue problems we study domain decomposition methods for fourth order plate problems the well known von karman equations are used as our model problem by exploiting the symmetry of the domain the solution of the original problem can be obtained by solving those associated reduced problems which are defined on subdomains with appropriate boundary conditions we show how nonoverlapping and overlapping domain decomposition methods can be used to solve the reduced problems for the linearized von karman equation we present preconditioners using both fourier analysis and probing techniques for the interface systems which are similar to those derived by chan et al finally we compare the efficiency of various domain decomposition preconditioners for solving the von karman equations

a novel domain decomposition framework for the ground state solution of bose einstein condensates abstract in this study we present a new type of domain decomposition algorithm to obtain the ground state solution for bose einstein condensates using our proposed algorithm instead of directly solving the nonlinear eigenvalue problem one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method and subsequently solve a small scale nonlinear eigenvalue problem on an enriched space simultaneously because solving large scale nonlinear eigenvalue problem directly is time intensive our algorithm can obviously improve the efficiency of producing simulation for bose einstein condensates in addition any domain decomposition algorithm for linear boundary value problems can be applied to our algorithm framework which makes the algorithm considerably flexible two numerical experiments are presented in the paper to demonstrate the efficiency and scalability of our proposed algorithm

on a class of free levy laws related to a regression problem abstract the free meixner laws arise as the distributions of orthogonal polynomials with constant coefficient recursions we show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum we apply this result to describe free levy processes with quadratic conditional variances and to prove a converse implication related to asymptotic freeness of random wishart matrices

free probability of type b analytic interpretation and applications in this paper we give an analytic interpretation of free convolution of type b and provide a new formula for its computation this formula allows us to show that free additive convolution of type b is essentially a re casting of conditionally free convolution we put in evidence several aspects of this operation the most significant being its apparition as an intertwiner between derivation and free convolution of type a we also show connections between several limit theorems in type a and type b free probability moreover we show that the analytical picture fits very well with the idea of considering type b random variables as infinitesimal deformations to ordinary non commutative random variables

functors of white noise associated to characters of the infinite symmetric group the characters of the infinite symmetric group are extended to multiplicative positive definite functions on pair partitions by using an explicit representation due to versik and kerov the von neumann algebra generated by the fields with f in an infinite dimensional real hilbert space is infinite and the vacuum vector is not separating for a family depending on an integer n 1 an exclusion principle is found allowing at most identical particles on the same state the algebras are type factors functors of white noise are constructed and proved to be non equivalent for different values of n

fock space associated to coxeter groups of type b abstract in this article we construct a generalized gaussian process coming from coxeter groups of type b it is given by creation and annihilation operators on an q fock space which satisfy the commutation relation b q x b q y q b q y b q x x y i x y q 2 n where x y are elements of a complex hilbert space with a self adjoint involution x x and n is the number operator with respect to the grading on the q fock space we give an estimate of the norms of creation operators we show that the distribution of the operators b q x b q x with respect to the vacuum expectation becomes a generalized gaussian distribution in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions our generalized gaussian distribution is associated to the orthogonal polynomials called the q meixner pollaczek polynomials yielding the q hermite polynomials when 0 and free meixner polynomials when q 0

a multi level correction scheme for eigenvalue problems abstractin this paper a new type of multi level correction scheme is proposed forsolving eigenvalue problems by nite element method with this new scheme the accuracy of eigenpair approximations can be improved after each correc tion step which only needs to solve a source problem on ner nite elementspace and an eigenvalue problem on the coarsest nite element space thiscorrection scheme can improve the e ciency of solving eigenvalue problemsby nite element method keywords eigenvalue problem multi level correction scheme nite ele ment method multi space multi grid ams subject classi cations 65n30 65b99 65n25 65l15 1 introduction the purpose of this paper is to propose a new type of multi level correction schemebased on nite element discretization to solve eigenvalue problems the two gridmethod for solving eigenvalue problems has been proposed and analyzed by xu andzhou in 21 the idea of the two grid comes from 19 20 for nonsymmetric orinde nite problems and nonlinear elliptic equations since then there have existedmany numerical methods for solving eigenvalue problems based on the idea of two grid method 1 6 17

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

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

classical versions of q gaussian processes conditional moments and bell s inequality we show that classical processes corresponding to operators which satisfy a q commutative relation have linear regressions and quadratic conditional variances from this we deduce that bell s inequality for their covariances can be extended from q 1 to the entire range 1 q 1

non crossing cumulants of type b we establish connections between the lattices of non crossing partitions of type b introduced by v reiner and the framework of the free probability theory of d voiculescu lattices of non crossing partitions of type a up to now have played an important role in the combinatorics of free probability primarily via the non crossing cumulants of r speicher here we introduce the concept of non crossing cumulant of type b the inspiration for its definition is found by looking at an operation of restricted convolution of multiplicative functions studied in parallel for functions on symmetric groups in type a and on hyperoctahedral groups in type b the non crossing cumulants of type b live in an appropriate framework of non commutative probability space of type b and are closely related to a type b analogue for the r transform of voiculescu which is the free probabilistic counterpart of the fourier transform by starting from a condition of vanishing of mixed cumulants of type b we obtain an analogue of type b for the concept of free independence for random variables in a non commutative probability space

a multilevel correction type of adaptive finite element method for eigenvalue problems an adaptive finite element method for eigenvalue problems is proposed based on the multilevel correction scheme different from the standard adaptive finite element method which requires solving ei

factoriality of q gaussian von neumann algebras we prove that the von neumann algebras generated by n q gaussian elements are factors for n 2

non injectivity of the q deformed von neumann algebra in this paper we prove that the von neumann algebra generated by q gaussians is not injective as soon as the dimension of the underlying hilbert space is greater than 1 our approach is based on a suitable vector valued khintchine type inequality for wick products the same proof also works for the more general setting of a yang baxter deformation our techniques can also be extended to the so called q araki woods von neumann algebras recently introduced by hiai in this latter case we obtain the non injectivity under some asssumption on the spectral set of the positive operator associated with the deformation

overlapping schwarz preconditioned eigensolvers for spectral element discretizations abstract model generalized eigenproblems associated with self adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered their numerical approximation is based on gauss lobatto legendre conforming spectral elements defined by gordon hall transfinite mappings the resulting discrete eigenproblems are solved iteratively with a locally optimal block preconditioned conjugate gradient lobpcg method accelerated by an overlapping schwarz preconditioner several numerical tests show the good convergence properties of the proposed preconditioned eigensolver such as its scalability and quasi optimality in the discretization parameters which are analogous to those obtained for overlapping schwarz preconditioners for linear systems