atoms and regularity for measures in a partially defined free convolution semigroup consider a borel probability measure on the real line and denote by t t 1 the free additive convolution semigroup defined by nica and speicher we show that the singular part of t is purely atomic and the density of t is locally analytic provided that t 1 the main ingredient is a global inversion theorem for analytic functions on a half plane

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

jack polynomials and free cumulants abstract we study the coefficients in the expansion of jack polynomials in terms of power sums we express them as polynomials in the free cumulants of the transition measure of an anisotropic young diagram we conjecture that such polynomials have nonnegative integer coefficients this extends recent results about normalized characters of the symmetric group

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

a note on relationship between two classes of integral inequalities this technical note firstly introduces two classes of integral inequalities with and without free matrices respectively and points out that they although in different forms are actually equivalent in the sense of conservatism i e the two corresponding ones produce the same tight upper bounds secondly the relationship between the method of integral inequalities with free matrices and the free weighting matrix technique is intensively investigated it is shown that these two different methods are actually equivalent in assessing the stability of time delay systems

sharp regularity for the degenerate doubly nonlinear parabolic equation abstract the aim of this paper is to provide sharp regularity estimates for locally bounded solutions of the degenerate doubly nonlinear parabolic equation u t div m u m 1 d u p 2 d u f where m 1 p 2 and f l q r more precisely we show that solutions are locally of class c 0 where depends explicitly only on the optimal holder exponent for solutions of the homogeneous case the integrability of f in space and time and the nonlinearity parameters p and m

affine bessel legendre inequality application to stability analysis for systems with time varying delays abstract recently some novel inequalities have been proposed such as the auxiliary function based integral inequality and the bessel legendre inequality which can be obtained from the former by choosing legendre polynomials as auxiliary functions these inequalities have been successfully applied to systems with constant delays but there have been some difficulties in application to systems with time varying delays since the resulting bounds contain the reciprocal convexity which may not be tractable as it is this paper proposes an equivalent form of the bessel legendre inequality which has the advantage of being easily applied to systems with time varying delays without the reciprocal convexity

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

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

boundedness of the gradient for a doubly nonlinear parabolic equation we prove the local boundedness of the gradient for positive solutions to a doubly nonlinear parabolic equation in the case when the standard lebesgue measure has been replaced by a doubling measure which supports a weak poincare inequality

convergence and equivalence results for the jensen s inequality application to time delay and sampled data systems the jensen s inequality plays a crucial role in the analysis of time delay and sampled data systems its conservatism is studied through the use of the gruss inequality it has been reported in the literature that fragmentation or partitioning schemes allow to empirically improve the results we prove here that the jensen s gap can be made arbitrarily small provided that the order of uniform fragmentation is chosen sufficiently large nonuniform fragmentation schemes are also shown to speed up the convergence in certain cases finally a family of bounds is characterized and a comparison with other bounds of the literature is provided it is shown that the other bounds are equivalent to jensen s and that they exhibit interesting well posedness and linearity properties which can be exploited to obtain better numerical results

existence of weak solutions of doubly nonlinear parabolic equations abstract we deal with a cauchy dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space time cylinder for doubly nonlinear parabolic equations whose prototype is t u div u m 1 d u p 2 d u f with a non negative lebesgue function f on the right hand side where p 2 n n 2 and m 0 the central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f the constructed solution is obtained by a limit of approximations i e we use solutions of regularized cauchy dirichlet problems and pass to the limit to receive a solution for the original cauchy dirichlet problem

robust h state feedback control for linear systems this paper investigates the problem of robust h control for linear systems first the state feedback closed loop control algorithm is designed second by employing the geometric progression theory a modified augmented lyapunov krasovskii functional lkf with the geometric integral interval is established then parameter uncertainties and the derivative of the delay are flexibly described by introducing the convex combination skill this technique can eliminate the unnecessary enlargement of the lkf derivative estimation which gives less conservatism in addition the designed controller can ensure that the linear systems are globally asymptotically stable with a guaranteed h performance in the presence of a disturbance input and parameter uncertainties a liquid monopropellant rocket motor with a pressure feeding system is evaluated in a simulation example it shows that this proposed state feedback control approach achieves the expected results for linear systems in the sense of the prescribed h performance

wirtinger based integral inequality application to time delay systems in the last decade the jensen inequality has been intensively used in the context of time delay or sampled data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities lmis however it is also well known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature reducing this gap is a relevant issue and always an open problem in this paper we propose an alternative inequality based on the fourier theory more precisely on the wirtinger inequalities it is shown that this resulting inequality encompasses the jensen one and also leads to tractable lmi conditions in order to illustrate the potential gain of employing this new inequality with respect to the jensen one two applications on time delay and sampled data stability analysis are provided

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

free matrix based integral inequality for stability analysis of systems with time varying delay the free weighting matrix and integral inequality methods are widely used to derive delay dependent criteria for the stability analysis of time varying delay systems because they avoid both the use of a model transformation and the technique of bounding cross terms this technical note presents a new integral inequality called a free matrix based integral inequality that further reduces the conservativeness in those methods it includes well known integral inequalities as special cases using it to investigate the stability of systems with time varying delays yields less conservative delay dependent stability criteria which are given in terms of linear matrix inequalities two numerical examples demonstrate the effectiveness and superiority of the method

comparison of some bounding inequalities applied in stability analysis of time delay systems abstract this paper gives comparisons of several bounding inequalities proposed recently for the estimation of integrals and sums of quadratic functions which plays an important role in stability analysis of time delay systems via the lyapunov krasovskii approach the equivalences of several known variants of the free matrix based inequalities and their generalized and simplified forms are shown secondly the relationship between the simplified free matrix based inequality and the combination of the bessel based inequality with different bounding inequalities affine in the length of the intervals are investigated finally we shall demonstrate the theoretical results with the application of the various estimations for the stability analysis of a linear system with time varying delay using the same lyapunov krasovskii functional

local behaviour of solutions to doubly nonlinear parabolic equations we give a relatively simple and transparent proof for harnack s inequality for certain degenerate doubly nonlinear parabolic equations we consider the case where the lebesgue measure is replaced with a doubling borel measure which supports a poincare inequality

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

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

quantum invariant families of matrices in free probability we consider self adjoint families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group for the free orthogonal and hyperoctahedral groups we obtain complete characterizations of the invariant families in terms of an operator valued r cyclicity condition this is a surprising contrast with the aldous hoover characterization of jointly exchangeable arrays

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

some minimization problems for the free analogue of the fisher information abstract we consider the free non commutative analogue introduced by d voiculescu of the concept of fisher information for random variables we determine the minimal possible value of a a if a is a non commutative random variable subject to the constraint that the distribution of a a is prescribed more generally we obtain the minimal possible value of a ij a ij 1 i j d if a ij 1 i j d is a family of non commutative random variables such that the distribution of a a is prescribed where a is the matrix a ij d i j 1 the d d generalization is obtained from the case d 1 via a result of independent interest concerning the minimal value of a ij a ij 1 i j d when the matrix a a ij d i j 1 and its adjoint have a given joint distribution a version of this result describes the minimal value of b ij 1 i j d when the matrix b b ij d i j 1 is selfadjoint and has a given distribution we then show how the minimization results obtained for lead to maximization results concerning the free entropy also defined by voiculescu

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

low complexity polynomial expansion detector with deterministic equivalents of the moments of channel gram matrix for massive mimo uplink we consider a low complexity polynomial expansion pe detector in a massive multiple input multiple output mimo uplink channel in contrast to most massive mimo systems in the literature where single antenna user equipments ues are assumed multiple antenna ues are employed in this paper moreover the channel between a base station bs and a ue is a jointly correlated rician fading channel the pe detector reduces the computational complexity of the minimum mean square error mmse detector by replacing the matrix inversion with an approximate matrix polynomial the coefficients of the approximate matrix polynomial are computed from the deterministic equivalents of the moments of the channel gram matrix we use operator valued free probability which is a more general version of free probability to derive the deterministic equivalents in particular we use the operator valued moment cumulant formula the proposed low complexity pe detector is easy to compute simulation results show that the proposed detector can achieve performance close to the mmse detector

a holder estimate for nonlinear parabolic systems of p laplacian type abstract we extend the holder regularity results for weak solutions of the p laplacian systems originally obtained by dibenedetto to a larger class of right hand side terms by appropriately arranging two scale parameters on the sizes of a local parabolic cylinder and of the gradient of the solutions we get the same holder exponent as in the scalar case our method can also recover the classical parabolic case p 2

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

reciprocally convex approach to stability of systems with time varying delays whereas the upper bound lemma for matrix cross product introduced by park 1999 and modified by moon park kwon and lee 2001 plays a key role in guiding various delay dependent criteria for delayed systems the jensen inequality has become an alternative as a way of reducing the number of decision variables it directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities resulting in a linear combination of positive functions weighted by the inverses of convex parameters this paper suggests the lower bound lemma for such a combination which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables comparable to those based on the jensen inequality lemma

cumulants in noncommutative probability theory i noncommutative exchangeability systems cumulants linearize convolution of measures we use a formula of good to define noncommutative cumulants in a very general setting it turns out that the essential property needed is exchangeability of random variables roughly speaking the formula says that cumulants are moments of a certain discrete fourier transform of a random variable this provides a simple unified method to understand the known examples of cumulants like classical free and various q cumulants

brief paper wirtinger s inequality and lyapunov based sampled data stabilization discontinuous lyapunov functionals appeared to be very efficient for sampled data systems fridman 2010 naghshtabrizi hespanha teel 2008 in the present paper new discontinuous lyapunov functionals are introduced for sampled data control in the presence of a constant input delay the construction of these functionals is based on the vector extension of wirtinger s inequality these functionals lead to simplified and efficient stability conditions in terms of linear matrix inequalities lmis the new stability analysis is applied to sampled data state feedback stabilization and to a novel sampled data static output feedback problem where the delayed measurements are used for stabilization

multiple integral inequalities and stability analysis of time delay systems abstract this paper is devoted to stability analysis of continuous time delay systems based on a set of lyapunov krasovskii functionals new multiple integral inequalities are derived that involve the famous jensen s and wirtinger s inequalities as well as the recently presented bessel legendre inequalities of seuret and gouaisbaut 2015 and the wirtinger based multiple integral inequalities of park et al 2015 and lee et al 2015 the present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of lmis establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods numerical examples illustrate the efficiency of the method

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 note on wirtinger type integral inequalities for time delay systems different integral inequalities play important role when problems of stability analysis and controller synthesis for time delay systems are considered the connection of jensen s inequality and extended jensen s inequality is well understood now to reduce the conservativeness introduced by the application of jensen s inequality several versions of the wirtinger integral inequality have been published recently this note presents a comparison between some of these inequalities

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

on equivalence and efficiency of certain stability criteria for time delay systems in recent years there have been a number of new delay dependent stability criteria based on linear matrix inequalities published in the literature this note aims to theoretically establish the equivalence of seven of these stability criteria moreover the efficiency of these stability criteria is assessed based on the number of unknowns in the linear matrix inequalities

stability analysis of sampled data systems via free matrix based time dependent discontinuous lyapunov approach in this paper a new time dependent discontinuous lyapunov functional namely free matrix based time dependent discontinuous fmbtdd lyapunov functional is introduced for stability analysis of sampled data systems first a modified free matrix based integral inequality mfmbii is derived based on the existing free matrix based integral inequality 1 and it is applied to develop a stability criterion for sampled data systems and then inspired by mfmbii fmbtdd term is established that leads to efficient stability conditions four numerical examples are given to demonstrate the effectiveness of the proposed methods

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

a unified fluctuation formula for one cut β ensembles of random matrices using a coulomb gas approach we compute the generating function of the covariances of power traces for one cut ensembles of random matrices in the limit of large matrix size this formula depends only on the support of the spectral density and is therefore universal for a large class of models this allows us to derive a closed form expression for the limiting covariances of an arbitrary one cut ensemble as particular cases of the main result we consider the classical gaussian wishart and jacobi ensembles for which we derive previously available results as well as new ones within a unified simple framework we also discuss the connections between the problem of trace fluctuations for the gaussian unitary ensemble and the enumeration of planar maps

harnack type inequalities for solutions of certain doubly nonlinear parabolic equations abstract here we prove an intrinsic version of the harnack inequality for locally bounded nonnegative weak solutions of nonlinear parabolic equations of the type u t div u m 1 du p 2 du in d t where p 1 m p 2 m p p n 3

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

improved delay dependent stability criteria for generalized neural networks with time varying delays abstract this paper is concerned with the problem of stability analysis for generalized neural networks with time varying delays a novel integral inequality which includes several existing inequalities as special cases is presented by employing a suitable lyapunov krasovskii functional lkf and using the proposed integral inequality to estimate the derivative of the lkf improved delay dependent stability criteria expressed in terms of linear matrix inequalities are derived finally four numerical examples are provided to demonstrate the effectiveness and the improvement of the proposed method

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

stability analysis of linear systems with interval time varying delays utilizing multiple integral inequalities this paper is devoted to stability analysis of continuous time delay systems with interval time varying delays having known bounds on the delay derivatives a parameterized family of lyapunov krasovskii functionals involving multiple integral terms is introduced and novel multiple integral inequalities are utilized to derive sufficient stability condition for systems with time varying delays the efficiency of the proposed method is illustrated by numerical examples

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

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

a note on equivalence between two integral inequalities for time delay systems jensen s inequality and extended jensen s inequality are two important integral inequalities when problems of stability analysis and controller synthesis for time delay systems are considered the extended jensen s inequality introduces two additional free matrices and is generally regarded to be less conservative than jensen s inequality the equivalence between jensen s inequality and extended jensen s inequality in bounding the quadratic term h t h t x t s z x s d s in lyapunov functional of time delay systems is presented and theoretically proved it is shown that the extended jensen s inequality does not decrease the lower bound of this quadratic term obtained using jensen s inequality and then it does not reduce the conservativeness though two additional free matrices m 1 and m 2 are involved

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

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

on operator valued bi free distributions abstract in this paper operator valued bi free distributions are investigated given a subalgebra d of a unital algebra b it is established that a two faced family z is bi free from b b op over d if and only if certain conditions relating the b valued and d valued bi free cumulants of z are satisfied using this we verify that a two faced family of matrices is r cyclic if and only if they are bi free from the scalar matrices over the scalar diagonal matrices furthermore the operator valued bi free partial r s and t transforms are constructed new proofs of results from free probability are developed in order to facilitate many of these bi free results

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

notes on stability of time delay systems bounding inequalities and augmented lyapunov krasovskii functionals the bounding inequalities and the lyapunov krasovskii functionals lkfs are important for the stability analysis of time delay systems much attention has been paid to develop tighter inequalities for improving stability criteria while the contribution of the lkfs has not been considered when discussing the relationship between the tightness of inequalities and the conservatism of criteria this note is concerned with this issue firstly it is proved that when a simple lkf is applied the stability criteria obtained by the wirtinger based inequality and the jensen inequality are equivalent although the wirtinger based inequality is tighter it means that the tighter inequality does not always lead to a less conservative criterion secondly it is found that a suitable augmented lkf with necessary integral vectors in its derivative is required to achieve the advantage of the wirtinger based inequality based on this observation two delay product type terms are introduced into the lkf to establish new stability criteria finally a numerical example is given to verify the equivalence statements and to show the benefit of the proposed criteria

levy type characterization of stable laws for free random variables we give a description of stable probability measures relative to free additive convolution the definition of domain of attraction is given and a proof is provided of the noncommutative analogue of levy theorem