raney distributions and random matrix theory recent works have shown that the family of probability distributions with moments given by the fuss catalan numbers permit a simple parameterized form for their density we extend this result to the raney distribution which by definition has its moments given by a generalization of the fuss catalan numbers such computations begin with an algebraic equation satisfied by the stieltjes transform which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the raney distribution for the fuss catalan distribution an equilibrium problem characterizing the density is identified the stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard gaussian matrices and s standard gaussian matrices is shown to satisfy a variant of the algebraic equation relating to the raney distribution supported on 0 infty we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization as an application the leading asymptotic form of the density as the endpoints of the support are approached is computed and is shown to have some universal features

numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

on the hausdorff continuity of free levy processes and free convolution semigroups abstract let denote a borel probability measure and let t t 1 denote the free additive convolution semigroup of nica and speicher we show that the support of these measures varies continuously in the hausdorff metric for t 1 we utilize complex analytic methods and in particular a characterization of the absolutely continuous portion of these supports due to huang

free ornstein uhlenbeck processes abstract free ornstein uhlenbeck processes are studied in finite von neumann algebras it is shown that a free self decomposable probability measure on r can be realized as the distribution of a stationary free ornstein uhlenbeck process driven by a free levy process a characterization of a probability measure on r to be the stationary distribution of a periodic free ornstein uhlenbeck process driven by a free levy process is given in terms of the levy measure of the measure finally the notion of a free fractional brownian motion is introduced it is proved that the free stochastic differential equation driven by a fractional free brownian motion has a unique solution we call the solution a fractional free ornstein uhlenbeck process

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

large deviations upper bounds for the laws of matrix valued processes and non communicative entropies using ito s calculus we study the large deviations properties of the law of the spectral measure of the hermitian brownian motion we extend this strategy to the symmetric unitary and wishart processes this dynamical approach is generalized to the study of the large deviations of the non commutative laws of several independent hermitian brownian motions as a consequence we can bound from above entropies defined in the spirit of the microstates entropy introduced by voiculescu

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

asymptotic properties of random matrices and pseudomatrices abstract we study the asymptotics of sums of matricially free random variables called random pseudomatrices and we compare it with that of random matrices with block identical variances for objects of both types we find the limit joint distributions of blocks and give their hilbert space realizations using operators called matricially free gaussian operators in particular if the variance matrices are symmetric the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks we also show that blocks of random pseudomatrices are asymptotically matricially free whereas the corresponding symmetric random blocks are asymptotically symmetrically matricially free where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization finally we show that row blocks of square block lower triangular and block diagonal pseudomatrices are asymptotically free monotone independent and boolean independent respectively

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

an analogue of hincin s characterization of infinite divisibility for operator valued free probability abstract let b be a finite separable von neumann algebra we prove that a b valued distribution that is the weak limit of an infinitesimal array is infinitely divisible the proof of this theorem utilizes the steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories we also develop weak topologies for this theory and prove the corresponding compactness and convergence results

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

on quantum statistical inference interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially in step with dramatic new developments in experimental techniques for studying small quantum systems furthermore developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory the present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context

modeling the large scale dynamics of saturated groundwater flow using spatial filtering theory 1 theoretical development we develop a model and a framework for understanding the explicitly resolved large scale dynamics of groundwater flow in fully saturated heterogeneous porous media the approach is based upon the large eddy simulation les methodology from geophysical fluid dynamics the idea behind the les approach is to explicitly resolve the larger scales of flow and provide a closure model for the effect of subgrid scales upon resolved scales the technical foundation of the method rests on a spatial filtering operation which allows one to precisely define the scale of the model we spatially filter darcy s law and the continuity equation to produce a model for the resolved scale dynamics we develop les approximations for the resulting flow equations and a closure model for unresolved subgrid scale terms we model the subgrid terms using resolved scale quantities only and thus do not require subgrid scale information the accuracy of these closure models depends upon the scale at which hydraulic conductivity is measured by field tests we characterize the approximations used to model subgrid scale effects asymptotically with a small parameter the dimensionless filter width defined as the ratio of the filter width to the dominant length scale of the explicitly resolved variables

strict limit types for monotone convolution we introduce the notion of strict domain of attraction in the context of monotone probability theory and prove that a law has a nonempty strict domain of attraction if and only if it is strictly stable we also show that the monotonic weak law of large numbers is equivalent to the classical and the free weak laws of large numbers finally a necessary condition for a measure to belong to the strict domain of attraction of a nondegenerate strictly stable law is given in terms of the absolute moments of

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

ideals of smooth functions and residue currents abstract let f f 1 f m be a holomorphic mapping in a neighborhood of the origin in c n we find sufficient condition in terms of residue currents for a smooth function to belong to the ideal in c or ck generated by f if f is a complete intersection the condition is necessary more generally we give a sufficient condition for an element of class c or ck in the koszul complex induced by f to be exact for the proofs we introduce explicit homotopy formulas for the koszul complex induced by f

addition of freely independent random variables abstract a direct proof is given of voiculescu s addition theorem for freely independent real valued random variables using resolvents of self adjoint operators in contrast to the original proof no assumption is made on the existence of moments above the second

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

brown measure support and the free multiplicative brownian motion abstract the free multiplicative brownian motion b t is the large n limit of brownian motion b t n on the general linear group gl n c we prove that the brown measure for b t which is an analog of the empirical eigenvalue distribution for matrices is supported on the closure of a certain domain t in the plane the domain t was introduced by biane in the context of the large n limit of the segal bargmann transform associated to gl n c we also consider a two parameter version b s t the large n limit of a related family of diffusion processes on gl n c introduced by the second author we show that the brown measure of b s t is supported on the closure of a certain planar domain s t generalizing t introduced by ho in the process we introduce a new family of spectral domains related to any operator in a tracial von neumann algebra the l p n spectrum for n n and p 1 a subset of the ordinary spectrum defined relative to potentially unbounded inverses we show that in general the support of the brown measure of an operator is contained in its l 2 2 spectrum

the levy ito decomposition in free probability in this paper we prove the free analog of the levy ito decomposition for levy processes a significant part of the proof consists of introducing free poisson random measures proving their existence and developing a theory of integration with respect to such measures the existence of free poisson random measures also yields via the free levy ito decomposition an alternative proof of the general existence of free levy processes in law

spectral rigidity for addition of random matrices at the regular edge abstract we consider the sum of two large hermitian matrices a and b with a haar unitary conjugation bringing them into a general relative position we prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of a and b as the dimension of the matrix increases this implies optimal rigidity of the eigenvalues and optimal rate of convergence in voiculescu s theorem our previous works 4 5 established these results in the bulk spectrum the current paper completely settles the problem at the spectral edges provided they have the typical square root behavior the key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge our results also hold if the haar unitary matrix is replaced by the haar orthogonal matrix

operator valued chordal loewner chains and non commutative probability abstract we adapt the theory of chordal loewner chains to the operator valued matricial upper half plane over a c algebra a we define an a valued chordal loewner chain as a subordination chain of analytic self maps of the a valued upper half plane such that each f t is the reciprocal cauchy transform of an a valued law t such that the mean and variance of t are continuous functions of t we relate a valued loewner chains to processes with a valued free or monotone independent increments just as was done in the scalar case by bauer 8 and schleisinger 48 we show that the loewner equation t f t z d f t z v t z when interpreted in a certain distributional sense defines a bijection between lipschitz mean zero loewner chains f t and vector fields v t z of the form v t z g t z where t is a generalized a valued law based on the loewner equation we derive a combinatorial expression for the moments of t in terms of t we also construct non commutative random variables on an operator valued monotone fock space which realize the laws t finally we prove a version of the monotone central limit theorem which describes the behavior of f t as t when t has uniformly bounded support

effects of first order approximations on head and specific discharge covariances in high contrast log conductivity the hydraulic head and the specific dischargefluctuations depend nonlinearly on the hydraulic conductivity however the methods most commonly used in the stochastic analysis of groundwaterflow are based upon the linearization of these relations in this paper we apply a numerical spectral approach to investigate the range of validity of the small perturbation approximation for head and specific discharge moments in two dimensionalfinite domains wefind that the small perturbation approximation tends to underestimate the variance of large scale head and specific dischargefluctuations and error increases with increasing log conductivity variance and increasing domain size the headfluctuations do not appear ergodic even when the small perturbation approximation predicts they will be ergodic the specific dischargefluctuations on the other hand do appear ergodic the small perturbation approximation performs well in estimating specific discharge variance in the longitudinal direction but significantly underestimates transverse specific discharge variance

the two parameter free unitary segal bargmann transform and its biane gross malliavin identification abstract motivated by a conditional expectation interpretation of the segal bargmann transform we derive the integral kernel for the large n limit of the two parameter segal bargmann hall transform over the unitary group u n and explore its limiting behavior we also extend the notion of circular systems to more general elliptic systems in order to give an alternate construction of our new two parameter free unitary segal bargmann hall transform via a biane gross malliavin type theorem

generators of some non commutative stochastic processes a fundamental result of biane math z 227 143 174 1998 states that a process with freely independent increments has the markov property but that there are two kinds of free levy processes the first kind has stationary increments while the second kind has stationary transition operators we show that a process of the first kind with mean zero and finite variance has the same transition operators as the free brownian motion with appropriate initial conditions while a process of the second kind has the same transition operators as a monotone levy process we compute an explicit formula for the generators of these families of transition operators in terms of singular integral operators and prove that this formula holds on a fairly large domain we also compute the generators for the q brownian motion and for the two state free brownian motions

matricial r transform abstract we study the addition problem for strongly matricially free random variables which generalize free random variables using operators of toeplitz type we derive a linearization formula for the matricial r transform related to the associated convolution it is a linear combination of voiculescu s r transforms in free probability with coefficients given by internal units of the considered array of subalgebras this allows us to view this formula as the matricial linearization property of the r transform since strong matricial freeness unifies the main types of noncommutative independence the matricial r transform plays the role of a unified noncommutative analog of the logarithm of the fourier transform for free boolean monotone orthogonal s free and c free independence

a khintchine decomposition for free probability let be a probability measure on the real line in this paper we prove that there exists a decomposition 0 1 n such that 0 is infinitely divisible and i is indecomposable for i 1 additionally we prove that the family of all divisors of a measure is compact up to translation analogous results are also proven in the case of multiplicative convolution

eigenvectors and eigenvalues in a random subspace of a tensor product given two positive integers n and k and a parameter t 0 1 we choose at random a vector subspace v n k n of dimension n tnk we show that the set of k tuples of singular values of all unit vectors in v n fills asymptotically as n tends to infinity a deterministic convex set k k t that we describe using a new norm in k

a distributed parameter approach for evaluating the accuracy of groundwater model predictions 2 application to groundwater flow the distributed parameter theory presented by mclaughlin and wood this issue is used to evaluate the accuracy of a groundwater flow model the special case investigated assumes that the model s prediction errors are due primarily to data limitations in this case approximate prediction error moments mean and covariance are obtained by solving two sets of coupled partial differential equations which have the same basic structure as the original flow equation solutions can be obtained with spectral green s function or numerical techniques depending on the assumptions made two examples are solved using a finite element approach the first two dimensional example confirms steady state infinite domain results obtained with spectral methods the second three dimensional example investigates the influence of spatial variability sampling strategy and suboptimal estimation on model prediction accuracy

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

limit theorems for free multiplicative convolutions we determine the distributional behavior for products of free random variables in a general infinitesimal triangular array the main theorems in this paper extend a result for measures supported on the positive half line and provide a new limit theorem for measures on the unit circle with nonzero first moment

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

asymmetric simple exclusion process with open boundaries and quadratic harnesses we show that the joint probability generating function of the stationary measure of a finite state asymmetric exclusion process with open boundaries can be expressed in terms of joint moments of markov processes called quadratic harnesses we use our representation to prove the large deviations principle for the total number of particles in the system we use the generator of the markov process to show how explicit formulas for the average occupancy of a site arise for special choices of parameters we also give similar representations for limits of stationary measures as the number of sites tends to infinity

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

limit distributions of random matrices abstract we study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation in particular we are concerned with the ensemble of symmetric blocks of independent hermitian random matrices which are asymptotically free asymptotically free from diagonal deterministic matrices and whose norms are uniformly bounded almost surely this class contains symmetric blocks of unitarily invariant hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability we show that the associated matricially free gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices including non hermitian gaussian matrices and matrices of wishart type

on the asymptotic distribution of block modified random matrices we study random matrices acting on tensor product spaces which have been transformed by a linear block operation using operator valued free probability theory under some mild assumptions on the linear map acting on the blocks we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution moreover using recent results on operator valued subordination we present an algorithm that computes numerically but in full generality the limiting eigenvalue distribution of the modified matrices our analytical results cover many cases of interest in quantum information theory we unify some known results and we obtain new distributions and various generalizations

linearization coefficients for orthogonal polynomials using stochastic processes given a basis for a polynomial ring the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients these coefficients have combinatorial significance for many classical families of orthogonal polynomials starting with a stochastic process and using the stochastic measures machinery introduced by rota and wallstrom we calculate and give an interpretation of linearization coefficients for a number of polynomial families the processes involved may have independent freely independent or q independent increments the use of noncommutative stochastic processes extends the range of applications significantly allowing us to treat hermite charlier chebyshev free charlier and rogers and continuous big q hermite polynomials we also show that the q poisson process is a markov process

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

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

multi variable subordination distributions for free additive convolution abstract let k be a positive integer and let d c k denote the space of joint distributions for k tuples of selfadjoint elements in c probability space the paper studies the concept of subordination distribution of with respect to for d c k where is the operation of free additive convolution on d c k the main tools used in this study are combinatorial properties of r transforms for joint distributions and a related operator model with operators acting on the full fock space multi variable subordination turns out to have nice relations to a process of evolution towards infinite divisibility on d c k that was recently found by belinschi and nica arxiv 0711 3787 most notably one gets better insight into a connection which this process was known to have with free brownian motion

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

what does an instrument measure empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports with the aid of linear filter theory we analyze 13 824 permeability measurements to empirically address the question what does an instrument measure by measure we mean the sample support or sample volume associated with an instrument as well as how the instrument spatially weights the heterogeneities comprising that sample support although the theoretical aspects of linear filter analysis are well documented physical data for testing the filtering behavior of an instrument particularly in the context of porous media flow are rare to nonexistent our exploration makes use of permeability data measured with a minipermeameter on a block of berea sandstone data were collected according to a uniform grid that was resampled with tip seals of increasing size i e increasing sample support spatial weighting filter functions characterizing the minipermeameter measurements were then calculated directly from the permeability data sets in this paper we limit our presentation to one of the six rock faces consisting of 2304 measurements as the general results for each rock face are similar we found that the empirical weighting functions are consistent with the basic physics of the minipermeameter measurement they decay as a nonlinear function of radial distance from the center of the tip seal consistent with the divergent flow geometry imposed by the minipermeameter the magnitude of the weighting function decreases while its breadth increases with increasing tip seal size reflecting the increasing sample support we further demonstrate both empirically and theoretically that nonadditive properties like permeability are amenable to linear filter analysis under certain limiting conditions i e small variances specifically the weighting function is independent of the power average employed in its calculation e g arithmetic versus harmonic average finally we examine the implications of these results for other instruments commonly employed in hydraulic testing e g slug and pump tests

scaling and universality in statistical physics abstract the twin concepts of scaling and universality have played an important role in the description of statistical systems hydrodynamics contains many applications of scaling including descriptions of the behavior of boundary layers prandtl blasius and of the fluctuating velocity in turbulent flow kolmogorov heisenberg onsager phenomenological theories of behavior near critical points of phase transitions made extensive use of both scaling to define the size of various fluctuations and universality to say that changes in the model would not change the answers these two ideas were combined via the statement that elimination of degrees of freedom and a concomitant scale transformation left the answers quite unchanged in wilson s hands this mode of thinking led to the renormalization group approach to critical phenomena subsequently feigenbaum showed how scaling universality and renormalization group ideas could be applied to dynamical systems specifically this approach enabled us to see how chaos first arises in those systems in which but a few degrees of freedom are excited in parallel libchaber developed experiments aimed at understanding the onset of chaos the results of which were subsequently used to show that feigenbaum s universal behavior was in fact realized in honest to goodness hydrodynamical systems more recently gemunu gunaratne mogens jensen and itamar procaccia have indicated that they believe that a different and weaker universality might hold for the fully chaotic behavior of low dimensional dynamical systems dynamically generated situations often seem to show kinds of scaling and universality quite different from that seen in critical phenomena a technical difference which seems to arise in these intrinsically dynamical processes is that instead of having a denumerable list of different critical quantities each with their critical index instead there is continuum of critical indices this so called multifractal behavior may nonetheless show some kinds of universality and indeed this might be the kind of scaling and universality shown by those hydrodynamical systems in which many degrees of freedom are excited

measurement scale network sampling scale and groundwater model parameters the scales at which model parameters are measured with local field tests distributed on a sampling network are examined two scales are defined to characterize the problem 1 the measurement scale associated with the resolution of a single field test and 2 the network scale associated with the separation between samples on a network using a spatial filtering approach it is shown that a network of measurements can only resolve a larger scale component of a parameter field the smaller scale component of the parameter field not seen by measurements here called the subgrid component can only vary on scales larger than the measurement scale and smaller than the network scale these unobserved subgrid scales give rise to the so called closure problem and consequent modeling errors when a significant proportion of the parameter variability is contained in the subgrid scales not only will the closure problem be significant but aliasing errors will also pollute the estimate of the large scale component of the parameter field these concepts are illustrated by performing simple analytical and numerical calculations

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

discussions around voiculescu s free entropies abstract in this article we set two analogous definitions of the free entropies and introduced by voiculescu invent math 118 1994 411 132 1998 189 we discuss their relations improving the preceding results obtained in cabanal duvillard and guionnet ann probab 2001 to appear where a bound on the microstates entropy was established

convergence rate for spectral distribution of addition of random matrices abstract let a and b be two n by n deterministic hermitian matrices and let u be an n by n haar distributed unitary matrix it is well known that the spectral distribution of the sum h a u b u converges weakly to the free additive convolution of the spectral distributions of a and b as n tends to infinity we establish the optimal convergence rate 1 n in the bulk of the spectrum

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

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

analytic subordination for bi free convolution in this paper we study some analytic properties of bi free additive convolution both scalar and operator valued we show that using properties of voiculescu s subordination functions associated to free additive convolution of operator valued distributions simpler formulas for bi free convolutions can be derived we use these formulas in order to prove a result about atoms of bi free additive convolutions

the universal structure of the groundwater flow equations most groundwater flow models are not sufficiently detailed to allow an explicit representation of all dynamical scales instead most models are constructed at such a coarse scale of resolution that unresolved subgrid variability often exists it is therefore important to understand the interaction between unresolved dynamics and explicitly resolved dynamics the concept of universality is central to this interaction and is here related to the problem of rescaling models of groundwater flow we examine when it is possible to construct an accurate model for the explicitly resolved large scales without an explicit description of the subgrid scale dynamics we demonstrate how unresolved subgrid scale dynamics interact with resolved scale dynamics we show that a universally valid resolved scale model can be constructed if the resolved dynamics are sufficiently independent of the details of the subgrid scale dynamics in that event a resolved scale model is composed of a universal structure and accompanying model parameters the model parameters represent the effect of unresolved dynamics upon resolved dynamics we examine two possible model structures for groundwater flow we show theoretically and numerically that a local darcy law is a universally valid resolved scale model if the resolved and subgrid scales of the hydraulic conductivity are separated in scale by a so called spectral gap if however the hydraulic conductivity possesses many scales of variability then a more general nonlocal darcy law is a more appropriate model structure when the nonlocal darcy s law is more appropriate numerical experiments suggest that errors using a local darcy s law with effective parameters are most significant at the smallest resolving scale of the model and are minimal at scales between 8 and 16 times the resolving scale

a concentration inequality and a local law for the sum of two random matrices let h n a n u n b n u n ast where an and bn are two n by n hermitian matrices and un is a haar distributed random unitary matrix and let mu h n mu a n mu b n be empirical measures of eigenvalues of matrices hn an and bn respectively then it is known see pastur and vasilchuk in commun math phys 214 249 286 2000 that for large n the measure mu h n is close to the free convolution of measures mu a n and mu b n where the free convolution is a non linear operation on probability measures the large deviations of the cumulative distribution function of mu h n from its expectation have been studied by chatterjee j funct anal 245 379 389 2007 in this paper we improve chatterjee s concentration inequality and show that it holds with the rate which is quadratic in n in addition we prove a local law for eigenvalues of h n n by showing that the normalized number of eigenvalues in an interval approaches the density of the free convolution of a and b provided that the interval has width log n 1 2

wigner chaos and the fourth moment we prove that a normalized sequence of multiple wigner integrals in a fixed order of free wigner chaos converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2 the fourth moment of the semicircular law this extends to the free probabilistic setting some recent results by nualart and peccati on characterizations of central limit theorems in a fixed order of gaussian wiener chaos our proof is combinatorial analyzing the relevant noncrossing partitions that control the moments of the integrals we can also use these techniques to distinguish the first order of chaos from all others in terms of distributions we then use tools from the free malliavin calculus to give quantitative bounds on a distance between different orders of chaos when applied to highly symmetric kernels our results yield a new transfer principle connecting central limit theorems in free wigner chaos to those in gaussian wiener chaos we use this to prove a new free version of an important classical theorem the breuer major theorem

almost one bit violation for the additivity of the minimum output entropy in a previous paper we proved that in the appropriate asymptotic regime the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic compact set kk t we also showed that the set kk t is obtained up to an intersection as the unit ball of the dual of a free compression norm in this paper we identify the maximum of ell p norms on the set kk t and prove that the maximum is attained on a vector of shape a b b where a b in particular we compute the precise limit value of the minimum output entropy of a single random quantum channel as a corollary we show that for any varepsilon 0 it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183 and that for appropriate choice of parameters the violation can be as large as log 2 varepsilon conversely our result implies that with probability one in the limit one does not obtain a violation of additivity using conjugate random quantum channels and the bell state in dimension 182 and less

the operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers the operational meaning of the representative elementary volume rev concept on which current foundational theories of water movement through porous media are based is analyzed critically it is concluded that the rev concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable in its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments the effects of resolution and precision differences among experimental measuring devices now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers can be incorporated naturally within the relativist concept but are excluded a priori in the rev concept the relativist point of view unlike the rev concept has a clear operational meaning and in principle extends theoretical validity to all local measurements made on porous media it is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated anisotropic deformable soil or aquifer

free convolution operators and free hall transform abstract we define an extension of the polynomial calculus on a w probability space by introducing an algebra c x i i i which contains polynomials this extension allows us to define transition operators for additive and multiplicative free convolution it also permits us to characterize the free segal bargmann transform and the free hall transform introduced by biane in a manner which is closer to classical definitions finally we use this extension of polynomial calculus to prove two asymptotic results on random matrices the convergence for each fixed time as n tends to of the distribution of the brownian motion on the linear group gl n c to the distribution of a free multiplicative circular brownian motion and the convergence of the classical hall transform on u n to the free hall transform

levy processes in free probability this is the continuation of a previous article that studied the relationship between the classes of infinitely divisible probability measures in classical and free probability respectively via the bercovici pata bijection drawing on the results of the preceding article the present paper outlines recent developments in the theory of levy processes in free probability

fluctuations at the edges of the spectrum of the full rank deformed gue we consider a full rank deformation of the gue frac w n sqrt n a n where a n is a full rank hermitian matrix of size n and w n is a gue the empirical eigenvalue distribution mu a n of a n converges to a probability distribution nu we identify all the possible limiting eigenvalue statistics at the edges of the spectrum including outliers edges and merging points of connected components of the limiting spectrum the results are stated in terms of a deterministic equivalent of the empirical eigenvalue distribution of w n a n namely the free convolution of the semi circle distribution and the empirical eigenvalues distribution of a n

limit theorems in free probability theory i based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line and on the unit circle mathbb t we prove analogs of limit theorems for nonidentically distributed random variables in classical probability theory

matricially free random variables we show that the framework developed by voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication the associated notion of independence called matricial freeness can be viewed as a concept which not only leads to a natural generalization of freeness but also underlies other fundamental types of noncommutative independence such as monotone independence and boolean independence at the same time the sums of matricially free random variables called random pseudomatrices are closely related to random matrices the main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial semicircle laws

a new example of independence and white noise we examine the notion of free independence according to voiculescu this form of independence is used for defining free white noise or process with stationary and freely independent increments we prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions we realize the free analogues of the wiener process and of the poisson process as processes on the full fock space ofl2

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