products of independent quaternion ginibre matrices and their correlation functions we discuss the product of independent induced quaternion 4 ginibre matrices and the eigenvalue correlations of this product matrix the joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single ginibre matrix but with a more complicated weight function we find the skew orthogonal polynomials corresponding to the weight function of the product matrix and use the method of skew orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size the radial behavior of the density of states is studied in the limit of large matrices and the macroscopic density is discussed the microscopic limit at the origin at the edge s and in the bulk is also discussed for the radial behavior of the density of states

exponential bounds for the support convergence in the single ring theorem abstract we consider an n n matrix of the form a u t v with u v some independent haar distributed unitary matrices and t a deterministic matrix we prove that for k n 1 6 and b 2 1 n tr t 2 as n tends to infinity we have e tr a k a k b 2 k and e tr a k 2 b 2 k this gives a simple proof with slightly weakened hypothesis of the convergence of the support in the single ring theorem improves the available error bound for this convergence from n to e c n 1 6 and proves that the rate of this convergence is at most n 1 6 log n

the single ring theorem we study the empirical measure lan of the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions lan converges towards a rotationally invariant measure on the complex plane whose support is a single ring in particular we provide a complete proof of feinberg zee single ring theorem 6 we also consider the case where un vn are independent haar distributed on the orthogonal group

on transitive algebras containing a standard finite von neumann subalgebra abstract let m be a finite von neumann algebra acting on a hilbert space h and a be a transitive algebra containing m in this paper we prove that if a is 2 fold transitive then a is strongly dense in b h this implies that if a transitive algebra containing a standard finite von neumann algebra in the sense of u haagerup the standard form of von neumann algebras math scand 37 1975 271 283 is 2 fold transitive then a is strongly dense in b h non selfadjoint algebras related to free products of finite von neumann algebras e g l f n and m 2 c 1 2 tr m 2 c 1 2 tr are studied brown measures of certain operators in m 2 c 1 2 tr m 2 c 1 2 tr are explicitly computed

strong convergence of an iterative method for nonexpansive and accretive operators abstract let x be a banach space and a an m accretive operator with a zero consider the iterative method that generates the sequence x n by the algorithm x n 1 n u 1 n j r n x n where n and r n are two sequences satisfying certain conditions and j r denotes the resolvent i r a 1 for r 0 strong convergence of the algorithm x n is proved assuming x either has a weakly continuous duality map or is uniformly smooth

on absolute moments of characteristic polynomials of a certain class of complex random matrices the integer moments of the spectral determinant det zi w 2 of complex random matrices w are obtained in terms of the characteristic polynomial of the positive semidefinite matrix ww for the class of matrices w au where a is a given matrix and u is random unitary this work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context

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

products of independent elliptic random matrices for fixed m 1 we study the product of m independent n times n elliptic random matrices as n tends to infinity our main result shows that the empirical spectral distribution of the product converges with probability 1 to the m th power of the circular law regardless of the joint distribution of the mirror entries in each matrix this leads to a new kind of universality phenomenon the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves our result also generalizes earlier results of gotze tikhomirov on the asymptotic spectrum of products of independent random matrices available at http arxiv org abs 1012 2710 and o rourke soshnikov j probab 16 81 2219 2245 2011 concerning the product of independent iid random matrices

strong convergence theorems for finitely many nonexpansive mappings and applications abstract let e be a uniformly convex banach space which satisfies opial s condition or whose norm is frechet differentiable recently takahashi and shimoji w takahashi k shimoji convergence theorems for nonexpansive mappings and feasibility problems math comput modelling 32 2000 1463 1471 introduced an iterative scheme given by finitely many nonexpansive mappings in e and proved weak convergence theorems which are connected with the problem of image recovery in this paper we introduce a new iterative scheme which includes their iterative scheme as a special case under the assumption that e is a reflexive banach space whose norm is uniformly g teaux differentiable and which has a weakly continuous duality mapping we prove strong convergence theorems which are connected with the problem of image recovery using the established results we consider the problem of finding a common fixed point of finitely many nonexpansive mappings

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

spectral relations between products and powers of isotropic random matrices we show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble is equal to the eigenvalue density of n th power of a single matrix from this ensemble in the limit when the size of the matrix tends to infinity using this observation one can derive the limiting density of the product of n independent identically distributed non hermitian matrices with unitary invariant measures in this paper we discuss two examples the product of n girko ginibre matrices and the product of n truncated unitary matrices we also provide evidence that the result holds also for isotropic orthogonal ensembles

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws

the early days of geometric nonlinear control around 1970 the study of nonlinear control systems took a sharp turn in part this was driven by the hope for a more inclusive theory which would be applicable to various newly emerging aerospace problems lying outside the scope of linear theory and also by the gradual realization that tools from differential geometry and lie theory in particular could be seen as providing a remarkably nice fit with what seemed to be needed for the wholesale extension of linear control theory into a nonlinear setting this paper discusses an initial phase of the development of geometric nonlinear control including material on the broader context from which it emerged we limit our account to developments occurring up to the early 1980s not because the field stopped developing at that point but rather to limit the scope of the project to something manageable even so because of the volume and diversity of the literature we have had to be selective even within the given time frame

hamiltonian structures and stability for rigid bodies with flexible attachments the dynamics of a rigid body with flexible attachments is studied a general framework for problems of this type is established in the context of poisson manifolds and reduction a simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration second the energy casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out the poisson structure and stability results take into account vibrations of the string rotations of the rigid body their coupling at the point of attachment and centrifugal and coriolis forces

lie poisson structures dual spin spacecraft and asymptotic stability on traite la dynamique d un astronef rigide portant trois rotors a partir des equations dynamiques on deduit les structures de lie poisson

resolvents of r diagonal operators we consider the resolvent a 1 of any diagonal operator a in a ii 1 factor our main theorem theorem 1 1 gives a universal asymptotic formula for the norm of such a resolvent en route to its proof we calculate the transform of the operator c 2 where c is voiculescu s circular operator and we give an asymptotic formula for the negative moments of a 2 for any diagonal a we use a mixture of complex analytic and combinatorial techniques each giving finer information where the other can give only coarse detail in particular we introduce partition structure diagrams in section 4 a new combinatorial structure arising in free probability

memory free dynamics for the thouless anderson palmer equations of ising models with arbitrary rotation invariant ensembles of random coupling matrices we propose an iterative algorithm for solving the thouless anderson palmer equations of ising models with arbitrary rotation invariant random coupling matrices in the thermodynamic limit we prove by means of the dynamical functional method that the proposed algorithm converges when the so called de almeida thouless criterion is fulfilled moreover we give exact analytical expressions for the rate of the convergence

iterative approximation of fixed points of nonexpansive mappings let k be a nonempty closed convex subset of a real banach space e which has a uniformly g teaux differentiable norm and t k k be a nonexpansive mapping with f t x k tx x for a fixed 0 1 define s k k by sx 1 x tx x k assume that zt converges strongly to a fixed point z of t as t 0 where zt is the unique element of k which satisfies zt tu 1 t tzt for arbitrary u k let n be a real sequence in 0 1 which satisfies the following conditions c1 lim n 0 c2 n for arbitrary x0 k let the sequence xn be defined iteratively by xn 1 nu 1 n sxn then xn converges strongly to a fixed point of t

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

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

classical and free infinitely divisible distributions and random matrices we construct a random matrix model for the bijection between classical and free infinitely divisible distributions for every d 1 we associate in a quite natural way to each infinitely divisible distribution a distribution p d on the space of d d hermitian matrices such that p d p d p d the spectral distribution of a random matrix with distribution p d converges in probability to when d tends to it gives among other things a new proof of the almost sure convergence of the spectral distribution of a matrix of the gue and a projection model for the marchenko pastur distribution in an analogous way for every d 1 we associate to each infinitely divisible distribution a distribution l d on the space of complex non hermitian d x d random matrices if is symmetric the symmetrization of the spectral distribution of m d when m d is l d distributed converges in probability to

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

lyapunov exponents of free operators abstract lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system this paper offers a definition of lyapunov exponents for a sequence of free linear operators the definition is based on the concept of the extended fuglede kadison determinant we establish the existence of lyapunov exponents derive formulas for their calculation and show that lyapunov exponents of free variables are additive with respect to operator product we illustrate these results using an example of free operators whose singular values are distributed by the marchenko pastur law and relate this example to c m newman s triangle law for the distribution of lyapunov exponents of large random matrices with independent gaussian entries as an interesting by product of our results we derive a relation between the extended fuglede kadison determinant and voiculescu s s transform

dt operators and decomposability of voiculescu s circular operator the dt operators are introduced one for every pair c consisting of a compactly supported borel probability measure on the complex plane and a constant c 0 these are operators on hilbert space that are defined as limits in moments of certain upper triangular random matrices the dt operators include voiculescu s circular operator and elliptic deformations of it as well as the circular free poisson operators we show that every dt operator is strongly decomposable we also show that a dt operator generates a ii1 factor whose isomorphism class depends only on the number and sizes of atoms of those dt operators that are also r diagonal are identified for a quasi nilpotent dt operator t we find the distribution of t t and a recursion formula for general moments of t

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

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

strong haagerup inequalities for free r diagonal elements abstract in this paper we generalize haagerup s inequality u haagerup an example of a nonnuclear c algebra which has the metric approximation property invent math 50 1978 1979 279 293 on convolution norm in the free group to a very general context of r diagonal elements in a tracial von neumann algebra moreover we show that in this holomorphic setting the inequality is greatly improved from its original form we give combinatorial proofs of two important special cases of our main result and then generalize these techniques en route we prove a number of moment and cumulant estimates for r diagonal elements that are of independent interest finally we use our strong haagerup inequality to prove a strong ultracontractivity theorem generalizing and improving the one in p biane free hypercontractivity comm math phys 184 1997 457 474

large deviation for the empirical eigenvalue density of truncated haar unitary matrices let u m be an m m haar unitary matrix and u m n be its n n truncation in this paper the large deviation is proven for the empirical eigenvalue density of u m n as m n and n the rate function and the limit distribution are given explicitly u m n is the random matrix model of quq where u is a haar unitary in a finite von neumann algebra q is a certain projection and they are free the limit distribution coincides with the brown measure of the operator quq

continuous family of invariant subspaces for r diagonal operators we show that every r diagonal operator x has a continuous family of invariant subspaces relative to the von neumann algebra generated by x this allows us to find the brown measure of x and to find a new conceptual proof that voiculescu s s transform is multiplicative our considerations base on a new concept of r diagonality with amalgamation for which we give several equivalent characterizations

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

randomized isometric linear dispersion space time block coding for the df relay channel this paper presents a randomized linear dispersion space time block code for decode and forward synchronous relays the coding matrices are obtained as a set of columns or rows of randomly generated haar distributed unitary matrices with respect to independent and identically distributed i i d generated codes this particular isometric structure reduces the intersymbol interference generated within each relay the gain over i i d codes in terms of spectral efficiency is analyzed for both the lmmse and the ml receivers under the assumption of frequency flat quasi static fading in this setting the spectral efficiency is a random quantity since it depends on the random coding matrices however it is proven that the spectral efficiency converges in probability to a deterministic quantity when the dimensions of the matrices tend to infinity while keeping constant their ratio i e the coding rate consequently when the random coding matrices are large enough the presented system behaves as a deterministic one this result is achieved by means of the rectangular r transform a powerful tool of free probability theory which allows determining the distribution of the singular values of a sum of rectangular matrices

strong convergence theorems for finitely many nonexpansive mappings abstract under the assumption that e is a reflexive banach space whose norm is uniformly geteaux differentiable and which has a weakly continuous duality mapping j with gauge function ceng cubiotti yao strong convergence theorems for finitely many nonexpansive mappings and applications nonlinear analysis 67 2007 1464 1473 introduced a new iterative scheme for a finite commuting family of nonexpansive mappings and proved strong convergence theorems about this iteration in this paper only under the hypothesis that e is a reflexive banach space which has a weakly continuous duality mapping j with gauge function and several control conditions about the iterative coefficient are removed we present a short and simple proof of the above theorem

phase flow of an axially symmetrical gyrostat with one constant rotor we analyze the attitude dynamics of an axially symmetric gyrostat under no external forces and one constant internal spin we introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere with these coordinates we realize that the problem belongs to a general class of hamiltonian systems namely the problem here considered is the one parameter hamiltonian that is a polynomial of at most degree two in a base of the lie algebra so 3 the parametric bifurcations are found for both cases when the rotor is spinning about the axis of symmetry of the gyrostat and when it is spinning about another axis of inertia the general solution for the global general flow is expressed in terms of the jacobian elliptic functions

commutative law for products of infinitely large isotropic random matrices ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left and right multiplication by an arbitrary unitary matrix we show that the multiplication of large isotropic random matrices is spectrally commutative and self averaging in the limit of infinite matrix size n the notion of spectral commutativity means that the eigenvalue density of a product abc of such matrices is independent of the order of matrix multiplication for example the matrix abcd has the same eigenvalue density as adcb in turn the notion of self averaging means that the product of n independent but identically distributed random matrices which we symbolically denote by aaa has the same eigenvalue density as the corresponding power a n of a single matrix drawn from the underlying matrix ensemble for example the eigenvalue density of abccabc is the same as that of a 2 b 2 c 3 we also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues 0 we show that the singularities at the origin of the eigenvalue density and of the singular value density are in one to one correspondence in the limit n the eigenvalue density of an isotropic random matrix has a power law singularity at the origin s with a power se 0 2 when and only when the density of its singular values has a power law singularity with a power s 4 s these results are obtained analytically in the limit n we supplement these results with numerical simulations for large but finite n and discuss finite size effects for the most common ensembles of isotropic random matrices

invertibility of random matrices unitary and orthogonal perturbations author s rudelson mark vershynin roman abstract we show that a perturbation of any fixed square matrix d by a random unitary matrix is well invertible with high probability a similar result holds for perturbations by random orthogonal matrices the only notable exception is when d is close to orthogonal as an application these results completely eliminate a hard to check condition from the single ring theorem by guionnet krishnapur and zeitouni

spectrum of markov generators on sparse random graphs we investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph this is the non hermitian random nxn matrix l defined by l j k x j k if k j and l j j sum l j k k j where x j k are i i d random weights under mild assumptions on the law of the weights we establish convergence as n tends to infinity of the empirical spectral distribution of l after centering and rescaling in particular our assumptions include sparse random graphs such as the oriented erd s renyi graph where each edge is present independently with probability p n 0 as long as np n log n 6 the limiting distribution is characterized as an additive gaussian deformation of the standard circular law in free probability terms this coincides with the brown measure of the free sum of the circular element and a normal operator with gaussian spectral measure the density of the limiting distribution is analyzed using a subordination formula furthermore we study the convergence of the invariant measure of l to the uniform distribution and establish estimates on the extremal eigenvalues of l

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

high order integration of smooth dynamical systems theory and numerical experiments this paper describes a new class of algorithms for integrating linear second order equations and those containing smooth non linearities the algorithms are based on a combination of ideas from standard newmark integration methods and extrapolation techniques for the algorithm to work the underlying newmark method must be stable second order accurate and produce asymptotic error expansions for response quantities containing only even ordered terms it is proved that setting the newmark parameter equal to 1 2 gives a desirable asymptotic expansion irrespective of the setting for numerical experiments are conducted for one linear and two non linear applications

rectangular random matrices related convolution we characterize asymptotic collective behavior of rectangular random matrices the sizes of which tend to infinity at different rates it appears that one can compute the limits of all noncommutative moments thus all spectral properties of the random matrices we consider because when embedded in a space of larger square matrices independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra therefore we can define a rectangular free convolution which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values this convolution is linearized by cumulants and by an analytic integral transform that we called the rectangular r transform

optimality reduction and collective motion the planar self steering particle model of agents in a collective gives rise to dynamics on the n fold direct product of se 2 the rigid motion group in the plane assuming a connected undirected graph of interaction between agents we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions the hamiltonian system associated with the necessary conditions for optimality is reducible to a lie poisson dynamical system possessing interesting structure in particular the strong coupling limit reveals additional hidden symmetry beyond the manifest one used in reduction this enables explicit integration of the dynamics and demonstrates the presence of a master clock that governs all agents to steer identically for finite coupling strength we show that special solutions exist with steering controls proportional across the collective these results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations

squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem we extend the so called single ring theorem feinberg and zee 1997 nucl phys b 504 579 also known as the haagerup larsen theorem haagerup and larsen 2000 j funct anal 176 331 we do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non hermitian matrix x being the spectral density weighted by the squared eigenvalue condition number is given by a simple formula involving only the radial spectral cumulative distribution function of x we show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number we give examples and provide a cross check of the analytic prediction by the large scale numerics

reduction of controlled lagrangian and hamiltonian systems with symmetry we develop reduction theory for controlled lagrangian and controlled hamiltonian systems with symmetry reduction theory for these systems is needed in a variety of examples such as a spacecraft with rotors a heavy top with rotors and underwater vehicle dynamics one of our main results shows the equivalence of the method of reduced controlled lagrangian systems and that of reduced controlled hamiltonian systems in the case of simple mechanical systems with symmetry

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

spectral density of generalized wishart matrices and free multiplicative convolution we investigate the level density for several ensembles of positive random matrices of a wishart like structure w xx where x stands for a non hermitian random matrix in particular making use of the cauchy transform we study the free multiplicative powers of the marchenko pastur mp distribution mp s which for an integer s yield fuss catalan distributions corresponding to a product of s independent square random matrices x x 1 x s new formulas for the level densities are derived for s 3 and s 1 3 moreover the level density corresponding to the generalized bures distribution given by the free convolution of arcsine and mp distributions is obtained we also explain the reason of such a curious convolution the technique proposed here allows for the derivation of the level densities for several other cases

applications of realizations aka linearizations to free probability abstract we show how the combination of new linearization ideas in free probability theory with the powerful realization machinery developed over the last 50 years in fields including systems engineering and automata theory allows solving the problem of determining the eigenvalue distribution or even the brown measure in the non selfadjoint case of noncommutative rational functions of random matrices when their size tends to infinity along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras e g type ii 1 von neumann algebras with a precise control of the domains of the rational expressions the paper provides sufficient background information with the intention that it should be accessible both to functional analysts and to algebraists

sample variance in free probability abstract let x 1 x 2 x n denote i i d centered standard normal random variables then the law of the sample variance q n i 1 n x i x 2 is the 2 distribution with n 1 degrees of freedom it is an open problem in classical probability to characterize all distributions with this property and in particular whether it characterizes the normal law in this paper we present a solution of the free analogue of this question and show that the only distributions whose free sample variance is distributed according to a free 2 distribution are the semicircle law and more generally so called odd laws by which we mean laws with vanishing higher order even cumulants in the way of proof we derive an explicit formula for the free cumulants of q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of r cyclicity

support convergence in the single ring theorem we study the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions all these eigenvalues lie in a single ring

r cyclic families of matrices in free probability we introduce the concept of r cyclic family of matrices with entries in a noncommutative probability space the definition consists in asking that only the cyclic noncrossing cumulants of the entries of the matrices are allowed to be nonzero let a1 as be an r cyclic family of d d matrices over a noncommutative probability space a we prove a convolution type formula for the explicit computation of the joint distribution of a1 as considered in md a with the natural state in terms of the joint distribution considered in the original space a of the entries of the s matrices several important situations of families of matrices with tractable joint distributions arise by application of this formula moreover let a1 as be a family of d d matrices over a noncommutative probability space a let d md a denote the algebra of scalar diagonal matrices and let c be the subalgebra of md a generated by a1 as d we prove that the r cyclicity of a1 as is equivalent to a property of c namely that c is free from md c with amalgamation over d

r diagonal dilation semigroups this paper addresses extensions of the complex ornstein uhlenbeck semigroup to operator algebras in free probability theory if a 1 a k are free fancyscript r diagonal operators in a ii1 factor then d t a i 1 cdots a i n e nt a i 1 cdots a i n defines a dilation semigroup on the non self adjoint operator algebra generated by a 1 a k we show that d t extends in two different ways to a semigroup of completely positive maps on the von neumann algebra generated by a 1 a k moreover we show that d t satisfies an optimal ultracontractive property d t colon l 2 to l infty sim t 1 for small t 0