an eigenvalue problem for hemivariational inequalities with combined nonlinearities on an infinite strip in this paper a class ofeigenvalue problems f hemivariational inequalities is studied which is defined on domains ofthe type r is a bounded open subset of r m m 1 and it involves

on the matsumoto yor property in free probability abstract we study the matsumoto yor property in free probability we prove that the limiting empirical eigenvalue distribution of the gig matrices and the marchenko pastur distribution have the free matsumoto yor property finally we characterize these distributions by regression properties in free probability

semilinear elliptic equations on unbounded domains on presente une nouvelle approche variationnelle a l existence des solutions pour le probleme aux valeurs limites u u r x f u x x x h 0 1 sur des domaines non bornes

free martingale polynomials abstract in this paper we investigate the properties of free sheffer systems which are certain families of martingale polynomials with respect to the free levy processes first we classify such families that consist of orthogonal polynomials these are the free analogs of the meixner systems next we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials finally we indicate how rota s finite operator calculus can be modified for the free context

laplace transforms which are negative powers of quadratic polynomials we find the distributions in r n for the independent random variables x and y such that e x x y a x y and e g x x y bq x y where q runs through the set of all quadratic forms on r n orthogonal to a given quadratic form v the essential part of this class is provided by distributions with laplace transforms 1 2 c s v s p that we describe completely obtaining a generalization of a gindikin theorem this leads to the classification of natural exponential families with the variance function of type 1 pm m m m v where m v is the symmetric matrix associated to the quadratic form v and m m is a real function these natural exponential families extend the classical wishart distributions on lorentz cones already considered by jensen and later on by faraut and koranyi

the characterizations of a semicircle law by the certain freeness in a c probability space in usual probability theory various characterizations of the gaussian law have been obtained for instance independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the gaussian law and the property of remaining independent under rotations characterizes the gaussian random variables in this paper we consider the free analogue of such a kind of characterizations replacing independence by freeness we show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables which covers the case for the sample mean and the sample variance characterizes the semicircle law moreover we give the alternative proof for nica s result that the property of remaining free under rotations characterizes a semicircular system our proof is more direct and straightforward one

on the limit distributions of random matrices with independent or free entries we state limit distribution results for random matrices with independent or free entries also addressing when we get freeness in the limit and semicircular and circular limits the results generalize some already known results about asymptotic freeness of large random matrices but our goal is to get a more optimal flavour on these results when having matrices with identically distributed entries we show that freeness in the limit is natural when we have free entries but unnatural when the entries are independent and restricted to the case of circular limits

a one parameter family of transforms linearizing convolution laws for probability distributions we study a family of transforms depending on a parameterq 0 1 which interpolate in an algebraic framework between a relative namely iz log iz of the logarithm of the fourier transform for probability distributions and its free analogue constructed by d voiculescu 16 17 the classical case corresponds toq 1 and the free one toq 0

an existence result for gradient type systems with a nondifferentiable term on unbounded strips in this paper we study the existence of nontrivial solutions for a class of gradient type systems on strip like domains where the nonlinear term is not necessarily continuously differentiable the proof of the main result is based on a nonsmooth version of the mountain pass theorem which involves the cerami compactness condition and on the principle of symmetric criticality for locally lipschitz functions

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

on the pair correlation of zeros of the riemann zeta function in 1972 montgomery 20 21 introduced a new method for studying the zeros of the riemann zeta function one of his main accomplishments was to determine partially the pair correlation of zeros and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros perhaps more importantly he conjectured on number theoretic grounds an asymptotic formula for the pair correlation of zeros and found that the form of this correlation exactly agreed with the gaussian unitary ensemble gue model for random hermitian matrices which had been studied earlier by physicists he was therefore able to formulate a general n correlation conjecture for zeros during the 1980s odlyzko 23 24 performed extensive numerical calculations of the correlations for zeros in ranges up to the 10th zero and found excellent agreement with the gue model more recently hejhal 17 was able to prove the same partial result for triple correlation as montgomery proved for pair correlation and rudnick and sarnak 25 have done the same for n correlation rudnick and sarnak extended their results to a large class of l functions and also showed that the riemann hypothesis rh is not needed for smoother forms of the asymptotic result very recently bogomolny and keating 1 2 have used a prime twin type conjecture to derive heuristically the n correlation conjecture beyond the range where the results of rudnick and sarnak apply the conclusion of all this work is to rmly establish but not prove the gue distribution for zeros of many zeta functions there is a dual relationship between zeros of the riemann zeta function and prime numbers following montgomery s work it was realized that information on pair correlation of zeros could be used to obtain information on primes this connection was developed by gallagher and mueller 10 and heath brown 15 later goldston and montgomery 14 found an equivalence under the riemann hypothesis between the pair correlation of zeros and the variance for the number of primes in short intervals this equivalence arises out of the explicit formula via a parseval relation together with a tauberian theorem from this work one sees that to obtain new information on primes from zeros will require some new insight on the zeros and while the connections to statistical physics mentioned above are a possible source of this insight so far no progress has been made on this fundamental problem the motivation for this paper is the observation that there are additional tools

a variational inequality on the half line abstract multiple solutions are obtained for a variational inequality defined on the half line 0 our approach is based on a key embedding result as well as on the non smooth critical point theory for szulkin type functionals

proof of a conjecture of m l eaton on the characteristic function of the wishart distribution let m geq 2 be a positive integer i m be the m times m identity matrix and sigma and a be symmetric m times m matrices where sigma is positive definite by proving that the function phi alpha a i m 2ia sigma alpha is a characteristic function only if alpha in 0 frac 1 2 1 frac 3 2 ldots m 2 2 cup lbrack m 1 2 infty we establish a conjecture of eaton a similar result is established for the rank 1 noncentral wishart distribution and is conjecture to also be valid for any greater rank

quadratic and inverse regressions for wishart distributions if u and v are independent random variables which are gamma distributed with the same scale parameter then there exist a and b in r such that e u u v a u v and e u 2 u v b u v 2 this in fact is characteristic of gamma distributions our paper extends this property to the wishart distributions in a suitable way by replacing the real number u 2 by a pair of quadratic functions of the symmetric matrix u this leads to a new characterization of the wishart distributions and to a shorter proof of the 1962 characterization given by olkin and rubin similarly if e u 1 exists there exists c in r such that e u 1 u v c u v 1 wesolowski has proved that this also is characteristic of gamma distributions we extend it to the wishart distributions finally things are explained in the modern framework of symmetric cones and simple euclidean jordan algebras

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains abstract we consider nonlinear elliptic eigenvalue problems on unbounded domains g r n using an extended ljusternik schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in l 2 g moreover we establish that the infimum of the spectrum of the linearized problem l is always a bifurcation point in addition there is an infinity of branches emanating at from the trivial line of solutions if belongs to the essential spectrum of l

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

the lukacs olkin rubin characterization of wishart distributions on symmetric cones we characterize the wishart distributions on a symmetric cone c if c 0 this has been done by lukacs in 1955 if c is the cone of positive definite symmetric matrices this has been done by olkin and rubin in 1962 we both shorten and extend the olkin rubin proof sometimes obscure by using three modern ideas i try to avoid artificial coordinates in differential geometry ii the variance function of a natural exponential family f characterizes f iii symmetric matrices are a particular example of a euclidean simple jordan algebra

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

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

linking and multiplicity results for the p laplacian on unbounded cylinders abstract we consider the p laplacian problem formula on unbounded cylinders rn m rn n m 2 where pu div u p 2 u is a constant in a certain range and a ln p l is nonnegative a 0 using the principle of symmetric criticality existence and multiplicity are proved under suitable conditions on a and f

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

cumulants in noncommutative probability theory ii we continue the investigation of noncommutative cumulants in this paper various characterizations of generalized gaussian random variables are proved

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

conditional moments of q meixner processes we show that stochastic processes with linear conditional expectations and quadratic conditional variances are markov and their transition probabilities are related to a three parameter family of orthogonal polynomials which generalize the meixner polynomials special cases of these processes are known to arise from the non commutative generalizations of the levy processes

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

minimization problems in l1 r3 abstract in this paper minimization problems in l1 ir3 are considered these problems arise in astrophysics for the determination of equilibrium configurations of axially symmetric rotating fluids rotating stars under nearly optimal assumptions a minimizer is proved to exist by a direct variational method which uses heavily the symmetry of the problem in order to get some compactness finally by looking directly at the euler equation we give some existence results of solutions of the euler equation even if the infimum is not finite author

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

a hemivariational inequality involving p laplacian on an infinite strip abstract this paper deals with an eigenvalue problem for hemivariational inequalities on domains of the type r is a bounded open subset of r n 1 n 2 and it involves concave convex nonlinearities under suitable conditions on the nonlinearities two nontrivial solutions are obtained which belong to a special closed convex cone of w 0 1 p r whenever the eigenvalues are of certain range our approach is variational based on the theories of non smooth analysis our results are a generalization of the case of laplacian from a kristaly et al to the case of p laplacian

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