constructing local generic formal fibers abstract let t m be a complete local noetherian unique factorization domain of dimension at least two such that the cardinality of the residue field t m is at least the cardinality of the real numbers suppose 0 p 0 p 1 p n m is a chain of distinct prime ideals of t such that p n intersected with the prime subring of t is the zero ideal we construct a chain of local unique factorization domains a n a n 1 a 1 a 0 t such that the completion of each a i is t and for every i the generic formal fiber of a i is local this means that the ring t a i k i is a local ring where k i is the quotient field of a i with p i a i k i its maximal ideal

symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

semilocal generic formal fibers abstract let t be a complete local ring and c a finite set of incomparable prime ideals of t we find necessary and sufficient conditions for t to be the completion of an integral domain whose generic formal fiber is semilocal with maximal ideals the elements of c in addition if the characteristic of t is zero we give necessary and sufficient conditions for t to be the completion of an excellent integral domain whose generic formal fiber is semilocal with maximal ideals the elements of c

complexity and tor on a complete intersection let r m be a complete intersection that is a local ring whose m adic completion is the quotient of a regular local ring by a regular sequence supposemis a finitely generatedr module it is known that the even and odd betti sequences ofmare eventually given by polynomials of the same degreen the complexity ofmis the nonnegative integern 1 we use this notion of complexity to study the vanishing of torri m n for finitely generated modulesmandnover a complete intersectionr we prove several theorems dealing with rigidity of tor which are generalizations and in certain situations improvements of known results the main idea of these rigidity theorems is that the number of consecutive vanishing tors required in the hypothesis of a rigidity theorem depends more on the minimum of the complexities ofmandnrather than on the codimension ofr we give examples showing that this dependence is sharp we also show that ifm rnhas finite length then for sufficiently high indices two consecutive vanishing tors force the vanishing of all higher tors

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

local rings of finite cohen macaulay type abstract let r m be a local cohen macaulay ring whose m adic completion r has an isolated singularity we verify the following conjecture of f o schreyer r has finite cohen macaulay type if and only if r has finite cohen macaulay type we also show that the hypersurface k x 0 x d f has finite cohen macaulay type if and only if k s x 0 x d f has finite cohen macaulay type where k s is the separable closure of the field k

preconditioned conjugate gradients for solving singular systems abstract in this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b where a is a singular symmetric positive semi definite matrix the method diverges if b is not exactly in the range r a of a if the null space n a of a is explicitly known then this divergence can be avoided by subtracting from b its orthogonal projection onto n a as well as analysing this subtraction conditions necessary for the existence of a nonsingular incomplete cholesky decomposition are given finally the theory is applied to the discretized semi definite neumann problem

some sphere theorems in linear potential theory in this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities whose equality cases are saturated by domains with spherical sy

classifying thick subcategories of the stable category of cohen macaulay modules abstract various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics in this paper as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p group due to benson carlson and rickard we consider classifying thick subcategories of the stable category of cohen macaulay modules over a gorenstein local ring the main result of this paper yields a complete classification of the thick subcategories of the stable category of cohen macaulay modules over a local hypersurface in terms of specialization closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus

noetherian rings between a semilocal domain and its completion we consider a general technique for constructing local noetherian integral domains letrbe a semilocal noetherian domain with jacobson radicalmand field of fractionsk letybe a nonzero element ofmand letr be the y adic completion ofr for elements 1 s yr algebraically independent overk we obtain a necessary and sufficient condition fora k 1 s r to be simultaneously noetherian and a directed union of localized polynomial rings insvariables overr we specify conditions in order that excellence be preserved and we use the construction to obtain a non noetherian ringaof the formk r which is a directed union of localized polynomial rings overr

the parallel subdomain levelset deflation method in reservoir simulation extreme and isolated eigenvalues are known to be harmful to the convergence of an iterative solver these eigenvalues can be produced by strong heterogeneity in the underlying physics we can improve the quality of the spectrum by deflating the harmful eigenvalues in this work deflation is applied to linear systems in reservoir simulation in particular large sudden differences in the permeability produce extreme eigenvalues the number and magnitude of these eigenvalues is linked to the number and magnitude of the permeability jumps two deflation methods are discussed firstly we state that harmonic ritz eigenvector deflation which computes the deflation vectors from the information produced by the linear solver is unfeasible in modern reservoir simulation due to high costs and lack of parallelism secondly we test a physics based subdomain levelset deflation algorithm that constructs the deflation vectors a priori numerical experiments show that both methods can improve the performance of the linear solver we highlight the fact that subdomain levelset deflation is particularly suitable for a parallel implementation for cases with well defined permeability jumps of a factor 104 or higher parallel physics based deflation has potential in commercial applications in particular the good scalability of parallel subdomain levelset deflation combined with the robust parallel preconditioner for deflated system suggests the use of this method as an alternative for amg

completions of local rings with an isolated singularity abstract it is shown that all equicharacteristic complete local rings can be realized as completions of local rings whose proper localizations are all regular the only restriction in the mixed characteristic case is that the prime integer cannot be a nonzero zerodivisor in any proper localization of the complete ring the constructed examples are unique factorization domains whenever possible and in general are as close to being ufds as the hypothesis will allow

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

characterization of completions of excellent domains of characteristic zero abstract we show that a complete local ring t containing the integers is the completion of a local excellent integral domain if and only if it is reduced equidimensional and no integer of t is a zero divisor

are complete intersections complete intersections abstract a commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence it has not previously been determined whether or not such a ring is necessarily itself the quotient of a regular ring by an ideal generated by a regular sequence in this article it is shown that if a complete intersection is a one dimensional integral domain then it is such a quotient however an example is produced of a three dimensional complete intersection domain which is not a homomorphic image of a regular local ring and so the property does not hold in general

on ricci curvature and volume growth in dimension three we prove that any complete metric on mathbb r 3 minus an open ball with non negative ricci curvature and quadratic ricci curvature decay has cubic volume growth

3 manifolds with nonnegative ricci curvature for a complete noncompact 3 manifold with nonnegative ricci curvature we prove that either it is diffeomorphic to 3 or the universal cover splits this confirms milnor s conjecture in dimension 3

on a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth on etudie le comportement asymptotique de varietes de riemann on construit des coordonnees a l infini sous certaines conditions intrinseques

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

descent and ascent of perinormality in some ring extension settings abstract in this paper we investigate some open questions regarding perinormal domains posed by neal epstein and jay shapiro in 6 more specifically we focus on the ascent descent property of perinormality between canonical integral domain extensions in particular a a x and a a we give special conditions under which perinormality ascends from a to the polynomial ring a x in the case of an universally catenary domain a whereas we have a characterizing result for when perinormality descends from a x to a the sufficient condition for the descent is cumbersome to check for this reason we turn to special cases for which perinormality descends from a x to a in the case of an analytically irreducible local domain a m and its m adic completion a m we refer to a technique for generating examples in which perinormality fails to ascend when a is perinormal we explore hypotheses under which a must be normal perinormal or weakly normal finally we make connexions between the concepts of semi normality weak normality relative and global perinormality and normality

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

deflated preconditioned conjugate gradient solvers for the pressure poisson equation extensions and improvements extensions and improvements to a deflated preconditioned conjugate gradient technique for the solution of the pressure poisson equation within an incompressible flow solver are described in particular the use of the technique for embedded grids for cases where volume of fluid or level set schemes are required and its implementation on parallel machines are considered several examples are included that demonstrate a considerable reduction in the number of iterations and a remarkable insensitivity to the number of groups regions chosen and or to the way the groups are formed copyright 2010 john wiley sons ltd

an efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs a mathematical model for the prediction of fluid pressures is given by a time dependent diffusion equation application of the finite element method leads to a system of linear equations a complication is that the underground consists of layers with very large differences in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the cg method has been observed moreover a classical termination criterion is not valid in this problem after diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability for the ic preconditioner the same behavior is observed to annihilate the effect of the extreme eigenvalues a deflated cg method is used the convergence rate improves considerably and the termination criterion becomes again reliable finally a cheap approximation of the eigenvectors is proposed

on uniqueness of tangent cones for einstein manifolds we show that for any ricci flat manifold with euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross section similarly for any noncollapsing limit of einstein manifolds with uniformly bounded einstein constants we show that local tangent cones are unique if one tangent cone has a smooth cross section

a comparison of deflation and coarse grid correction applied to porous media flow in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare a coarse grid correction preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the coarse grid correction this implies that the conjugate gradient method applied to the deflated preconditioned system is expected always to converge faster than the conjugate gradient method applied to the system preconditioned by the coarse grid correction numerical results for porous media flows emphasize the theoretical results

a comparison of deflation and the balancing preconditioner in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare the well known balancing preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always i e for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the balancing preconditioner even more we establish that both preconditioners lead to almost the same spectra the zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used moreover we prove that the a norm of the errors of the iterates built by the deflation preconditioner is always below the a norm of the errors of the iterates built by the balancing preconditioner depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system if the amount of work is equal both preconditioners are sensitive with respect to inexact computations finally we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors numerical results for porous media flows emphasize the theoretical results

c1 1 regularity of constrained area minimizing hypersurfaces where q c r is an open set and g is a continuous function on xj satisfying the lipschitz condition g p g q 1 p qj for all p q e aq this study was motivated by work of kohn and strang kos who showed that 1 arises as the relaxation of a nonconvex variational problem in optimal design the main purpose of swz was to show that the solution u to 1 can be explicitly constructed by equating the set u 3 r with the solution to the obstacle problem

unstructured additive schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients this paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes the effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a a1 ad and then considering certain positive coefficient sequences a m in which some of these values approach 0 or infty as m rightarrow infty we consider the performance of additive schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid we show that the preconditioned conjugate gradient method converges in both the energy and the euclidean norms with a number of iterations which grows only logarithmically in the size of the maximum jump cal j m max a k m a l m k l 1 ldots d as m rightarrow infty the result is obtained by a careful analysis of the preconditioned matrix it cannot be obtained by the usual procedure of estimating condition numbers a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in cal j m recent results of chan smith and zou have shown that by using this preconditioner together with the conjugate gradient method the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh our results now show that this method is also highly resilient to discontinuous coefficients even if no attention is paid to the coefficient discontinuity in the construction of the solver

ascent of finite cohen macaulay type in 1987 f o schreyer conjectured that a local ring r has finite cohen macaulay type if and only if the completion r has finite cohen macaulay type we prove the conjecture for excellent cohen macaulay local rings and also show by example that it can fail in general

new monotonicity formulas for ricci curvature and applications i we prove three new monotonicity formulas for manifolds with a lower ricci curvature bound and show that they are connected to rate of convergence to tangent cones in fact we show that the derivative of each of these three monotone quantities is bounded from below in terms of the gromov hausdorff distance to the nearest cone the monotonicity formulas are related to the classical bishop gromov volume comparison theorem and perelman s celebrated monotonicity formula for the ricci flow we will explain the connection between all of these moreover we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the green function that we prove this is parallel to the fact that perelman s monotonicity is closely related to the sharp gradient estimate for the heat kernel of li yau in cm4 one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross sections of einstein manifolds finally there are obvious parallelisms between our monotonicity and the positive mass theorem of schoen yau and witten

some results on tight closure and completion abstract we construct two examples of nonexcellent local noetherian domains which demonstrate that tight closure and completion do not commute the first example is a local normal domain a with a height one principal prime ideal p such that pa p a we also construct an example of a complete local normal gorenstein domain which is not f regular but is the completion of an f regular local ring

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

excellent rings with local generic formal fibers abstract let t m be a complete regular local ring of dimension at least 2 containing the rationals such that the cardinality of the residue field t m is at least the cardinality of the real numbers suppose p m is a prime ideal of t we construct an excellent regular local ring a such that the completion of a is t and the generic formal fiber of a is local this means that the ring t a k is a local ring where k is the quotient field of a with p a k its maximal ideal in addition if q is a prime ideal of t with q a p 0 then q pt

mean curvature flow with surgery of mean convex surfaces in mathbb r 3 we define a notion of mean curvature flow with surgery for two dimensional surfaces in mathbb r 3 with positive mean curvature our construction relies on the earlier work of huisken and sinestrari in the higher dimensional case one of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author invent math 2015

rigidity for multi taub nut metrics this paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity it proves that a complete hyperkahler manifold asymptotic to a circle fibration over the euclidean three space is either the standard r 3 s 1 or a multi taub nut manifold in par ticular the underlying complex manifold is either c c z or a minimal resolution of a cyclic kleinian singularity

a coarse space construction based on local dirichlet to neumann maps coarse grid correction is a key ingredient of scalable domain decomposition methods in this work we construct coarse grid space using the low frequency modes of the subdomain dirichlet to neumann maps and apply the obtained two level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

the construction of ale spaces as hyper kahler quotients on decrit la construction d une famille particuliere de 4 varietes hyper kahler les espaces asymptotiquement localement euclidiens ale on decrit une 4 variete de riemann avec juste une extremite qui a l infini ressemble au quotient r 4 de l espace euclidien r 4 par un groupe fini d identifications

a mass for alf manifolds we prove a positive mass theorem on alf manifolds i e complete noncompact manifolds that are asymptotic to a circle fibration over a euclidean base with fibers of asymptotically constant length

on the geometry of the level sets of bounded static potentials in this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity in the case where the static potential is bounded we introduce new quantities which are proven to be monotone along the level set flow of the potential function we then show how to use these properties to detect the rotational symmetry of the static solutions deriving a number of sharp inequalities among these we prove the validity without any dimensional restriction of the riemannian penrose inequality as well as of a reversed version of it in the class of asymptotically flat static metrics with connected horizon as a consequence of our analysis a simple proof of the classical 3 dimensional black hole uniqueness theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions

cotensor products of modules let c be a coalgebra over a field k and a its dual algebra the category of c comodules is equivalent to a category of a modules we use this to interpret the cotensor product m n of two comodules in terms of the appropriate hochschild cohomology of the a bimodule m n when a is finite dimensional profinite graded or differential graded the main applications are to galois cohomology comodules over the steenrod algebra and the homology of induced fibrations

on the approximation of the solution of the pressure equation by changing the domain the purpose of this paper is to study a well known technique for simplifying the pressure and velocity computations in models arising in reservoir simulation and metal casting in both cases the domain of the pressure equation is changed the equations considered are self adjoint second order elliptic problems with coefficient functions representing the permeability of the porous media of concern in the domain modification procedures low permeability areas are either removed or inserted in order to simplify the computations using the theory of sobolev spaces we prove that the approximations converge toward the correct solution as the permeability tends to zero in the proper regions finally we present a numerical experiment which indicates that the estimated rate of convergence is sharp

multilevel ilu decomposition in this paper the multilevel ilu mlilu decomposition is introduced during an incomplete gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels on these levels some smoothing steps are computed the mlilu decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices convergence rates independent of the number of unknowns are shown numerically for several examples many numerical experiments including unsymmetric and anisotropic problems problems with jumping coefficients as well as realistic problems are presented they indicate a very robust convergence behavior of the mlilu method

contracting convex hypersurfaces in riemannian manifolds by their mean curvature which shrink towards the center of the initial sphere in finite time it was shown in 3 that this behaviour is very typical if the initial hypersurface m o o r 1 is uniformly convex then the surfaces m t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction if the ambient space n is a general riemannian manifold the curvature of n will interfere with the mot ion of the surfaces m r we want to show here that the contract ion first to a small sphere and then to a single point is still

a torelli type theorem for gravitational instantons on montre que chaque 4 variete hyper kahler asymptotiquement localement euclidienne et isometrique a un membre d une des familles obtenues dans un article precedent

large scale behavior of kernels of schrodinger operators in this paper we study the asymptotic behavior of solutions to schrodinger equations on manifolds with nonnegative ricci curvature and euclidean volume growth as a first application we get a sharp asymptotic estimate on the green s function for the laplacian which sharpens results of li and yau next we establish a definite gap in the possible rates of decay of exterior harmonic functions on these manifolds when the manifold also has quadratic curvature decay we get supremum estimates on the c 2 norm of the green s function this has function theoretic implications for these manifolds we then give a generalization to manifolds with nonnegative ricci curvature outside a compact set finally we study schrodinger operators where the potential decays quadratically at infinity in this setting we obtain a geometric version at infinity of a classical theorem of bers

on the mass of static metrics with positive cosmological constant ii this is the second of two works in which we discuss the definition of an appropriate notion of mass for static metrics in the case where the cosmological constant is positive and the model solutions are compact in the first part we have established a positive mass statement characterising the de sitter solution as the only static vacuum metric with zero mass in this second part we prove optimal area bounds for horizons of black hole type and of cosmological type corresponding to riemannian penrose inequalities and to cosmological area bounds a la boucher gibbons horowitz respectively building on the related rigidity statements we also deduce a uniqueness result for the schwarzschild de sitter spacetime

support varieties and cohomology over complete intersections quillen s geometric approach to the cohomology of finite groups 30 has revolutionized modular representation theory the ideology and techniques involved have been generalized and extended to representations of various hopf algebras culminating in the recent work of friedlander and suslin 17 on finite group schemes in this paper we develop geometric methods for the study of finite modules over a local complete intersection r if r is such a ring and m is its maximal ideal then the m adic completion r has the form q f where f is a regular sequence and q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring the least number of equations needed to cut out r from a regular local ring equals the codimension of r where codim r r m dim r and r m denotes the minimal number of generators of an r module m in 5 a cone that is a homogeneous algebraic set vr m is attached to each finite r module m and used to study its minimal free resolution here we prove

higher regularity of the inverse mean curvature flow we prove higher regularity properties of inverse mean curvature flow in euclidean space a sharp lower bound for the mean curvature is derived for star shaped surfaces independently of the initial mean curvature it is also shown that solutions to the inverse mean curvature flow are smooth if the mean curvature is bounded from below as a consequence we show that weak solutions of the inverse mean curvature flow are smooth for large times beginning from the first time where a surface in the evolution is star shaped a classical solution of inverse mean curvature flow imcf in euclidean space is a smooth family f n n 0 t r n 1 of regular and

critical sets of elliptic equations given a solution u to a linear homogeneous second order elliptic equation with lipschitz coefficients we introduce techniques for giving improved estimates of the critical set u u x u x 0 as well as the first estimates on the effective critical set r u which roughly consists of points x such that the gradient of u is small somewhere on br x compared to the nonconstancy of u the results are new even for harmonic functions on n given such a u the standard first order stratification lk of u separates points x based on the degrees of symmetry of the leading order polynomial of u u x in this paper we give a quantitative stratification of u which separates points based on the number of almost symmetries of approximate leading order polynomials of u at various scales we prove effective estimates on the volume of the tubular neighborhood of each which lead directly to n 2 minkowski type estimates for the critical set of u with some additional regularity assumptions on the coefficients of the equation we refine the estimate to give new proofs of the uniform n 2 hausdorff measure estimate on the critical set and singular sets of u 2014 wiley periodicals inc

fast and robust solvers for pressure correction in bubbly flow problems we consider the numerical simulation of two phase fluid flow where bubbles or droplets of one phase move against a background of the other phase such flows are governed by the navier stokes equations the solution of which may be approximated using a pressure correction approach within such an approach the computational cost is often dominated by the solution of a linear system corresponding to a discrete poisson equation with discontinuous coefficients in this paper we explore the efficient solution of these linear systems using robust multilevel solvers such as deflated variants of the preconditioned conjugate gradient method or robust multigrid techniques we consider these families of methods in more detail and compare their performance in the simulation of bubbly flows some of these methods turn out to be very effective and reduce the amount of work to solve the pressure correction system substantially resulting in efficient calculations for two phase flows on highly resolved grids

monotonicity and its analytic and geometric implications in this expository article we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of function spaces and the geometry of the underlining spaces are intertwined after briefly discussing some of the well known analytical applications of monotonicity for parabolic operators we turn to their elliptic counterparts their geometric meaning and some geometric consequences

deflated preconditioned conjugate gradient solvers for the pressure poisson equation a deflated preconditioned conjugate gradient technique has been developed for the solution of the pressure poisson equation within an incompressible flow solver the deflation is done using a region based decomposition of the unknowns making it extremely simple to implement the procedure has shown a considerable reduction in the number of iterations for grids with large graph depth the savings exceed an order of magnitude furthermore the technique has shown a remarkable insensitivity to the number of groups regions chosen and to the way the groups are formed

the construction of projection vectors for a deflated iccg method applied to problems with extreme contrasts in the coefficients abstract to predict the presence of oil and natural gas in a reservoir it is important to know the fluid pressure in the rock formations a mathematical model for the prediction of the fluid pressure history is given by a time dependent diffusion equation application of the finite element method leads to systems of linear equations a complication is that the underground consists of layers with very large contrasts in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the iccg method has been observed and a classical termination criterion is not valid in this problem in 19 we have shown that the number of small eigenvalues of the diagonally scaled matrix is equal to the number of high permeability domains which are not connected to a dirichlet boundary in this paper the proof is extended to an incomplete cholesky decomposition to annihilate the bad effect of these small eigenvalues on the convergence the deflated iccg method is used in 19 we have shown how to construct a deflation subspace for the case of a set of more or less parallel layers that subspace proved to be a good approximation of the span of the small eigenvectors as a result of this the convergence of diccg is independent of the contrasts in the permeabilities in this paper it is shown how to construct deflation vectors even in the case of very irregular shaped layers and layers with so called inclusions a theoretical investigation and numerical experiments show that the diccg method is not sensitive to small perturbations of the deflation vectors the efficiency of the diccg method is illustrated by numerical experiments

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete