multi level monte carlo weak galerkin method with nested meshes for stochastic brinkman problem abstract this paper is devoted to the numerical analysis of a multi level monte carlo weak galerkin mlmcwg approximation with nested meshes for solving stochastic brinkman equations with two dimensional spatial domain with weak gradient operator and a stabilizer at hand the weak galerkin wg technique is a high order accurate and stable method which can easily handle deterministic partial differential equations with complex geometries flows with jump fluid viscosity coefficients or high contrast permeability fields given by each sample the multi level monte carlo mlmc technique with nested meshes balances the sampling error and the spatial approximation error where the computational cost can be sharply reduced to log linear complexity with respect to the degree of freedom in spatial direction the nested meshes requirement is introduced here in order to simplify the analysis which can be generalized to mlmc with non nested meshes error estimates are derived in terms of the spatial meshsize and the number of samples the numerical tests are provided to illustrate the behavior of the mlmcwg method and verify our theoretical results regarding optimal convergence of the approximate solutions

numerical simulation of creeping fluid flow in reconstruction models of porous media abstract in this paper we examine representative examples of realistic three dimensional models for porous media by comparing their geometry and permeability with those of the original experimental specimen the comparison is based on numerically exact evaluations of permeability porosity specific internal surface mean curvature euler number and local percolation probabilities the experimental specimen is a three dimensional computer tomographic image of fontainebleau sandstone the three models are stochastic reconstructions for which many of the geometrical characteristics coincide with those of the experimental specimen we find that in spite of the similarity in the geometrical properties the permeability and formation factor can differ greatly between models and the experiment

pore network extraction from micro computerized tomography images network models that represent the void space of a rock by a lattice of pores connected by throats can predict relative permeability once the pore geometry and wettability are known micro computerized tomography scanning provides a three dimensional image of the pore space however these images cannot be directly input into network models in this paper a modified maximal ball algorithm extending the work of silin and patzek d silin and t patzek physica a 371 336 2006 is developed to extract simplified networks of pores and throats with parametrized geometry and interconnectivity from images of the pore space the parameters of the pore networks such as coordination number and pore and throat size distributions are computed and compared to benchmark data from networks extracted by other methods experimental data and direct computation of permeability and formation factor on the underlying images good agreement is reached in most cases allowing networks derived from a wide variety of rock types to be used for predictive modeling

on double centralizer properties between quantum groups and ariki koike algebras abstract in this paper the double centralizer properties for tensor spaces as bimodules over quantum groups and ariki koike algebras as well as their relations with shoji s algebra h h see j algebra 226 2000 818 856 are studied we proved that under the separation condition see j reine angew math 513 1999 53 69 the natural homomorphism from the ariki koike algebra to the endomorphism algebra of tensor space as module over quantum group is surjective we also show that the natural homomorphism from shoji s algebra h h to that endomorphism algebra is always surjective and h h is actually isomorphic to a direct sum of some matrix algebras over some hecke algebras of type a

multiple burst error correcting cyclic product codes corresp let c be the cyclic product code of p single parity check codes of relatively prime lengths n 1 n 2 cdots n p n 1 it is proven that c can correct 2 p 2 2 p 3 1 bursts of length n 1 and lfloor max p 1 min 2 p s s 1 2 p s 2 p s 1 1 2 rfloor bursts of length n 1 n 2 cdots n s 2 leq s leq p 2 for p 3 this means that c is double burst n 1 correcting an efficient decoding algorithm is presented for this code

edge coloring and stopping sets analysis in product codes with mds components we consider non binary product codes with mds components and their iterative row column algebraic decoding on the erasure channel both independent and block erasures are considered in this paper a compact graph representation is introduced on which we define double diversity edge colorings via the rootcheck concept an upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size m then we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols the complexity of this algorithm per iteration is o m aleph for a given differential evolution parameter aleph where m aleph itself is small with respect to the huge cardinality of the coloring ensemble stopping sets of a product code are defined in the context of mds components and a relationship is established with the graph representation a full characterization of these stopping sets is given up to a size d 1 2 where d is the minimum hamming distance of the mds component code the performance of mds based product codes with and without double diversity coloring is analyzed in presence of both the block and the independent erasures in the latter case ml and iterative decoding are proven to coincide at small channel erasure probability furthermore numerical results show excellent performance in presence of unequal erasure probability due to double diversity colorings

modified ariki koike algebras and cyclotomic q schur algebras the modified ariki koike algebra open image in new window is a variation of the original ariki koike algebra open image in new window over an integral domain r when r is a rational function field over the independent parameters open image in new window but for general r open image in new window is not isomorphic to open image in new window and has a simpler structure than open image in new window in this paper we construct a cellular basis of open image in new window which has a similar property as the cellular basis of open image in new window introduced by dipper james mathas by comparing these two cellular bases we obtain some estimate on the decomposition numbers of open image in new window in terms of the decomposition numbers of open image in new window we also prove the integral form of the schur weyl reciprocity between a certain quantum algebra u q and open image in new window on the tensor space open image in new window

necessary and sufficient conditions for the two point phase probability function of two phase random media constructing realizations of random media with a specified two point phase probability function s2 has attracted considerable attention in the recent literature however little is known about cond

geometry models of porous media based on voronoi tessellations and their porosity permeability relations in this paper we present methods that directly model the random structure of porous media using voronoi tessellations three basic structures were generated and they correspond to porous medium geometries with intersecting fractures granular interconnected tubes tubular and fibers fibrous fluid flow through these models was solved by a massively parallelized lattice boltzmann code we established the porosity permeability relations for these basic geometry models it is found that for granular and tubular geometries the specific surface area is a critical structural parameter that can bring their porosity permeability relations together under a unified kozeny carman equation a connected fracture network superimposed on the basic voronoi structure increases the dimensionless permeability relative to the kozeny carman equation isolated large pores vugs on the other hand decreases the dimensionless permeability relative to the kozeny carman equation the kozeny carman equation however cannot distinguish a heterogeneous structure with an embedded partially penetrating fracture the porosity permeability relation for fibrous geometries in general agrees with those established for simple cubic body centered cubic and face centered cubic models in the dilute limit however the dependence on the solid fraction is weaker in voronoi geometries indicating weaker hydrodynamic interactions among randomly interconnected fibers than those in the idealized models

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

using convex quadratic programming to model random media with gaussian random fields excursion sets of gaussian random fields grfs have been frequently used in the literature to model two phase random media with measurable phase autocorrelation functions the goal of successful modeling is finding the optimal field autocorrelation function that best approximates the prescribed phase autocorrelation function in this paper we present a technique which uses convex quadratic programming to find the best admissible field autocorrelation function under a prescribed discretization unlike previous methods this technique efficiently optimizes over all admissible field autocorrelation functions instead of optimizing only over a predetermined parametrized family the results from using this technique indicate that the grf model is significantly more versatile than observed in previous studies an application to modeling a base catalyzed tetraethoxysilane aerogel system given small angle neutron scattering data is also presented

a stochastic ansatz and its relationship with the dichotomic markov process abstract it is shown that a simple stochastic ansatz for the solution of a stochastic differential equation leads to exact results for the dichotomic markov process this has implications for more complex problems where the ansatz leads to a simplified approach to the solution of stochastic equations in engineering and applied physics

development of optimal models of porous media by combining static and dynamic data the permeability and porosity distributions we describe a method for the development of the optimal spatial distributions of the porosity phi and permeability k of a large scale porous medium the optimal distributions are constrained by static and dynamic data the static data that we utilize are limited data for phi and k which the method honors in the optimal model and utilizes their correlation functions in the optimization process the dynamic data include the first arrival fa times at a number of receivers of seismic waves that have propagated in the porous medium and the time dependent production rates of a fluid that flows in the medium the method combines the simulated annealing method with a simulator that solves numerically the three dimensional 3d acoustic wave equation and computes the fa times and a second simulator that solves the 3d governing equation for the fluid s pressure as a function of time to our knowledge this is the first time that an optimization method has been developed to determine simultaneously the global minima of two distinct total energy functions as a stringent test of the method s accuracy we solve for flow of two immiscible fluids in the same porous medium without using any data for the two phase flow problem in the optimization process we show that the optimal model in addition to honoring the data also yields accurate spatial distributions of phi and k as well as providing accurate quantitative predictions for the single and two phase flow problems the efficiency of the computations is discussed in detail

on the action of ariki koike algebras on tensor space the action of the ariki koike algebra on tensor space is used to show an isomorphism theorem between the ariki koike algebra and a direct sum of matrix rings over tensorproducts of smaller ariki koike algebras as an application we get schur weyl duality for the action of the ariki koike algebra and the levi subalgebra of the q schur algebra when separation conditions holds mit hilfe der operation der ariki koike algebra auf dem tensorraum wird ein isomorphismus zwischen der ariki koike algebra und einer direkten summe von matrizenringen uber tensorprodukten kleinerer ariki koike algebren gezeigt als anwendung ergibt sich das die ariki koike algebra und die leviunteralgebra der q schur algebra schur weyl dualitat erfullen falls die separationsbedingung erfullt ist

microstructure reconstruction using entropic descriptors a multi scale approach to the inverse reconstruction of a pattern s microstructure is reported instead of a correlation function a pair of entropic descriptors eds is proposed for a stochastic optimization method the first of them measures a spatial inhomogeneity for a binary pattern or compositional one for a greyscale image the second one quantifies a spatial or compositional statistical complexity the eds reveal structural information that is dissimilar at least in part to that given by correlation functions at almost all discrete length scales the method is tested on a few digitized binary and greyscale images in each of the cases the persuasive reconstruction of the microstructure is found

schur weyl reciprocity for ariki koike algebras let g glm1 glmr be a levi subalgebra of glm with m ri 1mi and v the natural representation of the quantum group uq g we construct a representation of the ariki koike algebra hn r on the n fold tensor space of v commuting with the action of uq g and prove the schur weyl reciprocity for the actions of uq g and hn r on it

classification of missing values in spatial data using spin models a problem of current interest is the estimation of spatially distributed processes at locations where measurements are missing linear interpolation methods rely on the gaussian assumption which is often unrealistic in practice or normalizing transformations which are successful only for mild deviations from the gaussian behavior we propose to address the problem of missing value estimation on two dimensional grids by means of spatial classification methods based on spin ising potts and clock models the spin variables provide an interval discretization of the process values and the spatial correlations are captured in terms of interactions between the spins the spins at the unmeasured locations are classified by means of the energy matching principle the correlation energy of the entire grid including prediction sites is estimated from the sample based correlations we investigate the performance of the spin classifiers in terms of computational speed misclassification rate class histogram and spatial correlations reproduction using simulated realizations of spatial random fields real rainfall data and a digital test image we also compare the spin based methods with standard classifiers such as the k nearest neighbor the fuzzy k nearest neighbor and the support vector machine we find that the spin based classifiers provide competitive choices

list decoding tensor products and interleaved codes we design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes we show that for every code the ratio of its list decoding radius ldr to its minimum distance stays unchanged under the tensor product operation rather than squaring as one might expect this gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded we show that for every code its ldr remains unchanged under m wise interleaving for an integer m this generalizes a recent result of dinur et al in proceedings of the 40th acm symposium on theory of computing stoc 08 2008 pp 275 284 who proved such a result for interleaved hadamard codes equivalently linear transformations using the notion of generalized hamming weights we give better list size bounds for both the tensoring and interleaving of binary linear codes by analyzing the weight distribution of these codes we reduce the task of bounding the list size to one of bounding the number of close by low rank codewords for decoding linear transformations using rank reduction together with other ideas we obtain list size bounds that are tight over small fields our results give better bounds on the ldr than what is obtained from the johnson bound and yield rather general families of codes decodable beyond the johnson radius

effect of the image resolution on the statistical descriptors of heterogeneous media the characterization and reconstruction of heterogeneous materials such as porous media and electrode materials involve the application of image processing methods to data acquired by scanning electron microscopy or other microscopy techniques among them binarization and decimation are critical in order to compute the correlation functions that characterize the microstructure of the above mentioned materials in this study we present a theoretical analysis of the effects of the image size reduction due to the progressive and sequential decimation of the original image three different decimation procedures random bilinear and bicubic were implemented and their consequences on the discrete correlation functions two point line path and pore size distribution and the coarseness derived from the local volume fraction are reported and analyzed the chosen statistical descriptors correlation functions and coarseness are typically employed to characterize and reconstruct heterogeneous materials a normalization for each of the correlation functions has been performed when the loss of statistical information has not been significant for a decimated image its normalized correlation function is forecast by the trend of the original image reference function in contrast when the decimated image does not hold statistical evidence of the original one the normalized correlation function diverts from the reference function moreover the equally weighted sum of the average of the squared difference between the discrete correlation functions of the decimated images and the reference functions leads to a definition of an overall error during the first stages of the gradual decimation the error remains relatively small and independent of the decimation procedure above a threshold defined by the correlation length of the reference function the error becomes a function of the number of decimation steps at this stage some statistical information is lost and the error becomes dependent on the decimation procedure these results may help us to restrict the amount of information that one can afford to lose during a decimation process in order to reduce the computational and memory cost when one aims to diminish the time consumed by a characterization or reconstruction technique yet maintaining the statistical quality of the digitized sample

microstructural degeneracy associated with a two point correlation function and its information content a two point correlation function provides a crucial yet an incomplete characterization of a microstructure because distinctly different microstructures may have the same correlation function in an earlier letter gommes jiao and torquato phys rev lett 108 080601 2012 we addressed the microstructural degeneracy question what is the number of microstructures compatible with a specified correlation function we computed this degeneracy i e configurational entropy in the framework of reconstruction methods which enabled us to map the problem to the determination of ground state degeneracies here we provide a more comprehensive presentation of the methodology and analyses as well as additional results since the configuration space of a reconstruction problem is a hypercube on which a hamming distance is defined we can calculate analytically the energy profile of any reconstruction problem corresponding to the average energy of all microstructures at a given hamming distance from a ground state the steepness of the energy profile is a measure of the roughness of the energy landscape associated with the reconstruction problem which can be used as a proxy for the ground state degeneracy the relationship between this roughness metric and the ground state degeneracy is calibrated using a monte carlo algorithm for determining the ground state degeneracy of a variety of microstructures including realizations of hard disks and poisson point processes at various densities as well as those with known degeneracies e g single disks of various sizes and a particular crystalline microstructure we show that our results can be expressed in terms of the information content of the two point correlation functions from this perspective the a priori condition for a reconstruction to be accurate is that the information content expressed in bits should be comparable to the number of pixels in the unknown microstructure we provide a formula to calculate the information content of any two point correlation function which makes our results broadly applicable to any field in which correlation functions are employed

on the realizability of pair correlation functions abstract the pair correlation function g 2 r provides a basic geometric descriptor for many particle systems it must obey two necessary conditions i non negativity for all distances r and ii non negativity of its associated structure factor s k for all k here we utilize an improved stochastic construction algorithm for particle configurations to establish conditions in which i and ii are also sufficient i e g 2 r is in fact realizable two types of target pair correlation functions have been investigated in one two and three dimensions for hard core particles specifically a unit step function and a contact plus step pair correlation function results indicate that the former target function is realizable up to a terminal density set by necessary condition ii at which the particle core packing fraction equals 2 d in d dimensions furthermore results are consistent with the proposition that for d 1 the contact plus step function is realizable up to a terminal density due to condition ii at which the packing fraction of cores is d 2 2 d 1 torquato and stillinger j phys chem b 106 2000 8354 11406

q tensor space and q weyl modules we obtain the irreducible representations of the q schur algebra motivated by the fact that these representations give all the irreducible representations of gln q in the nondescribing characteristic the irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction the theory of polynomial representations of general linear groups is equivalent to the representation theory of schur algebras see green s book 5 and the bibliography therein in 4 we defined q analogues of schur algebras when q 1 these are the usual schur algebras and when q is a prime power representations of q schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case including all irreducible representations of these groups and important information about decomposition numbers it is natural to ask what features of the classical schur algebras have qanalogues in this paper we define q analogues of tensor space of weyl modules and of weight spaces thereby generalizing the main reslllts which appear in green s book 5 for example we classify the irreducible modules for qschur algebras we determine bases for q weyl modules compatible with weight spaces and we give results on composition multiplicities of the irreducible modules in q weyl modules the proofs are largely self contained so by specializing q to 1 we recover the corresponding results in 5 this paper therefore is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 the q schur algebra let r be a natural number let r be an integral domain and let q be a unit in r we denote the symmetric group on r letters by er the hecke algebra z is the r free r algebra with basis twlw e er where the multiplication is determined by the following rule if a i i 1 is a basic transposition in received by the editors august 15 1989 1980 mathematics subject classification 198 5 revision primary 1 6a64 1 6a6 5 secondary 20c30 this research was supported in part by nsf grant no dms 8802290 the authors gratefully acknowledge support received from nato grant no 0222 87 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

efficient uncertainty quantification for permeability of three dimensional porous media through image analysis and pore scale simulations in this paper we propose an efficient coupled approach for uncertainty quantification of permeability for randomly reconstructed three dimensional 3d pore images where the porosity and two point correlations of a realistic sandstone sample are honored the joshi quiblier adler approach and karhunen loeve expansion are utilized for quick reconstruction of 3d pore images with reduced random dimensionality the eigenvalue problem for the covariance matrix of 3d intermediate gaussian random fields is solved equivalently by a kernel method then the lattice boltzmann method is adopted to simulate fluid flow in reconstructed pore space and evaluate permeability lastly the sparse polynomial chaos expansion sparse pce integrated with a feature selection method is employed to predict permeability distributions incurred by the randomness in microscopic pore structures the feature selection process which is intended to discard redundant basis functions is carried out by the least absolute shrinkage and selection operator modified least angle regression along with cross validation the competence of our proposed approach is validated by the results from monte carlo simulation it reveals that a small number of samples is sufficient for sparse pce with feature selection to produce convincing results then we utilize our method to quantify the uncertainty of permeability under different porosities and correlation parameters it is found that the predicted permeability distributions for reconstructed 3d pore images are close to experimental measurements of berea sandstones in the literature in addition the results show that porosity and correlation length are the critical influence factors for the uncertainty of permeability

boolean reconstructions of complex materials integral geometric approach we show that for the boolean model of random composite media one can from a single image of a system at any particle fraction define a set of parameters which allows one to accurately reconstruct the medium for all other phase fractions the morphological characterization is based on a family of measures known in integral geometry which provides powerful formulas for the boolean model the percolation thresholds of either phase are obtained with good accuracy from the reconstructions one can subsequently predict property curves for the material across all phase fractions from the single three dimensional image we illustrate this for transport and mechanical properties of complex boolean systems and for experimental sandstone samples

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

multi level monte carlo weak galerkin method for elliptic equations with stochastic jump coefficients in this paper we present a multi level monte carlo weak galerkin method for solving elliptic equations with stochastic jump coefficients the multi level monte carlo technique balances the spatial approximation error and the sampling error the weak galerkin technique is a stable and high order accurate method which can easily handle deterministic partial differential equations with complex geometries or jump coefficients given by each sample and this method is also able to capture highly complex solutions exhibiting discontinuities or oscillations with high resolution comparing with the standard monte carlo method by using the multi level monte carlo weak galerkin method the computational cost can be sharply reduced to log linear complexity with respect to the degree of freedom in spatial direction the numerical experiments verify the efficiency of our algorithms

entropic descriptor of a complex behaviour abstract we propose a new type of entropic descriptor that is able to quantify the statistical complexity a measure of complex behaviour by taking simultaneously into account the average departures of a system s entropy s from both its maximum possible value s max and its minimum possible value s min when these two departures are similar to each other the statistical complexity is maximal we apply the new concept to the variability over a range of length scales of spatial or grey level pattern arrangements in simple models the pertinent results confirm the fact that a highly non trivial length scale dependence of the entropic descriptor makes it an adequate complexity measure able to distinguish between structurally distinct configurational macrostates with the same degree of disorder a feature that makes it a good tool for discerning structures in complex patterns

precise algorithms to compute surface correlation functions of two phase heterogeneous media and their applications the quantitative characterization of the microstructure of random heterogeneous media in d dimensional euclidean space r d via a variety of n point correlation functions is of great importance since the respective infinite set determines the effective physical properties of the media in particular surface surface f ss and surface void f sv correlation functions obtainable from radiation scattering experiments contain crucial interfacial information that enables one to estimate transport properties of the media e g the mean survival time and fluid permeability and complements the information content of the conventional two point correlation function however the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use we first present a concise derivation of the small r behaviors of these functions which are linked to the mean curvature of the system then we demonstrate that one can reduce the computational complexity of the problem without sacrificing accuracy by extracting the necessary interfacial information from a cut of the d dimensional statistically homogeneous and isotropic system with an infinitely long line accordingly we devise algorithms based on this idea and test them for two phase media in continuous and discrete spaces specifically for the exact benchmark model of overlapping spheres we find excellent agreement between numerical and exact results we compute surface correlation functions and corresponding local surface area variances for a variety of other model microstructures including hard spheres in equilibrium decorated stealthy patterns as well as snapshots of evolving pattern formation processes e g spinodal decomposition it is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures

a frobenius formula for the characters of ariki koike algebras abstract let h n r be the ariki koike algebra associated to the complex reflection group wn r g r 1 n in this paper we give a new presentation of h n r by making use of the schur weyl reciprocity for h n r established by m sakamoto and t shoji 1999 j algebra 221 293 314 this allows us to construct various non parabolic subalgebras of h n r we construct all the irreducible representations of h n r as induced modules from such subalgebras we show the existence of a partition of unity in h n r which is specialized to a partition of unity in the group algebra c wn r then we prove a frobenius formula for the characters of h n r which is an analogy of the frobenius formula proved by a ram 1991 invent math 106 461 488 for the iwahori hecke algebra of type a

estimation of the influence of second and third order moments on random sets reconstructions abstract to characterize a random image we often limit ourselves to the use of two functions the first one is the histogram or grey level distribution the other is the covariance function which allows the study of the spatial grey level distribution one evaluation of the relevance of these two functions to describe the morphology of images is presented especially we show the importance of having recourse to the centered third order moment in addition to these two functions to obtain a fine characterization of images this study is done for binary random images generated from the same primary grains poisson polygons or discs that are implanted in different ways we simulate with the gagalowicz s procedure some textures respecting the same second or third order moments we study the resemblance in term of the morphology described with erosion and dilation curves and with granulometry by opening and pseudo granulometry by closing with squares the analysis of the results of these measurements is carried out with the help of correspondence analysis that allows a quantitative and qualitative approach of the differences of morphology

on the locality of codeword symbols consider a linear n k d q code c we say that the ith coordinate of c has locality r if the value at this coordinate can be recovered from accessing some other r coordinates of c data storage applications require codes with small redundancy low locality for information coordinates large distance and low locality for parity coordinates in this paper we carry out an in depth study of the relations between these parameters we establish a tight bound for the redundancy n k in terms of the message length the distance and the locality of information coordinates we refer to codes attaining the bound as optimal we prove some structure theorems about optimal codes which are particularly strong for small distances this gives a fairly complete picture of the tradeoffs between codewords length worst case distance and locality of information symbols we then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes using our structure theorem we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings we prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance

modeling multiscale diffusion processes in random heterogeneous media a simple stochastic up scaling strategy is presented for incorporating the effects of micro scale variations of thermal conductivity in the thermal analysis of macro scale systems incomplete characterization of the fine scale variation necessitates posing the thermal conductivity as a stochastic multiscale random field a data driven approach coupled with a non linear dimension reduction algorithm is used to construct a low dimensional stochastic descriptor of the fine scale variation of the stochastic conductivity a representative volume element rve argument is utilized to construct a simple model for up scaling the stochastic thermal conductivity in this model the effective stochastic thermal conductivity is estimated by virtually interrogating the stochastic rv a realistic example of the proposed stochastic multiscale diffusion strategy is illustrated utilizing limited experimental data

compressing random microstructures via stochastic wang tilings this rapid communication presents a stochastic wang tiling based technique to compress or reconstruct disordered microstructures on the basis of given spatial statistics unlike the existing approaches based on a single unit cell it utilizes a finite set of tiles assembled by a stochastic tiling algorithm thereby allowing to accurately reproduce long range orientation orders in a computationally efficient manner although the basic features of the method are demonstrated for a two dimensional particulate suspension the present framework is fully extensible to generic multidimensional media

modeling heterogeneous materials via two point correlation functions ii algorithmic details and applications in the first part of this series of two papers we proposed a theoretical formalism that enables one to model and categorize heterogeneous materials media via two point correlation functions s2 and introduced an efficient heterogeneous medium reconstruction algorithm called the lattice point algorithm here we discuss the algorithmic details of the lattice point procedure and an algorithm modification using surface optimization to further speed up the reconstruction process the importance of the error tolerance which indicates to what accuracy the media are reconstructed is also emphasized and discussed we apply the algorithm to generate three dimensional digitized realizations of a fontainebleau sandstone and a boron carbide aluminum composite from the two dimensional tomographic images of their slices through the materials to ascertain whether the information contained in s2 is sufficient to capture the salient structural features we compute the two point cluster functions of the media which are superior signatures of the microstructure because they incorporate topological connectedness information we also study the reconstruction of a binary laser speckle pattern in two dimensions in which the algorithm fails to reproduce the pattern accurately we conclude that in general reconstructions using s2 only work well for heterogeneous materials with single scale structures however two point information via s2 is not sufficient to accurately model multiscale random media moreover we construct realizations of hypothetical materials with desired structural characteristics obtained by manipulating their two point correlation functions

exterior statistics based boundary conditions for establishing statistically equivalent representative volume elements of statistically nonhomogeneous elastic microstructures abstract the exterior statistics based boundary conditions or esbcs have been developed as optimally effective boundary conditions that can be applied to statistically equivalent representative volume elements or serves of heterogeneous microstructures for predicting homogenized response functions in ghosh and kubair 2016 however the initial development was for nonuniform but statistically homogeneous microstructures with no localized features like clustering in this case the radial distribution function s2 r is adequate for the statistically informed green s functions needed for the development of esbcs however when the microstructure includes statistical inhomogeneities in the form of fiber clusters or matrix rich regions the distance based radial distribution function becomes ineffective this paper overcomes this shortcoming by introducing the joint distance radial and orientation based two point correlation functions s2 r in the statistically informed green s functions needed for the esbcs the efficacy of the esbcs is illustrated through validation simulations that compare its results with those generated by affine transformation based displacement and periodicity boundary conditions additionally comparisons are made with the statistical volume elements or sve methods it is concluded that the simulations with esbcs prescribed on the serves have a definite advantage over other methods in defining optimal sized serves without any iteration

pore space reconstruction of vuggy carbonates using microtomography and multiple point statistics 1 to reconstruct complex porous media such as carbonates we propose a two step approach to combine different types of images microtomography at the resolution of a few microns to resolve large pores and vugs with statistically reconstructed high resolution images for smaller features two dimensional 2 d thin sections provide multiple point statistics which describe the statistical relation between multiple spatial locations that can be used to generate 3 d images with accurate connectivity these statistical reconstructions are combined with images directly measured by microtomography the integrated method is tested on carbonates for which 3 d images of disconnected vugs are captured while 2 d thin sections characterize the small scale structure the integrated images have permeabilities computed using the lattice boltzmann method that are similar to laboratory measured values

improved reconstructions of random media using dilation and erosion processes by using the most sensitive two point correlation functions introduced to date we reconstruct the microstructures of two phase random media with heretofore unattained accuracy such media arise in a host of contexts including porous and composite media ecological structures biological media and astrophysical structures the aforementioned correlation functions are special cases of the so called canonical n point correlation function h n and generalize the ones that have been recently employed to advance our ability to reconstruct complex microstructures y jiao f h stillinger and s torquato proc natl acad sci 106 17634 2009 the use of these generalized correlation functions is tantamount to dilating or eroding a reference phase of the target medium and incorporating the additional topological information of the modified media thereby providing more accurate reconstructions of percolating filamentary and other topologically complex microstructures we apply our methods to a multiply connected donut medium and a dilute distribution of cracks a set of essentially zero measure demonstrating improved accuracy in both cases with implications for higher dimensional and biconnected two phase systems the high information content of the generalized two point correlation functions suggests that it would be profitable to explore their use to characterize the structural and physical properties of not only random media but also molecular systems including structural glasses

prediction of permeability for porous media reconstructed using multiple point statistics to predict multiphase flow through geologically realistic porous media it is necessary to have a three dimensional 3d representation of the pore space we use multiple point statistics based on two dimensional 2d thin sections as training images to generate geologically realistic 3d pore space representations thin section images can provide multiple point statistics which describe the statistical relation between multiple spatial locations and use the probability of occurrence of particular patterns assuming that the medium is isotropic a 3d image can be generated that preserves typical patterns of the void space seen in the thin sections the method is tested on berea sandstone for which a 3d image from micro ct computerized tomography scanning is available and shows that the use of multiple point statistics allows the long range connectivity of the structure to be preserved in contrast to two point statistics methods that tend to underestimate the connectivity furthermore a high resolution 2d thin section image of a carbonate reservoir rock is used to reconstruct 3d structures by the proposed method the permeabilities of the statistical images are computed using the lattice boltzmann method lbm the results are similar to the measured values to the permeability directly computed on the micro ct image for berea and to predictions using analysis of the 2d images and the effective medium approximation

morita equivalences of ariki koike algebras we prove that every ariki koike algebra is morita equivalent to a direct sum of tensor products of smaller ariki koike algebras which have q connected parameter sets a similar result is proved for the cyclotomic q schur algebras combining our results with work of ariki and uglov the decomposition numbers for the ariki koike algebras defined over fields of characteristic zero are now known in principle

limiting efficiencies of burst correcting array codes the author evaluates the limiting efficiencies e s of burst correcting array codes a n sub 1 n sub 2 s for all negative readouts s as n sub 2 tends to infinity and n sub 1 is properly chosen to maximize the efficiency specializing the result to the products of the first i primes donated by s sub i 1 or 4 5 and e 1 or 2 3 this result reveals the existence of burst correcting array codes with efficiencies arbitrarily close to 1 and with rates also arbitrarily close to 1

reconstructing random media we formulate a procedure to reconstruct the structure of general random heterogeneous media from limited morphological information by extending the methodology of rintoul and torquato j colloid interface sci bold 186 467 1997 developed for dispersions the procedure has the advantages that it is simple to implement and generally applicable to multidimensional multiphase and anisotropic structures furthermore an extremely useful feature is that it can incorporate any type and number of correlation functions in order to provide as much morphological information as is necessary for accurate reconstruction we consider a variety of one and two dimensional reconstructions including periodic and random arrays of rods various distribution of disks debye random media and a fontainebleau sandstone sample we also use our algorithm to construct heterogeneous media from specified hypothetical correlation functions including an exponentially damped oscillating function as well as physically unrealizable ones copyright ital 1998 ital the american physical society

fractal heterogeneous media a method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm the obtained structures are invasive stochastic fractals with the hurst exponent varying as a continuous parameter as opposed to lacunar deterministic fractals such as the menger sponge by employing the detrending moving average algorithm a carbone phys rev e 76 056703 2007 the hurst exponent of the generated structure can be subsequently checked the fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself consequently in this framework the hurst exponent should be intended as an estimator of compactness rather than of roughness applications can be envisaged for simulating and quantifying complex systems characterized by self similar heterogeneity across space for example exploitation areas range from the design and control of multifunctional self assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology environmental sciences geomorphological sciences etc

on the generation of periodic discrete structures with identical two point correlation strategies for the generation of periodic discrete structures with identical two point correlation called 2pc equivalent are developed it is shown that starting from a set of 2pc equivalent root structures 2pc equivalent child structures of arbitrary resolution and number of phases e g material phases can be generated based on phase extension through trivial embeddings kernel based extension and phase coalescence proofs are provided by means of discrete fourier transform theory a python 3 implementation is offered for reproduction of examples and future applications

geometrical ambiguity of pair statistics point configurations point configurations have been widely used as model systems in condensed matter physics materials science and biology statistical descriptors such as the n body distribution function g n are usually employed to characterize point configurations among which the most extensively used is the pair distribution function g 2 an intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration although it is known that the pair distance information contained in g 2 is in general insufficient to uniquely determine a point configuration this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies in this paper we present the idea of the distance space called the mathbb d space the pair distances of a specific point configuration are then represented by a single point in the mathbb d space we derive the conditions on the pair distances that can be associated with a point configuration which are equivalent to the realizability conditions of the pair distribution function g 2 moreover we derive the conditions on the pair distances that can be assembled into distinct configurations i e with structural degeneracy these conditions define a bounded region in the mathbb d space by explicitly constructing a variety of degenerate point configurations using the mathbb d space we show that pair information is indeed insufficient to uniquely determine the configuration in general we also discuss several important problems in statistical physics based on the mathbb d space including the reconstruction of atomic structures from experimentally obtained g 2 and a recently proposed decorrelation principle the degenerate configurations have relevance to open questions involving the famous traveling salesman problem

development of optimal models of porous media by combining static and dynamic data the porosity distribution this paper is part of a project the goal of which is the development of the optimal spatial distributions of the porosity and permeability of a large scale porous medium by using complementary static and dynamic data for the medium the data include limited measurements of the porosity which the method honors preserves in the optimal model and utilizes its correlation function together with the first arrival fa times at a certain number of receivers of seismic waves that have propagated in the medium and the time dependence of the pressure of a fluid flowing in the medium the method uses the simulated annealing sa technique in order to develop the optimal model in the present paper we utilize the porosity and fa times data in order to develop the optimal spatial distribution of the porosity this is accomplished by combining the sa method with a simulator that solves for the numerical solution of the acoustic wave equation from which the fa times are estimated limited porosity and fa times data we show that the optimal model not only honors the data but also provides accurate estimates of the porosities in the rest of the porous medium the efficiency of the computations is discussed in detail

modeling heterogeneous materials via two point correlation functions basic principles heterogeneous materials abound in nature and man made situations examples include porous media biological materials and composite materials diverse and interesting properties exhibited by these materials result from their complex microstructures which also make it difficult to model the materials yeong and torquato phys rev e 57 495 1998 introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions in this first part of a series of two papers we collect the known necessary conditions on the standard two point correlation function s2 r and formulate a conjecture in particular we argue that given a complete two point correlation function space s2 r of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space we provide examples of realizable two point correlation functions and suggest a set of analytical basis functions we also discuss an exact mathematical formulation of the re construction problem and prove that s2 r cannot completely specify a two phase heterogeneous material alone moreover we devise an efficient and isotropy preserving construction algorithm namely the lattice point algorithm to generate realizations of materials from their two point correlation functions based on the yeong torquato technique subsequent analysis can be performed on the generated images to obtain desired macroscopic properties these developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two point correlation functions we will mainly focus on basic principles in this paper the algorithmic details and applications of the general scheme are given in the second part of this series of two papers

geometrical ambiguity of pair statistics ii heterogeneous media in the first part of this series of two papers y jiao f h stillinger and s torquato phys rev e 81 011105 2010 we considered the geometrical ambiguity of pair statistics associated with point configurations here we focus on the analogous problem for heterogeneous media materials heterogeneous media are ubiquitous in a host of contexts including composites and granular media biological tissues ecological patterns and astrophysical structures the complex structures of heterogeneous media are usually characterized via statistical descriptors such as the n point correlation function s n an intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two point correlation function s 2 of a target medium recently general claims of the uniqueness of reconstructions using s 2 have been made based on numerical studies which implies that s 2 suffices to uniquely determine the structure of a medium within certain numerical accuracy in this paper we provide a systematic approach to characterize the geometrical ambiguity of s 2 for both continuous two phase heterogeneous media and their digitized representations in a mathematically precise way in particular we derive the exact conditions for the case where two distinct media possess identical s 2 i e they form a degenerate pair the degeneracy conditions are given in terms of integral and algebraic equations for continuous media and their digitized representations respectively by examining these equations and constructing their rigorous solutions for specific examples we conclusively show that in general s 2 is indeed not sufficient information to uniquely determine the structure of the medium which is consistent with the results of our recent study on heterogeneous media reconstruction y jiao f h stillinger and s torquato proc natl acad sci u s a 106 17634 2009 the analytical examples include complex patterns composed of building blocks bearing the letter t and the word water as well as degenerate stacking variants of the densest sphere packing in three dimensions barlow films several numerical examples of degeneracy e g reconstructions of polycrystal microstructures laser speckle patterns and sphere packings are also given which are virtually exact solutions of the degeneracy equations the uniqueness issue of multiphase media reconstructions and additional structural information required to characterize heterogeneous media are discussed including two point quantities that contain topological connectedness information about the phases

a random field model for generating synthetic microstructures of functionally graded materials this article presents a new level cut inhomogeneous filtered poisson random field model for representing two phase microstructures of statistically inhomogeneous functionally graded materials with fully penetrable embedded particles the model involves an inhomogeneous filtered poisson random field comprising a sum of deterministic kernel functions that are scaled by random variables and a cut of the filtered poisson field above a specified level the resulting level cut field depends on the poisson intensity level kernel functions random scaling variables and random rotation matrices a reconstruction algorithm including model calibration and monte carlo simulation is presented for generating samples of two phase microstructures of statistically inhomogeneous media numerical examples demonstrate that the model developed is capable of producing a wide variety of two and three dimensional microstructures of functionally graded composites containing particles of various sizes shapes densities gradations and orientations an example involving finite element analyses of random microstructures leading to statistics of effective properties of functionally graded composites illustrates the usefulness of the proposed model copyright 2008 john wiley sons ltd