stochastic analysis of transport in tubes with rough walls flow and transport in tubes with rough surfaces play an important role in a variety of applications often the topology of such surfaces cannot be accurately described in all of its relevant details due to either insufficient data or measurement errors or both in such cases this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we use a computational framework which is based on stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a generalized polynomial chaos and monte carlo simulations to solve the transformed stochastic problem we use our approach to describe transport of a passive scalar in stokes flow and to quantify the corresponding predictive uncertainty

multiresolution wavelet coarsening and analysis of transport in heterogeneous media abstract modeling transport in a disordered medium with broadly distributed and correlated heterogeneities has been hampered by the fact that a very large set of equations must be solved if the heterogeneities are distributed over several distinct length scales we recently proposed a novel method based on the application of wavelet transformations phys rev lett 79 1997 4385 that can be used for simulation of transport processes in such disordered media in this paper we extend the method so as to be able to study transport in two classes of highly heterogeneous media that are otherwise very difficult to simulate namely disordered anisotropic media and those that contain more than one distinct family of transport paths for example natural rock with both pores and fractures we report the results of extensive computer simulations of transport in such disordered media in which the heterogeneities are broadly distributed and contain long range correlations we show that even for the type of disordered media that we consider in this paper the wavelet based method drastically reduces by at least two orders of magnitude the number of equations to be solved without neglecting any important information about the media s morphology hence providing highly accurate estimates of their transport properties thus we conclude that the wavelet based method is applicable to any disordered medium with any type of heterogeneities

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

when are multidegrees positive abstract let k be an arbitrary field p p k m 1 k k p k m p be a multiprojective space over k and x p be a closed subscheme of p we provide necessary and sufficient conditions for the positivity of the multidegrees of x as a consequence of our methods we show that when x is irreducible the support of multidegrees forms a discrete algebraic polymatroid in algebraic terms we characterize the positivity of the mixed multiplicities of a standard multigraded algebra over an artinian local ring and we apply this to the positivity of mixed multiplicities of ideals furthermore we use our results to recover several results in the literature in the context of combinatorial algebraic geometry

log concavity of independence polynomials of some kinds of trees abstract an independent set in a graph g is a set of pairwise non adjacent vertices let ik g denote the number of independent sets of cardinality k in g then its generating function i g x k 0 g i k g x k is called the independence polynomial of g gutman and harary 1983 alavi et al 1987 conjectured that the independence polynomial of any tree or forest is unimodal this conjecture is still open in this paper after obtaining recurrence relations and giving factorizations of independence polynomials for certain classes of trees we prove the log concavity of their independence polynomials thus our results confirm the conjecture of alavi et al for some special cases

fluid flow in porous media with rough pore solid interface quantifying fluid flow through porous media hinges on the description of permeability a property of considerable importance in many fields ranging from oil and gas exploration to hydrology a common building block for modeling porous media permeability is consideration of fluid flow through tubes with circular cross section described by poiseuille s law in which flow discharge is proportional to the fourth power of the tube s radius in most natural porous media pores are neither cylindrical nor smooth they often have an irregular cross section and rough surfaces this study presents a theoretical scaling of poiseuille s approximation for flow in pores with irregular rough cross section quantified by a surface fractal dimension ds2 the flow rate is a function of the average pore radius to the power 2 3 ds2 instead of 4 in the original poiseuille s law values of ds2 range from 1 to 2 hence the power in the modified poiseuille s approximation varies between 4 and 2 indicating that flow rate decreases as pore surface roughness and surface fractal dimension ds2 increases we also proposed pore length radius relations for isotropic and anisotropic fractal porous media the new theoretical derivations are compared with standard approximations and with experimental values of relative permeability the new approach results in substantially improved prediction of relative permeability of natural porous media relative to the original poiseuille equation

simple and prime crossed products of c algebras by compact quantum group coactions let g a be a compact quantum group and a coaction of g on a c algebra b we give necessary and sufficient conditions for the simplicity and primeness of the crossed product b g in terms of certain fixed point algebras

locally compact group actions on c algebras and compact subgroups be a compact subgroup in this paper we shall give necessary and sufficient conditions in order that the crossed product g x a be simple or prime our conditions are in terms of the dual 2 of the com pact subgroup k the paper is organized as follows in the second section we summarize some preliminary definitions and results about continuous banach representations of compact groups the same section contains the definitions of the subspaces of spherical functions sn nl l 7 n e e those of the algebras s i and the relations between these subspaces section 3 contains a study of saturated actions of g on a in terms of the ideals s s we mention that for compact groups the notion of saturated action has been defined by rieffel see 19 chap 73 the same section contains conditions in order that the crossed product be simple for compact abelian groups g our result reduces to the following g x

transfer matrix methods meet ehrhart theory abstract transfer matrix methods originated in physics where they were used to count the number of allowed particle states on a structure whose width n is a parameter typically the number of states is exponential in n one mathematical instance of this methodology is to enumerate the proper vertex colorings of a graph of growing size by a fixed number of colors in ehrhart theory lattice points in the dilation of a fixed polytope by a factor k are enumerated by inclusion exclusion relevant conditions on how the lattice points interact with hyperplanes are enforced typically the number of points are quasi polynomial in k the text book example is that for a fixed graph the number of proper vertex colorings with k colors is polynomial in k this paper investigates the joint enumeration problem with both parameters n and k free we start off with the classical graph colorings and then explore common situations in combinatorics related to ehrhart theory we show how symmetries can be explored to reduce calculations and explain the interactions with discrete geometry

dependency of percolation critical exponents on the exponent of power law size distribution abstract the standard percolation theory uses objects of the same size moreover it has long been observed that the percolation properties of the systems with a finite distribution of sizes are controlled by an effective size and consequently the universality of the percolation theory is still valid in this study the effect of power law size distribution on the critical exponents of the percolation theory of the two dimensional models is investigated two different object shapes i e stick shaped and square are considered these two shapes are the representative of the fractures in fracture reservoirs and the sandbodies in clastic reservoirs the finite size scaling arguments are used for the connectivity to determine the dependency of the critical exponents on the power law exponent in particular the deviations of percolation exponents from their universal values as well as the connectivity behavior of such systems are investigated numerically as a result this extends the applicability of the conventional percolation approach to study the connectivity of systems with a very broad size distribution

a strong connes spectrum for finite group actions of simple rings abstract in this paper we give necessary and sufficient conditions for a skew group ring to be simple our conditions are in terms of the irreducible representations of the group g we also introduce a version of the strong connes spectrum for finite group actions of simple rings

spectra for compact quantum group coactions and crossed products we present definitions of both connes spectrum and strong connes spectrum for actions of compact quantum groups on c algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple c algebra our results extend to the case of compact quantum ac tions the results in 8 which in turn generalize results by connes olesen and pedersen and kishimoto for abelian group actions we prove in addition that the connes spectra are closed under tensor products these results are new for compact nonabelian groups as well

multilevel upscaling through variational coarsening 1 a new efficient multilevel upscaling procedure for single phase saturated flow in porous media is presented while traditional approaches to this problem have focused on the computation of an upscaled hydraulic conductivity here the coarse scale model is created explicitly from the fine scale model through the application of operator induced variational coarsening this technique which originated with robust multigrid solvers has been shown to accurately capture the influence of fine scale heterogeneous structure over the complete hierarchy of coarse scale models that it generates moreover implicit in this hierarchy is the construction of interpolation operators that provide a natural and complete multiscale basis for the fine scale problem thus this new multilevel upscaling methodology is similar to the multiscale finite element method and indeed it attains similar accuracy in computations of the fine scale hydraulic head and coarse scale normal flux on a variety of problems yet it is an order of magnitude faster on the examples considered here

mixed multiplicities of ideals versus mixed volumes of polytopes the main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as samuel s multiplicities in particular we can give a purely algebraic proof of bernstein s theorem which asserts that the number of common zeros of a system of laurent polynomial equations in the torus is bounded above by the mixed volume of their newton polytopes

a galois correspondence for compact group actions on c algebras abstract in this paper we prove a galois correspondence for compact group actions on c algebras in the presence of a commuting minimal action namely we show that there is a one to one correspondence between the c subalgebras that are globally invariant under the compact action and the commuting minimal action that in addition contain the fixed point algebra of the compact action and the closed normal subgroups of the compact group

on the connectivity of three dimensional fault networks natural fault networks involve a very broad range of fault lengths modeled in general by a power law length distribution n l al such a scaling law does not allow to define any a priori pertinent scale of observation for hydraulic field experiments in fractured media to investigate the relative effects of faults depending on their length we undertake in the spirit of percolation theory a theoretical and numerical study of the connectivity of three dimensional fault networks following power law length distributions we first establish the correct analytical expression of a percolation parameter p which describes the connectivity of the system the parameter p is found to be dependent on the third moment of the length distribution for fault planes it allows us to identify different regimes of connectivity depending on a the exponent of the fault length distribution the geometrical properties of the infinite cluster which partly control transport properties are also established at the percolation threshold for natural fault networks our theoretical analysis suggests that faults larger than a critical length scale may form a well connected network while smaller faults may be not connected on average this result which implies an increase of the connectivity with scale is consistent with scaling effects observed on permeability measurements

numerical calculations of critical densities for lines and planes critical percolation densities have been calculated numerically for large systems of lines in two dimensions and planes in three dimensions results for lines of varying length have been successfully predicted from the constant length case in special cases very large systems of up to 1 4 million lines have been studied by calculating critical densities for a range of system sizes and using a finite size scaling argument a prediction of the infinite critical density is made for the two dimensional system with fixed length orthogonal lines this corresponds to 3 11 intersections per line for the three dimensional system of fixed size orthogonal planes the prediction is 2 00 intersections per plane

maximal subalgebras of c algebras associated with periodic flows we find necessary and sufficient conditions for the subalgebra of analytic elements associated with a periodic c dynamical system to be a maximal norm closed subalgebra our conditions are in terms of the arveson spectrum of the action we also describe equivalent properties of the system in terms of the strong connes spectrum and the simplicity of the crossed product c 2012 elsevier inc all rights reserved

milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs the chromatic polynomial g q of a graph g counts the num ber of proper colorings of g we give an affirmative answer to the conjecture of read and rota heron welsh that the absolute values of the coefficients of the chromatic polynomial form a log concave sequence the proof is obtained by identifying g q with a sequence of numerical invariants of a projective hypersurface analogous to the milnor number of a local analytic hypersurface as a by product of our approach we obtain an analogue of kouchnirenko s

log concavity of characteristic polynomials and the bergman fan of matroids in a recent paper the first author proved the log concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0 answering a long standing conjecture of read in graph theory we extend the proof to all realizable matroids making progress towards a more general conjecture of rota heron welsh our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety the log concavity then follows from an inequality of hodge type

connectivity consistent mapping method for 2 d discrete fracture networks 1 we present a new flow computation method in 2 d discrete fracture networks dfn intermediary between the classical dfn flow simulation method and the projection onto continuous grids the method divides the simulation complexity by solving for flows successively at a local mesh scale and at the global domain scale at the local mesh scale flows are determined by classical dfn flow simulations and approximated by an equivalent hydraulic matrix ehm relating heads and flow rates discretized on the mesh borders assembling the equivalent hydraulic matrices provides for a domain scale discretization of the flow equation the equivalent hydraulic matrices transfer the connectivity and flow structure complexities from the local mesh scale to the domain scale compared to existing geometrical mapping or equivalent tensor methods the ehm method broadens the simulation range of flow to all types of 2 d fracture networks both below and above the representative elementary volume rev additional computation linked to the derivation of the local mesh scale equivalent hydraulic matrices increases the accuracy and reliability of the method compared to dfn methods the ehm method first provides a simpler domain scale alternative permeability model second it enhances the simulation capacities to larger fracture networks where flow discretization on the dfn structure yields system sizes too large to be solved using the most advanced multigrid and multifrontal methods we show that the ehm method continuously moves from the dfn method to the tensor representation as a function of the local mesh scale discretization the balance between accuracy and model simplification can be optimally controlled by adjusting the domain scale and local mesh scale discretizations

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

connectivity of fracture systems a percolation theory approach critical percolation densities have been found numerically for various systems of lines uniformly distributed in the plane the average number of intersections per line at percolation has also been found and varies only slightly over the cases considered it may therefore provide a useful rule of thumb for deciding whether a system percolates an estimate of the critical percolation density from the lattice percolation probability is presented possible extensions of the techniques described to three dimensions are discussed

positivity of mixed multiplicities let r u v n2r u v be a standard bigraded algebra over an artinian local ring k r 0 0 standard means r is generated over k by a finite number of elements of degree 1 0 and 0 1 since the length r u v of r u v is finite we can consider r u v as a function in two variables u and v this function was first studied by van der waerden w and bhattacharya b who proved that there is a polynomial pr u v of total degree dimr 2 such that r u v pr u v for u and v large enough katz mandal andverma kmv found out that the degree of pr u v is equal to rdimr 2 where rdimr is the relevant dimension of r defined as follows let r denote the ideal generated by the homogeneous elements of degree u v with u 1 v 1 let projr be the set of all homogeneous prime ideals r of r then rdimr max dimr projr if projr and rdimr can be any negative integer if projr if we write

compact actions commuting with ergodic actions and applications to crossed products let a k be a c dynamical system with k compact in this paper we prove a duality result for saturated actions theorem 3 3 the proof of this result can also be considered as an alternate proof of the corresponding result for von neumann algebras due to araki haag kastler and takesaki 14 we also obtain results concerning the simplicity and the primeness of the crossed product a k in terms of the ergodicity of the commutant of propositions 5 3 and 5 4

the entropic discriminant abstract the entropic discriminant is a non negative polynomial associated to a matrix it arises in contexts ranging from statistics and linear programming to singularity theory and algebraic geometry it describes the complex branch locus of the polar map of a real hyperplane arrangement and it vanishes when the equations defining the analytic center of a linear program have a complex double root we study the geometry of the entropic discriminant and we express its degree in terms of the characteristic polynomial of the underlying matroid singularities of reciprocal linear spaces play a key role in the corank one case the entropic discriminant admits a sum of squares representation derived from the discriminant of a characteristic polynomial of a symmetric matrix

connectivity properties of two dimensional fracture networks with stochastic fractal correlation 1 we present a theoretical and numerical study of the connectivity of fracture networks with fractal correlations in addition to length distribution this spatial property observed on most fracture networks conveys long range correlation that may be crucial on network connectivity we especially focus on the model that comes out relevant to natural fracture network a fractal density distribution for the fracture centers dimension d and a power law distribution for the fracture lengths exponent a n l l a three different regimes of connectivity are identified depending on d and on a for a d 1 the connectivity is ruled by fractures much smaller than the system size with a strong control of spatial correlation the connectivity now decreases with system scale finally for the self similar case a d 1 which corresponds to the transition between the two previous regimes connectivity properties are scale invariant percolation threshold corresponds to a critical fractal density and the average number of intersections per fracture at threshold is a scale invariant as well

permeability of self affine rough fractures the permeability of two dimensional fractures with self affine fractal roughness is studied via analytic arguments and numerical simulations the limit where the roughness amplitude is small compared with average fracture aperture is analyzed by a perturbation method while in the opposite case of narrow aperture we use heuristic arguments based on lubrication theory numerical simulations using the lattice boltzmann method are used to examine the complete range of aperture sizes and confirm the analytic arguments

spectral theory for dynamics on graphs containing attractive and repulsive interactions many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions examples include synchronization of nonlinear oscillator networks the behavior of groups or cliques in social networks and the study of optimal convergence for consensus algorithm it is important to determine the index of a matrix i e the number of positive and negative eigenvalues and the dimension of the kernel in this paper we consider the common examples where the matrix takes the form of a signed graph laplacian we show that the there are topological constraints on the index of the laplacian matrix related to the dimension of a certain homology group when the homology group is trivial the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions in general these constraints give bounds on the number of positive and negative eigenvalues with the dimension of the homology group counting the number

fourier acceleration of iterative processes in disordered systems technical details are given on how to use fourier acceleration with iterative processes such as relaxation and conjugate gradient methods these methods are often used to solve large linear systems of equations but become hopelessly slow very rapidly as the size of the set of equations to be solved increases fourier acceleration is a method designed to alleviate these problems and result in a very fast algorithm the method is explained for the jacobi relaxation and conjugate gradient methods and is applied to two models the random resistor network and the random central force network in the first model acceleration works very well in the second little is gained we discuss reasons for this we also include a discussion of stopping criteria

connectivity of random fault networks following a power law fault length distribution we present a theoretical and numerical study of the connectivity of fault networks following power law fault length distributions n l l a as expected for natural fault networks different regimes of connectivity are identified depending on a for a 3 faults smaller than the system size rule the network connectivity and classical laws of percolation theory apply on the opposite for a 1 the connectivity is ruled by the largest fault in the system for 1 a 3 both small and large faults control the connectivity in a ratio which depends on a the geometrical properties of the fault network and of its connected parts density scaling properties are established at the percolation threshold finally implications are discussed in the case of fault networks with constant density in particular we predict the existence of a critical scale at which fault networks are always connected whatever a smaller than 3 and whatever their fault density

arithmetic matroids the tutte polynomial and toric arrangements abstract we introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group in particular we study the representability of its dual providing an extension of the gale duality to this setting guided by the geometry of generalized toric arrangements we provide a combinatorial interpretation of the associated arithmetic tutte polynomial which can be seen as a generalization of crapo s formula for the classical tutte polynomial

algebraic analogs of the connes spectrum abstract in this paper we obtain algebraic analogs of certain known results on crossed products of von neumann algebras and of c algebras of particular interest here is the connes spectrum introduced by a connes in his classification of factors of type iii in addition we are concerned with certain von neumann algebra results contained in the work of connes takesaki and nakagami and with certain more algebraic c algebra results from the work of olesen pederson and kishimoto let g be a finite group and let a be a g graded ring with 1 then there exists an extension ring of a determined by this structure called the smash product of a by g this overring comes from the theory of hopf algebras and we denote it by a g in the first half of this paper we introduce and study the connes and strong connes subgroups of g and we relate them to the ideal structure of a g in particular we obtain criteria for a g to be prime or simple in a different direction suppose a is a k algebra and g is a finite abelian group of k automorphisms of a if the field k contains appropriate roots of unity then the skew group ring ag has a natural smash product structure a g where g hom g k is the dual group of g the problem is to explicitly determine the connes subgroup of g in terms of the action of g on a this is achieved in the second half of the paper under the additional assumption that a is a prime ring a more detailed discussion of the operator algebra sources of our results will be given in the appendix

h vectors of matroids and logarithmic concavity abstract let m be a matroid on e representable over a field of characteristic zero we show that h vectors of the following simplicial complexes are log concave 1 the matroid complex of independent subsets of e 2 the broken circuit complex of m relative to an ordering of e the first implies a conjecture of colbourn on the reliability polynomial of a graph and the second implies a conjecture of hoggar on the chromatic polynomial of a graph the proof is based on the geometric formula for the characteristic polynomial of denham garrousian and schulze

lattice boltzmann simulation of fluid flow in fracture networks with rough self affine surfaces using the lattice boltzmann method we study fluid flow in a two dimensional 2d model of fracture network of rock each fracture in a square network is represented by a 2d channel with rough self affine internal surfaces various parameters of the model such as the connectivity and the apertures of the fractures the roughness profile of their surface as well as the reynolds number for flow of the fluid are systematically varied in order to assess their effect on the effective permeability of the fracture network the distribution of the fractures apertures is approximated well by a log normal distribution which is consistent with experimental data due to the roughness of the fractures surfaces and the finite size of the networks that can be used in the simulations the fracture network is anisotropic the anisotropy increases as the connectivity of the network decreases and approaches the percolation threshold the effective permeability k of the network follows the power law k approximately beta where is the average aperture of the fractures in the network and the exponent beta may depend on the roughness exponent a crossover from linear to nonlinear flow regime is obtained at a reynolds number re approximately o 1 but the precise numerical value of the crossover re depends on the roughness of the fractures surfaces

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed