the blow up rate for a system of heat equations with nonlinear boundary conditions this paper deals with the blow up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions the upper and lower bounds of blow up rate are obtained

gaussian limit for determinantal random point fields author s soshnikov alexander abstract we prove that under fairly general conditions properly rescaled determinantal random point field converges to a generalized gaussian random process

central limit theorems from the roots of probability generating functions abstract for each n let x n 0 n be a random variable with mean n standard deviation n and let p n z k 0 n p x n k z k be its probability generating function we show that if none of the complex zeros of the polynomials p n z is contained in a neighborhood of 1 c and n n e for some e 0 then x n x n n n 1 is asymptotically normal as n that is it tends in distribution to a random variable z n 0 1 on the other hand we show that there exist sequences of random variables x n with n c log n for which p n z has no roots near 1 and x n is not asymptotically normal these results disprove a conjecture of pemantle and improve upon various results in the literature we go on to prove several other results connecting the location of the zeros of p n z and the distribution of the random variable x n

gaussian fluctuations of young diagrams under the plancherel measure we obtain the central limit theorem for fluctuations of young diagrams around their limit shape in the bulk of the spectrum of partitions lwn2n under the plancherel measure thus settling a long standing problem posed by logan shepp namely under normalization growing like ffiffiffiffiffiffiffiffiffiffiffi n p the corresponding random process in the bulk is shown to converge in the sense of finite dimensional distributions to a gaussian process with independent values while local correlations in the vicinity of each point measured on various power scales possess certain self similarity the proofs are based on the poissonization techniques and use costin lebowitz soshnikov s central limit theorem for determinantal random point processes our results admit a striking reformulation after the rotation of young diagrams by 458 whereby the normalization no longer depends on the location in the spectrum in addition we explain heuristically the link with an earlier result by kerov on the convergence to a generalized gaussian process

steady state groundwater flow simulation with imprecise parameters a methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models in this case fuzzy numbers are used to represent parameter imprecision as such they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters the imprecise input parameters may come from indirect measurements subjective interpretation and expert judgment of available information in the methodology a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model this fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations two numerical solution methods are used to solve the fuzzy groundwater flow model the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics the iterative method is simple but may overestimate the uncertainty of hydraulic heads the groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads a numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem it is also applied to a simplified two dimensional heterogeneous flow case to demonstrate the methodology

blow up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition this paper is devoted to a nonlinear parabolic equation subject to nonlocal boundary condition and with nonlocal reaction source some criteria on this problem which determine whether the solutions blow up in a finite time or exist globally are given moreover the blow up rate estimates are also obtained

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

spectral statistics of large dimensional spearman s rank correlation matrix and its application between two independent random variables in this paper we will establish a clt for the linear spectral statistics of this non parametric random matrix model in the scenario of high dimension supposing that p p n and p n c2 0 1 as n 1 we propose a novel evaluation scheme to estimate the core quantity in anderson and zeitouni s cumulant method in 1 to bypass the so called joint cumulant summability in addition we raise a two step comparison approach to obtain the explicit formulae for the mean and covariance functions in the clt relying on this clt we then construct a distribution free statistic to test complete independence for components of random vectors owing to the non parametric property we can use this test on generally distributed random variables including the heavy tailed ones

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

the circular unitary ensemble and the riemann zeta function the microscopic landscape and a new approach to ratios we show in this paper that after proper scalings the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros which are on the real line form a determinantal point process with sine kernel our scaling is performed at the so called microscopic level that is we consider the characteristic polynomial at points whose distance to 1 has order 1 n we prove that the rescaled characteristic polynomial does not even have a moment of order one hence making the classical techniques of random matrix theory difficult to apply in order to deal with this issue we couple all the dimensions n on a single probability space in such a way that almost sure convergence occurs when n goes to infinity the strong convergence results in this setup provide us with a new approach to ratios we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form exp 2 i pi alpha n and related objects such as the logarithmic derivative we also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale on the number theory side inspired by the work by keating and snaith we conjecture some new limit theorems for the value distribution of the riemann zeta function on the critical line at the level of stochastic processes

central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients we study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles we study those biorthogonal ensembles for which the underlying biorthogonal family s

more on stochastic models in a recent review article g sposito et al 1986 examined the various stochastic theories which are concerned with transport of solutes in porous media in this short note we expand on their discussion to include several topics which had been omitted we begin by looking at two definitions of probability theory and their relation to the concept of an ensemble an rev ensemble of soils is defined and examined the concept of ergodicity is reviewed and it is pointed out that most stochastic models are theoretically unverifiable the relationship between scale of measurement and stochasticity is briefly reviewed and an equation that combines the two concepts is presented

mechanics of heterogeneous porous media with several spatial scales by extending the theory of homogenization we consider a heterogeneous porous medium whose material structure is characterized by multiple periodicity over several disparate length scales because for geological and engineering applications the driving force e g the pressure gradient and the global motion of primary interest is usually at a scale much larger than the largest of structural periodicity we derive the phenomenological equations of motion by using the perturbation technique of multiple scales all the effective coefficients are defined by boundary value problems on unit cells in smaller scales and no constitutive assumptions are added aside from the basic equations governing the mechanics of the pore fluid and the solid matrix in part i the porous matrix is assumed to be rigid the case of three scales is treated first symmetry and positiveness of the effective darcy permeability tensor are proven extensions to four and more scales are then discussed in part ii we allow the solid matrix to be deformable and deduce the equations of consolidation of a two phase medium the coupling of fluid flow and the quasi static elastic deformation of the matrix is considered when there are three disparate length scales effective coefficients of various kinds are deduced in terms of cell problems on the scale of the pores several general symmetry relations as well as specific properties of a medium composed of layers of porous matrix are discussed

solving the steady state groundwater flow equation for finite linear aquifers using a generalized fourier series approach in two dimensional domains a method to solve steady linear groundwater flow problems using generalized fourier series is developed and particularized for multiple fourier series in two dimensional domains it leads to a linear vector equation whose solution provides a finite number of generalized fourier coefficients approximating the hydraulic head field its implementation is shown and two relevant properties are found for the system matrix it is always symmetric and once computed if additional fourier terms are needed for a better approximation of the hydraulic head field previously computed matrix elements remain invariant i e only new rows and columns are added to the system matrix the method is demonstrated in three simple cases with different geometries and transmissivity fields where solutions are compared with analytical and finite element method results thus the method is verified as an alternative to other flow solvers additionally it provides a direct way to obtain the spectral form of the flow equation solution given a spectral representation of transmissivity and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions copyright 2009 john wiley sons ltd

on random matrices from the compact classical groups if m is a matrix taken randomly with respect to normalized haar measure on u n o n or sp n then the real and imaginary parts of the random variables tr mk k 1 converge to independent normal random variables with mean zero and variance k 2 as the size n of the matrix goes to infinity for the unitary group this is a direct consequence of the strong szeg6 theorem for toeplitz determinants we will prove a conjecture of diaconis

large time behavior of macrodispersion in heterogeneous trending aquifers 1 this paper presents a stochastic analysis of the large time behavior of macrodispersion in a three dimensional heterogeneous aquifer with a linear trend in the mean log hydraulic conductivity to solve the problem analytically focus is placed on the particular case where the linear trend is aligned in the direction of mean hydraulic head gradient a spectral approach based on fourier stieltjes representations for the perturbed quantities is used to develop closed form expressions that describe variability of flow velocity the second order mean flow and asymptotic macrodispersion the impact of the mean log hydraulic conductivity gradient on these results is examined it is found that the asymptotic longitudinal and transverse macrodispersion coefficients decrease with the increasing trend gradient of mean log hydraulic conductivity in the case of finite peclet numbers this feature is a consequence of the reduction in variability of flow velocity with the trend gradient

deviations from the circular law consider ginibre s ensemble of n n non hermitian random matrices in which all entries are independent complex gaussians of mean zero and variance open image in new window as n the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane in this note we describe fluctuations about this circular law first we obtain finite n formulas for the covariance of certain linear statistics of the eigenvalues asymptotics of these objects coupled with a theorem of costin and lebowitz then result in central limit theorems for a variety of these statistics

the nevai condition and a local law of large numbers for orthogonal polynomial ensembles we consider asymptotics of orthogonal polynomial ensembles in the macroscopic and mesoscopic scales we prove both global and local laws of large numbers analogous to the recently proven local semicircle law for wigner matrices under fairly weak conditions on the underlying measure our main tools are a general concentration inequality for determinantal point processes with a kernel that is a selfadjoint projection and a strengthening of the nevai condition from the theory of orthogonal polynomials

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

scale and directional dependence of macrodispersivities in colonnade networks macrodispersion in discrete colonnade network models is investigated using a series of monte carlo type numerical experiments the numerical simulations consider fluid flow and advective transport through a square flow region of a two dimensional network of hexagonal colonnades macrodispersivities are calculated as a function of the scale and orientation of the square flow region within the larger parent geometry particle tracking with flux weighted tracer injection and monitoring is used to generate experimental residence time distributions rtds the method of moments is used to characterize the longitudinal tracer spreading the simulated rtds are utilized to examine the macrodispersive behavior in colonnade networks column diameter 1 m with lognormally distributed fracture apertures b in millimeters the network models are assumed to consist of open fractures with ln b 1 945 and ln b 0 896 this translates to an equivalent continuum log conductivity variance ln k2 of 3 67 based on an ensemble average of 100 realizations the slope for spatial variance versus scale of observation ranges from 3 15 to 3 36 m for varying orientations of the hydraulic gradient for ensemble averaged data the varying orientations appear to have little effect on the macrodispersive behavior for single realization experiments the computed macrodispersivities are directionally dependent at a length scale as large as 30 times the column diameter and probably much beyond the computed asymptotic and preasymptotic macrodispersivities are compared with available stochastic solutions for two dimensional isotropic heterogeneity in the horizontal plane the ensemble based numerical data are in excellent agreement with the solutions of dagan 1984 1988 and gelhar and axness 1983 however for individual realizations nonergodic behavior is clearly apparent in the near source evolving region of transport and the numerical data are quite variable between realizations the study provides important insight on applicability of stochastic continuum theories to discrete colonnade network models having ln k2 much greater than 1

determinants of airy operators and applications to random matrices the purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of wiener hopf operators the determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

blowup of solutions to a porous medium equation with nonlocal boundary condition in this paper we investigate the blowup properties of the positive solutions to a porous medium equation with nonlocal boundary we obtain the blowup condition and its blowup rate estimate

asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions abstract this paper is concerned with some dynamical property of a reaction diffusion equation with nonlocal boundary condition under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function the asymptotic behavior of the time dependent solution is characterized in relation to a finite or an infinite set of constant steady state solutions this characterization is determined solely by the initial function and it leads to the stability and instability of the various steady state solutions in the case of finite constant steady state solutions the time dependent solution blows up in finite time when the initial function in greater than the largest constant solution also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

stochastic analysis of nonstationary subsurface solute transport 1 unconditional moments this paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media partial differential equations for three unconditional ensemble moments the concentration mean concentration covariance and velocity concentration cross covariance are derived by applying perturbation techniques to the governing transport equation for a conservative solute concentration uncertainty is assumed to be the result of unmodeled small scale fluctuations in a steady state velocity field the moment expressions which describe how each moment evolves over time and space resemble the classic deterministic advection dispersion equation and can be solved using similar methods a solution procedure based on a galerkin finite element algorithm is illustrated with a hypothetical two dimensional example for this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation the perturbation solution is shown to reproduce the statistics obtained from a monte carlo simulation quite well for a natural log conductivity standard deviation of 0 5 and moderately well for a natural log conductivity standard deviation of 1 0 the computational effort required for a perturbation solution is significantly less than that required for a monte carlo solution of acceptable accuracy sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases

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

an l blow up estimate for a nonlinear heat equation on decrit une borne superieure au taux d explosion des solutions de l equation de la chaleur non lineaire u t u u p ou u 0 p 1

a fast fourier transform stochastic analysis of the contaminant transport problem a three dimensional stochastic analysis of the contaminant transport problem is developed in the spirit of naff 1990 the new derivation is more general and simpler than previous analysis the fast fourier transformation is used extensively to obtain numerical estimates of the mean concentration and various spatial moments data from both the borden and cape cod experiments are used to test the methodology results are comparable to results obtained by other methods and to the experiments themselves

blow up rate and profile for a degenerate parabolic system coupled via nonlocal sources this paper deals with a degenerate parabolic system with nonlocal nonlinear sources subject to homogeneous dirichlet condition we establish the critical exponent as well as the blow up rate and profile for the system it is interesting to observe that the blow up exponent is determined by the interaction between the nonlinear diffusion and the nonlinear reaction while the blow up rate is independent of the nonlinear diffusion the effects from the nonlocality of the sources are substatial

roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition in this paper we investigates the blow up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition we obtain the blow up condition and its blow up rate estimate

numerical solutions of reaction diffusion equations with nonlocal boundary conditions abstract the purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction diffusion equations with nonlocal boundary conditions using the finite difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time dependent and the steady state finite difference systems each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence comparison theorem for the corresponding finite difference system the existence comparison theorems are used to investigate the asymptotic behavior of the discrete time dependent solution in relation to the discrete maximal and minimal solutions of the steady state problem numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

rigidity of the three dimensional hierarchical coulomb gas a random set of points in euclidean space is called rigid or hyperuniform if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i i d random points this phenomenon has received considerable attention in recent years due to its appearance in random matrix theory the theory of coulomb gases and zeros of random analytic functions however most of the published results are in dimensions one and two this paper gives the first proof of hyperuniformity in a coulomb type system in dimension three known as the hierarchical coulomb gas this is a simplified version of the actual 3d coulomb gas the interaction potential in this model inspired by dyson s hierarchical model of the ising ferromagnet has a hierarchical structure and is locally an approximation of the coulomb potential hyperuniformity is proved at both macroscopic and microscopic scales with upper and lower bounds for the order of fluctuations that match up to logarithmic factors the fluctuations have cube root behavior in agreement with a well known prediction for the 3d coulomb gas for completeness analogous results are also proved for the 2d hierarchical coulomb gas and the 1d hierarchical log gas

stochastic analysis of spatial variability in two dimensional steady groundwater flow assuming stationary and nonstationary heads two dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity t is analyzed stochastically using spectral analysis and the theory of intrinsic random functions conditions that ensure a stationary statistically homogeneous head process are derived and using two convenient forms for the covariance function of the ln t process the head covariance function is studied in addition the head variogram is obtained for a particular nonstationary case and the asymptotic head variogram is derived under very general conditions results are compared to those obtained by gelhar 1976 for one and two dimensional phreatic flow and bakr et al 1978 for one and three dimensional confined flow multidimensional flow analysis results in a significantly reduced head variance the head correlation remains high over much greater distances than the ln t correlation the variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2 times the correlation scale of the log transmissivity the variogram for nonstationary heads continues to grow logarithmically as lag distance increases independent of the form of the input covariance in the nonstationary case the conditions for stationarity are contrasted with the corresponding results obtained for the one and three dimensional cases of gutjahr and gelhar 1981 the head variance calculated from the stationary theory is found to agree with that of previous monte carlo simulations

strong szegő asymptotics and zeros of the zeta function assuming the riemann hypothesis we prove the weak convergence of linear statistics of the zeros of the riemann function to a gaussian field with covariance structure corresponding to the h1 2 norm of the test functions for this purpose we obtain an approximate form of the explicit formula relying on selberg s smoothed expression for and the helffer sjostrand functional calculus our main result is an analogue of the strong szeg theorem known for toeplitz operators and random matrix theory 2014 wiley periodicals inc

the asymptotic distribution of a single eigenvalue gap of a wigner matrix we show that the distribution of a suitable rescaling of a single eigenvalue gap lambda i 1 m n lambda i m n of a random wigner matrix ensemble in the bulk is asymptotically given by the gaudin mehta distribution if the wigner ensemble obeys a finite moment condition and matches moments with the gue ensemble to fourth order this is new even in the gue case as prior results establishing the gaudin mehta law required either an averaging in the eigenvalue index parameter i or fixing the energy level u instead of the eigenvalue index the extension from the gue case to the wigner case is a routine application of the four moment theorem the main difficulty is to establish the approximate independence of the eigenvalue counting function n infty x tilde m n where tilde m n is a suitably rescaled version of m n with the event that there is no spectrum in an interval x x s in the case of a gue matrix this will be done through some general considerations regarding determinantal processes given by a projection kernel

limiting laws of linear eigenvalue statistics for hermitian matrix models we study the variance and the laplace transform of the probability law of linear eigenvalue statistics of unitary invariant matrix models of n n hermitian matrices as n assuming that the test function of statistics is smooth enough and using the asymptotic formulas by deift et al commun pure appl math 52 1325 1425 1999 for orthogonal polynomials with varying weights we show first that if the support of the density of states of the model consists of q 2 intervals then in the global regime the variance of statistics is a quasiperiodic function of n as n generically in the potential determining the model we show next that the exponent of the laplace transform of the probability law is not in general 1 2 variance as it should be if the central limit theorem would be valid and we find the asymptotic form of the laplace transform of the probability law in certain cases

the spatial and temporal moments of tracer concentration in disordered porous media this study develops a methodology for deriving the spatial and temporal moments of tracer concentration in disordered geological formations and suggests a geostatistical formalism for interpolation and prediction of concentration conditional to concentration measurements the required spatial and temporal moments of the concentration are obtained by a biparticle tracking scheme which yields the bivariate probability distribution function of particle displacements in terms of the input log conductivity random field and the prevailing flow conditions the structure of the moments of the concentration is discussed and we show that 1 the concentration correlation function is anisotropic nonstationary and nonsymmetrical 2 there is a larger correlation between concentrations along the mean flow direction than normal to it 3 the correlation between the concentration at any two points with fixed coordinates on a moving lagrangian coordinate system increases with travel time 4 the correlation function necessarily assumes negative values beyond a certain threshold distance and 5 the concentration variance is finite even when neglecting pore scale dispersion application of the geostatistical prediction algorithm using the concentration spatial temporal moments shows a good potential for reducing prediction uncertainty

a quasi linear theory of non fickian and fickian subsurface dispersion 1 theoretical analysis with application to isotropic media a theory is presented which accounts for nonlinearity caused by the deviation of plume particles from their mean trajectory in three dimensional statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities existing linear theories predict that in the absence of local dispersion transverse dispersivities tend asymptotically to zero as fickian conditions are reached according to our new quasi linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero fickian asymptotes which are proportional to y4 when the log hydraulic conductivity variance y2 is much less than 1 all existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to y2 when y2 1 and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied however the quasi linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting y2 it predicts that when y 1 in isotropic media both the longitudinal and transverse dispersivities ascend monotonically toward fickian asymptotes proportional to y

flow in heterogeneous media displaying a linear trend in the log conductivity the purpose of this study is to derive a general solution for the problem of flow in nonstationary geological formations where the nonstationary manifests itself in the form of a spatial trend in the mean log conductivity a stochastic frame of reference is adopted to account for the spatial variability of the hydraulic conductivity for a complete stochastic description we derive the expected values and spatial covariances of the hydraulic head and the fluid flux vector as well as a relation between the expected values of the head and the fluxes these expressions are obtained using a perturbation expansion of the log conductivity about its nonstationary mean and they are correct to the first order in the variance of the log conductivity the expressions we derive are applicable for any space dimensionality and for arbitrary orientation of the trend in space a general methodology is outlined for derivation of these expressions for any type of spatial covariance and for demonstration explicit results are obtained for a gaussian isotropic covariance 16 refs 6 figs

limit theorems for biorthogonal ensembles and related combinatorial identities abstract we study the fluctuations of certain biorthogonal ensembles for which the underlying family p q satisfies a finite term recurrence relation of the form x p x j p x for polynomial linear statistics of such ensembles we reformulate the cumulant method introduced in 53 in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix j in the spirit of 12 we show that the asymptotic fluctuations are described by the right limits of the matrix j moreover whenever the right limit is a laurent matrix we show that the clt is equivalent to soshnikov s main combinatorial lemma we discuss several applications to unitary invariant hermitian random matrices in particular we provide a general central limit theorem clt and a law of large numbers in the one cut regime we also prove a clt for the square singular values of the product of independent complex rectangular ginibre matrices as well as for the laguerre and jacobi biorthogonal ensembles introduced in 7 and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients finally we discuss the connection with the strong szeg limit theorem where this combinatorial method originates

large deviations for the fermion point process associated with the exponential kernel for the fermion point process on the whole complex plane associated with the exponential kernel e z bar w we show the central limit theorem for the random variable d r the number of points inside the ball d r of radius r as r and we establish the large deviation principle for the random variables r 2 d r r 0

macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media analytical solutions using spectral stochastic theory macrodispersivity tensors are developed for one two and three dimensional isotropic and anisotropic fractal porous media since natural geologic aquifers are always bounded we introduce the concept of a maximum length scale lmax in the one dimensional case the asymptotic long time longitudinal macrodispersivity a is proportional to lmax2 and in the multidimensional cases a lmax in the multidimensional cases as the fractal dimension increases a decreases because a larger fractal dimension means the tortuosity of solute particles is larger leading to more mixing in the transverse direction which in turn reduces longitudinal spreading the transverse macrodispersivities are shown to be analogous to traditional spectral stochastic results in the multidimensional cases lmax is shown to be controlled by the aquifer thickness as a result ergodic conditions may be reached in relatively thin aquifers allowing the use of the asymptotic macrodispersivity expressions obtained here for single realizations

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

global existence and blow up for a porous medium system with nonlocal boundary conditions and nonlocal sources abstract this paper is devoted to a porous medium system subject to nonlocal boundary conditions and with nonlocal sources we investigate the global existence and blow up in finite time of a nonnegative solution by using a sub super solution method moreover under appropriate hypotheses for the conditions we obtain asymptotic properties of the solutions

pattern densities in non frozen planar dimer models in this paper we introduce a family of observables for the dimer model on a bi periodic bipartite planar graph called pattern density fields we study the scaling limit of these objects for non frozen gibbs measures of the dimer model and prove that they converge to a linear combination of a derivative of the gaussian massless free field and an independent white noise

using the nonstationary spectral method to analyze flow through heterogeneous trending media this paper describes a nonstationary spectral theory for analyzing flow in a heterogeneous porous medium with a systematic trend in log hydraulic conductivity this theory relies on a linearization of the groundwater flow equation but does not require the stationarity assumptions used in classical spectral theories the nonstationary theory is illustrated with a two dimensional analysis of a linear trend aligned with the mean flow direction in this case closed form solutions can be obtained for the effective hydraulic conductivity head covariance and log conductivity head cross covariance the effective hydraulic conductivity decreases from the geometrical mean as the mean slope of the log conductivity increases trending leads to a reduction of head variance and a structural change in the head covariance and the log conductivity head cross covariance such changes have important implications for measurement conditioning or cokriging methods which rely on the head covariance and log conductivity head covariance the nonstationary spectral analysis is also compared with classical spectral analysis this comparison indicates that the classical spectral method correctly predicts the normalized head covariance in a linear trending media the stationary spectral method fails to capture the qualitative influence of trends on the effective hydraulic conductivity and the log conductivity head cross covariance although the magnitude of the error is relatively small for realistic values of the mean log conductivity slope the stationary and nonstationary results are the same when there is no trend in log conductivity the trending conductivity example illustrates that the nonstationary spectral method has all the capabilities of the classical spectral approach while not requiring as many restrictive assumptions

incomplete determinantal processes from random matrix to poisson statistics we study linear statistics of a class of determinantal processes which interpolate between poisson and gue ginibre statistics in dimension 1 or 2 these processes are obtained by performing an independent bernoulli percolation on the particle configuration of a log gas confined in a general potential we show that depending on the expected number of deleted particles there is a universal transition for mesoscopic linear statistics namely at small scales the point process behave according to random matrix theory while at large scales it needs to be renormalized because the variance of any linear statistic diverges the crossover is explicitly characterized as the superposition of a h 1 or h 1 2 correlated gaussian noise depending on the dimension and an independent poisson process the proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process

gaussian fluctuations of eigenvalues in wigner random matrices we study the fluctuations of eigenvalues from a class of wigner random matrices that generalize the gaussian orthogonal ensemble we begin by considering an n n matrix from the gaussian orthogonal ensemble goe or gaussian symplectic ensemble gse and let xk denote eigenvalue number k under the condition that both k and n k tend to infinity as n w e show thatxk is normally distributed in the limit we also consider the joint limit distribution of eigenvalues xk1 x km from the goe or gse where k1 n km and ki 1 ki 1 i m 1 tend to infinity with n the result in each case is an m dimensional normal distribution using a recent universality result by tao and vu we extend our results to a class of wigner real symmetric matrices with non gaussian entries that have an exponentially de caying distribution and whose first four moments match the gaussian moments

transfer current and pattern fields in spanning trees when a simply connected domain d subset mathbb r d d ge 2 is approximated in a good way by embedded connected weighted graphs we prove that the transfer current matrix defined on the edges of the graph viewed as an electrical network converges up to a local weight factor to the differential of green s function on d this observation implies that properly rescaled correlations of the spanning tree model and correlations of minimal subconfigurations in the abelian sandpile model have a universal and conformally covariant limit we further show that on a periodic approximation of the domain all pattern fields of the spanning tree model as well as the minimal pattern e g zero height fields of the sandpile converge weakly in distribution to gaussian white noise

on the dagan model of solute transport in groundwater foundational aspects the convection dispersion equation cde developed by dagan 1984 to predict the ensemble average concentration of a conservative solute in groundwater has been used successfully to interpret recent tracer experiments in a sand aquifer at the borden site in canada this successful application encouraged further investigation of the dagan model in respect to its physical basis and predictive characteristics it is shown in the present paper which deals with the first topic that the dagan model cde can be derived through an extension of the cumulant expansion technique applied previously by chu and sposito 1980 to develop a mean cde for solute transport in homogeneous porous media this technique also leads directly to the mean cde developed by gelhar and axness 1983 as an asymptotic large time limit and to model dispersion coefficients derived by mather on and de marsily 1980 and guven and moiz 1986 the dagan cde then is considered in detail in respect to which conceptualization of the solute concentration resident or flux the model may utilize in predictive applications general mathematical expressions relating the two conceptualizations are derived for an arbitrary solute transport problem and then are applied to the dagan model for point and prism source inputs finally the dagan model is interpreted physically using data from the recent tracer experiments of roberts and mackay 1986 at the borden site it is shown that model predictions of resident and flux concentrations are numerically indistinguishable on any time scale over which field solute concentration measurements typically are made the model prediction of dispersion coefficients on the other hand leads to dramatic differences in predicted plume behavior depending on whether finite time expressions or their asymptotic limits are used according to the dagan model the asymptotic transverse dispersion coefficient cannot replace the time dependent coefficient on any realistic time scale for solute movement at the borden site

on spectral measure and monte carlo approaches to flow in heterogeneous media a pedagogical approach to the spectral integral method as applied to flow problems is presented the deterministic fourier analog to the random spectral integral is highlighted thus illustrating the close ties between the two the general solution to the deterministic flow problem is used to illustrate that a weakly stationary conductivity field does not in general lead to a weakly stationary head field several alternative monte carlo methods are presented and compared to illustrate the problems associated with dropping terms of the form h from the flow equation where and h are respectively perturbations in the conductivity and head fields

stochastic modeling of groundwater flow by unconditional and conditional probabilities 2 the solute transport the problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two and three dimensional flows approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity the solute concentration regarded as a random variable is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty the uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale in any case the concentration expectation does not obey a diffusion type equation in the case of two dimensional flows unless the solute body has traveled a distance larger than a few tens transmissivity integral scales this distance may be exceedingly large in many conceivable applications

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

a monte carlo assessment of eulerian flow and transport perturbation models monte carlo studies of flow and transport in two dimensional synthetic conductivity fields are employed to evaluate first order flow and eulerian transport theories hydraulic conductivity is assumed to obey fractional brownian motion fbm statistics with infinite integral scale or to have an exponential covariance structure with finite integral scale the flow problem is solved via a block centered finite difference scheme and a random walk approach is employed to solve the transport equation for a conservative tracer the model is tested for mass conservation and convergence of computed statistics and found to yield accurate results it is then used to address several issues in the context of flow and transport the validity of the first order relation between the fluctuating velocity covariance and the fluctuating log conductivity is examined the simulations show that for exponential covariance this approximation is justified in the mean flow direction for log conductivity variance f2 of the order of unity however as f2 increases the relation for the transverse velocity component deviates from the fully nonlinear monte carlo results eulerian transport models neglect triplet correlation functions that appear in the nonlocal macroscopic flux the relative importance of the triplet correlation term for conservative chemicals is examined this term appears to be small relative to the convolution flux term in mildly heterogeneous media as f2 increases or the integral scale grows the triplet correlation becomes significant in purely convective transport the triplet correlation term is significant if the heterogeneity is evolving the exact nonlocal macroscale flux for the purely convective case significantly differs from that of the convective dispersive transport this is in agreement with recent theoretical analysis and numerical studies and it suggests that neglecting local scale dispersion may lead to large errors localization errors in the flux term are evaluated using monte carlo simulations the nonlocal in time model significantly differs from the fully nonlocal model for small variance and integral scale there is a slight difference between the fully localized flux and the fully nonlocal convolution flux this is also in agreement with recent theories that suggest that moments through the second for the two models are nearly identical for conservative tracers the fully localized model does not perform well in the purely convective cases

fluctuations of eigenvalues of random normal matrices in this article we consider a fairly general potential in the plane and the corresponding boltzmann gibbs distribution of eigenvalues of random normal matrices as the order of the matrices tends to infinity the eigenvalues condensate on a certain compact subset of the plane the droplet we prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a gaussian field and we discuss various ramifications of this result

monte carlo studies of flow and transport in fractal conductivity fields comparison with stochastic perturbation theory a monte carlo simulation of flow and transport is employed to study tracer migration in porous media with evolving scales of heterogeneity fractal media transport is studied with both conservative and reactive chemicals in media that possess physical as well as chemical heterogeneity linear kinetic equations are assumed to relate the sorbed phase and the aqueous phase concentrations the fluctuating log conductivity possesses the power law spectrum of a fractional brownian motion fbm chemical heterogeneity is represented as spatially varying reaction rates that also are assumed to obey fbm statistics and may be correlated to the conductivity field the model is based on a finite difference approximation to the flow problem and a random walk particle tracking approach for solving the solute transport equation the model is used to make comparisons with the nonlocal transport equations recently developed by deng et al 1993 and hu et al 1995 1997 the results presented herein support these nonlocal models for a wide range of heterogeneous systems however the infinite integral scale associated with the fractal conductivity has a significant effect on the prediction of the nonlocal theories this suggests that integral scale should play a role in stochastic eulerian perturbation theories the importance of the local scale dispersion depends to a great extent on the magnitude of the local dispersivities the effect of neglecting local dispersion decreases with the decrease in local dispersivity

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

random point fields associated with certain fredholm determinants ii fermion shifts and their ergodic and gibbs properties we construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants under a mild condition they turn out unique gibbs measures also some ergodic properties including the entropy positivity are discussed in the lattice case

high resolution monte carlo simulation of flow and conservative transport in heterogeneous porous media 1 methodology and flow results in this the first of two papers concerned with the use of numerical simulation to examine flow and transport parameters in heterogeneous porous media via monte carlo methods various aspects of the modelling effort are examined in particular the need to save on core memory causes one to use only specific realizations that have certain initial characteristics in effect these transport simulations are conditioned by these characteristics also the need to independently estimate length scales for the generated fields is discussed the statistical uniformity of the flow field is investigated by plotting the variance of the seepage velocity for vector components in the x y and z directions finally specific features of the velocity field itself are illuminated in this first paper in particular these data give one the opportunity to investigate the effective hydraulic conductivity in a flow field which is approximately statistically uniform comparisons are made with first and second order perturbation analyses the mean cloud velocity is examined to ascertain whether it is identical to the mean seepage velocity of the model finally the variance in the cloud centroid velocity is examined for the effect of source size and differing strengths of local transverse dispersion

permeability estimates of self affine fracture faults based on generalization of the bottleneck concept we propose a method for calculating the effective permeability of two dimensional self affine permeability fields based on generalizing the one dimensional concept of a bottleneck we test the method on fracture faults where the local permeability field is given by the cube of the aperture field the method remains accurate even when there is substantial mechanical overlap between the two fracture surfaces the computational efficiency of the method is comparable to calculating a simple average and is more than two orders of magnitude faster than solving the reynolds equations using a finite difference scheme pacs numbers in many low permeability geological formations flow occurs primarily through fracture networks 19 in order to model such systems and to predict their behavior there is a need for reliable modeling of the hydromechanical behavior of fracture we consider in this note the situation where the shear displacement between the fracture walls strongly affects its permeability because of its relevance this situation has been considered in many recent hydromechanical studies 2 3 10 18 20 27 28 laboratory tests report that the shearing process results in a significant channelization of the flow and an enhancement of the permeability in the direction normal to the shear this behavior is found to be related to the longrange spatial organization of the void space and efforts have been undertaken to modelize such system in order to provide upscaled value for the fracture permeability recently mallikanas and rajaram 14 determined analytically using perturbation analysis of the reynolds equation to the lowest non trivial order the fracture permeability this model however does not take into account the role of contact areas and will fail if they appear the effect of contacts may however be taken into account by introducing an empirical parameter 29 that is strongly influenced by the number and the spatial distribution of the contacts 13 we present in this note a computational method for calculating the permeability of such fracture faults even in the presence of contacts this new method scales linearly with the number of grid points and is more than two orders of magnitude faster than solving the finitedifferenced reynolds equations through lu decomposition

stochastic models of subsurface flow infinite versus finite domains and stationarity stochastic solutions of the differential equation describing one dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary statistically homogeneous random process are developed using several techniques the analytical approximations using first order analysis propagation of error covariance differential equations and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning using the theory of intrinsic random functions stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three dimensional stationary random field in contrast to the one dimensional case it is found that the resulting three dimensional head perturbation will be locally stationary under very general conditions results from the one dimensional analytical solutions are found to be in agreement with previous monte carlo simulations for a flow system of finite length the solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust showing reasonable agreement with monte carlo results even for the largest input standard deviation of lnk 4 6

correlation functions for random complex zeroes strong clustering and local universality we prove strong clustering of k point correlation functions of zeroes of gaussian entire functions in the course of the proof we also obtain universal local bounds for k point functions of zeroes of arbitrary nondegenerate gaussian analytic functions in the second part of the paper we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of gaussian entire functions in particular of the number of zeroes in measurable domains of large area this complements our recent results from the paper fluctuations in random complex zeroes

global blow up for a localized nonlinear parabolic equation with a nonlocal boundary condition abstract this paper deals with the blow up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition under certain conditions the blowup criteria is established furthermore when f u u p 0 p 1 the global blowup behavior is shown and the blowup rate estimates are also obtained