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

the l component of the unipotent albanese map we prove a finiteness theorem for the local l p component of the qp unipotent albanese map for curves as an application we refine the non abelian selmer varieties arising in the study of global points and deduce thereby a new proof of siegel s theorem for affine curves over q of genus one with mordell weil rank at most one

massey products and ideal class groups we consider certain massey products in the cohomology of a galois extension of fields with coefficients in p power roots of unity we prove formulas for these products both in general and in the special case that the galois extension in question is the maximal extension of a number field unramified outside a set of primes s including those above p and any archimedean places we then consider those zp kummer extensions l of the maximal p cyclotomic extension k of a number field k that are unramified outside s we show that massey products describe the structure of a certain decomposition free quotient of a graded piece of the maximal unramified abelian pro p extension of l in which all primes above those in s split completely with the grading arising from the augmentation filtration on the group ring of the galois group of l k we explicitly describe examples of the maximal unramified abelian pro p extensions of unramified outside p kummer extensions of the cyclotomic field of all p power roots of unity for irregular primes p

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

a non abelian conjecture of tate shafarevich type for hyperbolic curves let x denote a hyperbolic curve over mathbb q and let p denote a prime of good reduction the third author s approach to integral points introduced in kim invent math 161 629 656 2005 publ res inst math sci 45 89 133 2009 endows x mathbb z p with a nested sequence of subsets x mathbb z p n which contain x mathbb z these sets have been computed in a range of special cases balakrishnan et al j am math soc 24 281 291 2011 dan cohen and wewers proc lond math soc 110 133 171 2015 dan cohen and wewers int math res not imrn 17 5291 5354 2016 kim j am math soc 23 725 747 2010 there is good reason to believe them to be practically computable in general in 2012 the third author announced the conjecture that for n sufficiently large x mathbb z x mathbb z p n this conjecture may be seen as a sort of compromise between the abelian confines of the bsd conjecture and the profinite world of the grothendieck section conjecture after stating the conjecture and explaining its relationship to these other conjectures we explore a range of special cases in which the new conjecture can be verified

appendix and erratum to massey products for elliptic curves of rank 1 the paper 6 contains a few errors in the basic assumptions as well as in the formula of corollary 0 2 first of all it should have been made clear at the outset that the regular model e for the elliptic curve e must be the minimal regular model and x the complement of the origin in the regular minimal model similarly the tangential base point b must be integral in that it is a z basis of the relative tangent space ete z it could also be an integral two torsion point for the arguments of the paper to hold verbatim the most significant error is in the contribution of the local terms at l 6 p that is lemma 1 2 the problem is that a point that is integral on x may not be integral on a smooth model over a field of good reduction as it stands the lemma will only apply to points that are integral in this stronger sense however to get immediate examples one can replace the lemma by

li p service an algorithm for computing p adic polylogarithms we describe an algorithm for computing coleman s p adic poly logarithms up to a given precision 2007 american mathematical society

torsion points on curves and p adic abelian integrals theorem a let f c j be an albanese morphism defined over a number field k of a s mooth curve of genus g into its jacobian suppose j has potential complex multiplication let s denote the set of primes p of k satisfying i p does not divide 2 or 3 ii k q is unramified at p iii c has good ordinary reduction over k then the set t of torsion points of a defined over an algebraic closure of k which lie on the image of c is defined over an extension of k unramified above s moreover to pg

multiscale finite element methods for high contrast problems using local spectral basis functions in this paper we study multiscale finite element methods msfems using spectral multiscale basis functions that are designed for high contrast problems multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem this local spectral problem strongly depends on the choice of initial partition of unity functions the resulting space enriches the initial multiscale space using eigenvectors of local spectral problem the eigenvectors corresponding to small asymptotically vanishing eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small asymptotically vanishing eigenvalues we present a convergence study that shows that the convergence rate in energy norm is proportional to h 1 2 where is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space thus we would like to reach to a larger eigenvalue with a smaller coarse space this is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems numerical results are presented to back up our theoretical results and to show higher accuracy of msfems with spectral multiscale basis functions we also present a hierarchical construction of the eigenvectors that provides cpu savings

hida families and rational points on elliptic curves let e be an elliptic curve over q of conductor n mp having a prime p n of multiplicative reduction because e is modular it corresponds to a normalised weight two eigenform on 0 n whose q expansion is denoted f n anq letx hom z p z p z p 1 z zp which contains z as a dense subset by associating to k z the character x xk 2 denote by a u the ring of cp valued p adic analytic functions on a compact open subset u of x hida s theory associates to f a neighborhood u of 2 x which can be assumed for simplicity to be contained in the residue class of 2 modulo p 1 and a formal q expansion

adaptive generalized multiscale finite element methods for h curl elliptic problems with heterogeneous coefficients abstract in this paper we construct an adaptive multiscale method for solving h curl elliptic problems in highly heterogeneous media our method is based on the generalized multiscale finite element method we will first construct a suitable snapshot space and a dimensional reduction procedure to identify important modes of the solution we next develop and analyze an a posteriori error indicator and the corresponding adaptive algorithm in addition we will construct a coupled offline online adaptive algorithm which provides an adaptive strategy to the selection of offline and online basis functions our theory shows that the convergence is robust with respect to the heterogeneities and contrast of the media we present several numerical results to illustrate the performance of our method

p adic multiple zeta values ii tannakian interpretations we establish a tannakian formalism of p adic multiple polylogarithms and p adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de rham fundamental torsor and a rigid fundamental torsor of the projective line minus three points and also discuss its hodge and e tale analogues as an application we give a way to erase log poles of p adic multiple polylogarithms and introduce overconvergent p adic multiple polylogarithms which might be p adic multiple analogue of zagier s single valued complex polylogarithms

the rate distortion function on classes of sources determined by spectral capacities the quantity sup a in cal r r a d is considered where cal r is a class of homogeneous n parameter sources and r a d denotes the single letter mean square error mse rate distortion function for the individual source a in particular the case in which the class cal r is specified in terms of spectral information is treated for general classes of spectral measures whose upper measures are capacities in the sense of choquet alternating of order two this type of class includes many common models for spectral uncertainty such as mixture models spectral band models and neighborhoods generated by variation and prohorov metrics it is shown that each such class contains a worst case source whose rate distortion function achieves the supremum over the class for each value of distortion this source is characterized as having a spectral density that is a derivative in the sense of huber and strassen of the upper spectral measure with respect to lebesgue measure on pi pi n moreover it is shown that the spectral measure of the worst case source is closest in a sense defined by directed divergence to lebesgue measure which corresponds to a memoryless source numerical results are presented for the particular case in which the source spectral measure is a mixture of a gauss markov spectrum and an unknown contaminating component

quadratic chabauty for modular curves and modular forms of rank one in this paper we provide refined sufficient conditions for the quadratic chabauty method on a curve x to produce an effective finite set of points containing the rational points x mathbb q with the condition on the rank of the jacobian of x replaced by condition on the rank of a quotient of the jacobian plus an associated space of chow heegner points we then apply this condition to prove the effective finiteness of x mathbb q for any modular curve x x 0 n or x mathrm ns n of genus at least 2 with n prime the proof relies on the existence of a quotient of their jacobians whose mordell weil rank is equal to its dimension and at least 2 which is proven via analytic estimates for orders of vanishing of l functions of modular forms thanks to a kolyvagin logachev type result

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media it is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs however the computational cost of conventional numerical modeling methods such as finite difference method and finite element method becomes prohibitively expensive when applied to very large models we propose a generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element the application of multiscale basis functions can capture the fine scale medium property variations and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite element method for wave equation while restricting the error to low values we formulate the continuous galerkin and discontinuous galerkin formulation of the multiscale method both of which have pros and cons applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system

the cuspidalisation of sections of arithmetic fundamental groups abstract in this paper which is a sequel to 7 we investigate the theory of cuspidalisation of sections of arithmetic fundamental groups of hyperbolic curves to cuspidally i th and 2 p th step prosolvable arithmetic fundamental groups as a consequence we exhibit two necessary and sufficient conditions for sections of arithmetic fundamental groups of hyperbolic curves over p adic local fields to arise from rational points we also exhibit a class of sections of arithmetic fundamental groups of p adic curves which are orthogonal to pic and which satisfy unconditionally one of the above conditions

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

fast online generalized multiscale finite element method using constraint energy minimization abstract local multiscale methods often construct multiscale basis functions in the offline stage without taking into account input parameters such as source terms boundary conditions and so on these basis functions are then used in the online stage with a specific input parameter to solve the global problem at a reduced computational cost recently online approaches have been introduced where multiscale basis functions are adaptively constructed in some regions to reduce the error significantly in multiscale methods it is desired to have only 1 2 iterations to reduce the error to a desired threshold using generalized multiscale finite element framework 10 it was shown that by choosing sufficient number of offline basis functions the error reduction can be made independent of physical parameters such as scales and contrast in this paper our goal is to improve this using our recently proposed approach 4 and special online basis construction in oversampled regions we show that the error reduction can be made sufficiently large by appropriately selecting oversampling regions our numerical results show that one can achieve a three order of magnitude error reduction which is better than our previous methods we also develop an adaptive algorithm and enrich in selected regions with large residuals in our adaptive method we show that the convergence rate can be determined by a user defined parameter and we confirm this by numerical simulations the analysis of the method is presented

a new heterogeneous multiscale method for time harmonic maxwell s equations in this paper we suggest a new heterogeneous multiscale method hmm for the time harmonic maxwell equations in locally periodic media the method is constructed by using a divergence regularization in one of the cell problems this allows us to introduce fine scale correctors that are not subject to a cumbersome divergence free constraint and which can hence easily be implemented to analyze the method we first revisit classical homogenization theory for time harmonic maxwell equations and derive a new homogenization result that makes use of the divergence regularization in the two scale homogenized equation we then show that the hmm is equivalent to a discretization of this equation in particular writing both problems in a fully coupled two scale formulation is the crucial starting point for a corresponding numerical analysis of the method with this approach we are able to prove rigorous a priori error estimates in the mathbf h curl and the h 1 norm and we derive reliable and efficie

p adic multiple zeta values i our main aim in this paper is to give a foundation of the theory of p adic multiple zeta values we introduce one variable p adic multiple polylogarithms by coleman s p adic iterated integration theory we define p adic multiple zeta values to be special values of p adic multiple polylogarithms we consider the formal p adic kz equation and introduce the p adic drinfel d associator by using certain two fundamental solutions of the p adic kz equation we show that our p adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p adic kz equation and our p adic multiple zeta values appear as coefficients of the p adic drinfel d associator we show various properties of p adic multiple zeta values which are sometimes analogous to the complex case and are sometimes peculiar to the p adic case via the p adic kz equation

on robust wiener signal estimation a minimax formulation is considered for the problem of designing robust linear causal estimators of continuous time signals in noise it is shown that the minimax estimation problem can generally be reduced to finding a least favorable pair of spectra in particular robust predictors filters and finite lag smoothers are shown to be just the optimal predictors filters and smoothers respectively for a least favorable pair this result can often greatly simplify the design of robust estimators

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

minimax linear smoothing for capacities the case of correlated signals and noise results of an earlier paper giving minimax linear smoothers for the problem of estimating a homogeneous signal field in an additive orthogonal noise field when both have uncertain spectral properties are extended to the case in which the signal and noise fields are arbitrarily correlated as before spectral uncertainty is modeled by assuming that the spectral measures of the signal and noise fields lie in classes of measures generated by two alternating choquet capacities it is demonstrated that this problem admits a general solution in terms of the huber strassen derivative between the capacities that generate the uncertainty sets and that the least favorable spectra for smoothing in orthogonal noise are also the least favorable marginal spectra for smoothing in correlated noise the resulting filter is seen to be a zonal filter that also arises as the solution to an analogous problem in nonparametric minimax hypothesis testing these new results extend the applicability of minimax robust smoothing techniques to application involving signal dependent noise phenomena such as multipath and clutter which are usually difficult to model precisely

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

massey products for elliptic curves of rank 1 for an elliptic curve over q of analytic rank 1 we use the level two selmer variety and secondary cohomology products to find explicit analytic defining equations for global integral points inside the set of p adic points

residual driven online generalized multiscale finite element methods the construction of local reduced order models via multiscale basis functions has been an area of active research in this paper we propose online multiscale basis functions which are constructed using the offline space and the current residual online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators we derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error this error decrease is independent of physical parameters such as the contrast and multiple scales in the problem the offline spaces are constructed using generalized multiscale finite element methods gmsfem we show that if one chooses a sufficient number of offline basis functions one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast we note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects using the residual information we can incorporate the distant information provided the offline approximation satisfies certain properties in the paper theoretical and numerical results are presented our numerical results show that if the offline space is sufficiently large in terms of the dimension such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues then the error reduction by adding online multiscale basis function is independent of the contrast we discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces

robust wiener filters for random signals in correlated noise corresp minimax robust wiener filtering is considered for the case in which the signal and noise spectral density matrix is not completely specified results are obtained for spectral density matrix classes which are defined by upper and lower bounds on the components of the matrix these results form an extension of earlier results on robust wiener filtering for the case of uncorrelated signals and noise

tangential localization for selmer varieties this paper proposes a tangential version of the theory of selmer varieties together with a formulation of cohomological duality in families of lie algebras indexed by nonabelian cohomology this theory allows one to consider deformations of cohomology classes as one moves over the selmer variety and suggests an approach for generalizing to number fields the homotopical techniques for proving diophantine finiteness that were developed over q the utility of this perspective is demonstrated by way of a new proof of siegel s theorem on finiteness of s integral points for the projective line minus three points over a totally real field

minimax linear smoothing for capacities abstract minimax linear smoothers are considered for the problem of estimating a homogeneous signal field in an additive orthogonal noise field a minimax game with the quadratic mean estimation error as an objective function is used to formulate this problem uncertainty in signal and noise field spectra is modeled using general nonparametric classes of measures proposed by huber and strassen for the problem of minimax hypothesis testing these classes which are described in terms of choquet alternating capacities of order 2 include the conventional models for spectral uncertainty and admit a general solution to the minimax linear smoothing problem author

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

p adic l functions and selmer varieties associated to elliptic curves with complex multiplication we show how the finiteness of integral points on an elliptic curve over q with complex multiplication can be accounted for by the nonvanishing of l functions that leads to bounds for dimensions of selmer varieties

the motivic fundamental group of p1 0 1 and the theorem of siegel in this paper we establish a link between the structure theory of the pro unipotent motivic fundamental group of the projective line minus three points and diophantine geometry in particular we give a p adic proof of siegel s theorem