accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

an extended pressure finite element space for two phase incompressible flows with surface tension we consider a standard model for incompressible two phase flows in which a localized force at the interface describes the effect of surface tension if a level set or vof method is applied then the interface which is implicitly given by the zero level of the level set function is in general not aligned with the triangulation that is used in the discretization of the flow problem this non alignment causes severe difficulties w r t the discretization of the localized surface tension force and the discretization of the flow variables in cases with large surface tension forces the pressure has a large jump across the interface in standard finite element spaces due to the non alignment the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure in many simulations these effects cause large oscillations of the velocity close to the interface so called spurious velocities in this paper for a simplified model problem we give an analysis that explains why known standard methods for discretization of the localized force term and for discretization of the pressure variable often yield large spurious velocities in the paper s grosz a reusken finite element discretization error analysis of a surface tension force in two phase incompressible flows preprint 262 igpm rwth aachen siam j numer anal accepted for publication we introduce a new and accurate method for approximation of the surface tension force in the present paper we use the extended finite element space xfem presented in n moes j dolbow t belytschko a finite element method for crack growth without remeshing int j numer meth eng 46 1999 131 150 t belytschko n moes s usui c parimi arbitrary discontinuities in finite elements int j numer meth eng 50 2001 993 1013 for the discretization of the pressure we show that the size of spurious velocities is reduced substantially provided we use both the new treatment of the surface tension force and the extended pressure finite element space

karoubianness of a triangulated category abstract we prove that triangulated categories with bounded t structures are karoubian consequently for an ext finite abelian category over a commutative noetherian complete local ring its bounded derived category is krull schmidt

a level set approach for diffusion and stefan type problems with robin boundary conditions on quadtree octree adaptive cartesian grids we present a numerical method for simulating diffusion dominated phenomena on irregular domains and free moving boundaries with robin boundary conditions on quadtree octree adaptive meshes in particular we use a hybrid finite difference and finite volume framework that combines the level set finite difference discretization of min and gibou 2007 13 with the treatment of robin boundary conditions of papac et al 2010 19 on uniform grids we present numerical results in two and three spatial dimensions on the diffusion equation and on a stefan type problem in addition we present an application of this method to the case of the simulation of the ehrlich schwoebel barrier in the context of epitaxial growth

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

a comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems in l adams and z li siam j sci comput 24 2002 pp 463 479 a multigrid method was designed specifically for interface problems that have been discretized using the methods described in l adams and z li siam j sci comput 24 2002 pp 463 479 and in z li and k ito siam j sci comput 23 2001 pp 339 361 for elliptic interface problems using the maximum principle preserving schemes in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 a new method was introduced that utilizes a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse grid matrices are m matrices this paper compares the immersed interface multigrid methods introduced in l adams and z li siam j sci comput 24 2002 pp 463 479 and l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 with algebraic multigrid which uses no geometric information to set up the multigrid components for coarse grid correction we show that algebraic multigrid is a robust solver for our test problems it outperforms the method in l adams and z li siam j sci comput 24 2002 pp 463 479 and performs nearly as well as the method in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 which is shown to be the most efficient for all problem parameters and sizes

the happel functor and homologically well graded iwanaga gorenstein algebras abstract happel constructed a fully faithful functor h d b mod mod z t for a finite dimensional algebra he also showed that this functor h gives an equivalence precisely when gldim thus if h gives an equivalence then it provides a canonical tilting object h of mod z t in this paper we generalize the happel functor h in the case where t is replaced with a finitely graded ig algebra a we study when this functor is fully faithful or is an equivalence for this purpose we introduce the notion of homologically well graded hwg ig algebra which can be characterized as an algebra posses a homological symmetry which a posteriori guarantee that the algebra is ig we prove that hwg ig algebras is precisely the class of finitely graded ig algebras that the happel functor is fully faithful we also identify the class that the happel functor gives an equivalence as a consequence of our result we see that if h gives an equivalence then it provides a canonical tilting object h t of cm z a for some special classes of finitely graded ig algebras our tilting objects h t coincide with tilting object constructed in previous works

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

tensor decompositions for learning latent variable models this work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models including gaussian mixture models hidden markov models and latent dirichlet allocation which exploits a certain tensor structure in their low order observable moments typically of second and third order specifically parameter estimation is reduced to the problem of extracting a certain orthogonal decomposition of a symmetric tensor derived from the moments this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices although tensor decompositions are generally intractable to compute the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches including power iterations and maximization approaches similar to the case of matrices a detailed analysis of a robust tensor power method is provided establishing an analogue of wedin s perturbation theorem for the singular vectors of matrices this implies a robust and computationally tractable estimation approach for several popular latent variable models

new enhancements of derived categories of coherent sheaves and applications abstract we introduce new enhancements for the bounded derived category d b coh x of coherent sheaves on a suitable scheme x and for its subcategory p e r f x of perfect complexes they are used for translating fourier mukai functors to functors between derived categories of dg algebras for relating homological smoothness of p e r f x to geometric smoothness of x and for proving homological smoothness of d b coh x moreover we characterize properness of p e r f x and d b coh x geometrically

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

tilting modules and auslander s gorenstein property abstract the notion of generalized tilting modules is introduced over arbitrary rings it is shown that many results on tilting modules over artin algebras are naturally generalized the auslander s gorenstein property for tilting modules is considered over noetherian rings

xgfm recovering convergence of fluxes in the ghost fluid method abstract we introduce a simple but effective correction to the ghost fluid method gfm introduced by liu et al 2000 30 for capturing sharp interface conditions when solving poisson equations with discontinuities our method is shown to alleviate the gfm lack of convergence for the flux across the interfaces pointwise convergence of the gradient is recovered by ensuring consistent jump conditions with an iterative process compared to the gfm the discretization stencil and the symmetric positive definiteness of the resulting linear system are not modified only the right hand side of the linear system is affected we first show that the original gfm is consistent with the second order accurate symmetric scheme of gibou et al 2002 17 for dirichlet conditions in separate regions of the computational domain where the parameters and the solution are smooth but with matching conditions at the interface that are not fully consistent in general our proposed method recovers consistency and is shown to produce convergent gradients through several numerical examples in two and three spatial dimensions

efficient symmetric positive definite second order accurate monolithic solver for fluid solid interactions we introduce a robust and efficient method to simulate strongly coupled monolithic fluid rigid body interactions we take a fractional step approach where the intermediate state variables of the fluid and of the solid are solved independently before their interactions are enforced via a projection step the projection step produces a symmetric positive definite linear system that can be efficiently solved using the preconditioned conjugate gradient method in particular we show how one can use the standard preconditioner used in standard fluid simulations to precondition the linear system associated with the projection step of our fluid solid algorithm overall the computational time to solve the projection step of our fluid solid algorithm is similar to the time needed to solve the standard fluid only projection step the monolithic treatment results in a stable projection step i e the kinetic energy does not increase in the projection step numerical results indicate that the method is second order accurate in the l norm and demonstrate that its solutions agree quantitatively with experimental results

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

rings with auslander dualizing complexes abstract a ring with an auslander dualizing complex is a generalization of an auslander gorenstein ring we show that many results which hold for auslander gorenstein rings also hold in the more general setting on the other hand we give criteria for existence of auslander dualizing complexes which show these occur quite frequently the most powerful tool we use is the local duality theorem for connected graded algebras over a field filtrations allow the transfer of results to nongraded algebras we also prove some results of a categorical nature most notably the functoriality of rigid dualizing complexes

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

a probabilistic analysis of em for mixtures of separated spherical gaussians we show that given data from a mixture of k well separated spherical gaussians in d a simple two round variant of em will with high probability learn the parameters of the gaussians to near optimal precision if the dimension is high d ln k we relate this to previous theoretical and empirical work on the em algorithm

on convergence properties of the em algorithm for gaussian mixtures we build up the mathematical connection between the expectation maximization em algorithm and gradient based approaches for maximum likelihood learning of finite gaussian mixtures we show that the em step in parameter space is obtained from the gradient via a projection matrix p and we provide an explicit expression for the matrix we then analyze the convergence of em in terms of special properties of p and provide new results analyzing the effect that p has on the likelihood surface based on these mathematical results we present a comparative discussion of the advantages and disadvantages of em and other algorithms for the learning of gaussian mixture models

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

matched interface and boundary mib method for elliptic problems with sharp edged interfaces elliptic problems with sharp edged interfaces thin layered interfaces and interfaces that intersect with geometric boundary are notoriously challenging to existing numerical methods particularly when the solution is highly oscillatory this work generalizes the matched interface and boundary mib method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems we classify these problems into five distinct topological relations involving the interfaces and the cartesian mesh lines flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point the concepts of primary and secondary fictitious values are introduced to deal with sharp edged interfaces for corner singularity or tip singularity an appropriate polynomial is multiplied to the solution to remove the singularity extensive numerical experiments confirm the designed second order convergence of the proposed method

poisson equations in irregular domains with robin boundary conditions solver with second order accurate gradients abstract we present a conservative method for solving the poisson equation in irregular domains with robin boundary conditions second order accurate solutions and gradients in the l norm are obtained

relative singularity categories i auslander resolutions abstract let r be an isolated gorenstein singularity with a non commutative resolution a end r r m in this paper we show that the relative singularity category r a of a has a number of pleasant properties such as being hom finite moreover it determines the classical singularity category d s g r of buchweitz and orlov as a certain canonical quotient category if r has finite cm type which includes for example kleinian singularities then we show the much more surprising result that d s g r determines r aus r where aus r is the corresponding auslander algebra the proofs of these results use dg algebras a koszul duality and the new concept of dg auslander algebras which may be of independent interest

derived equivalences via hrs tilting abstract let a be an abelian category and b be the happel reiten smalo tilt of a with respect to a torsion pair we give necessary and sufficient conditions for the existence of a derived equivalence between a and b which is compatible with the inclusion of b into the derived category of a as applications we obtain new derived equivalences related to splitting torsion pairs ttf triples and two term silting subcategories respectively we prove that for the realization functor of any bounded t structure its denseness implies its fully faithfulness

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

generalized serre duality abstract we introduce a notion of generalized serre duality on a hom finite krull schmidt triangulated category t this duality induces the generalized serre functor on t which is a linear triangle equivalence between two thick triangulated subcategories of t moreover the domain of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the third term of an auslander reiten triangle in t dually the range of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the first term of an auslander reiten triangle in t we compute explicitly the generalized serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of artin and zhang we obtain a characterization of gorenstein algebras an artin algebra a is gorenstein if and only if the bounded homotopy category of finitely generated projective a modules has serre duality in the sense of bondal and kapranov

finite difference ghost point multigrid methods on cartesian grids for elliptic problems in arbitrary domains in this paper we present a numerical method for solving elliptic equations in an arbitrary domain described by a level set function with general boundary conditions dirichlet neumann robin etc on cartesian grids using finite difference discretization and non eliminated ghost values a system of n i n g equations in n i n g unknowns is obtained by finite difference discretization on the n i internal grid points and second order interpolation to define the conditions for the n g ghost values the resulting large sparse linear system is then solved by a multigrid technique the novelty of the papers can be summarized as follows general strategy to discretize the boundary condition to second order both in the solution and its gradient a relaxation of inner equations and boundary conditions by a fictitious time method inspired by the stability conditions related to the associated time dependent problem with a convergence proof for the first order scheme an effective geometric multigrid which maintains the structure of the discrete system at all grid levels it is shown that by increasing the relaxation step of the equations associated to the boundary conditions a convergence factor close to the optimal one is obtained several numerical tests including variable coefficients anisotropic elliptic equations and domains with kinks show the robustness efficiency and accuracy of the approach

localization of triangulated categories and derived categories the notion of quotient and localization of abelian categories by dense subcategories i e serre classes was introduced by gabriel and plays an important role in ring theory 6 131 the notion of triangulated categories was introduced by grothendieck and developed by verdier 9 16 and is recently useful in representation theory 8 4 143 the quotient of triangulated categories by epaisse subcategories is constructed in 16 both quotients were indicated by grothendieck and they resemble each other in this paper we will consider triangulated categories and derived categories from the point of view of localization of abelian categories verdier gave a condition which is equivalent to the one that a quotient functor has a right adjoint and considered a relation between tpaisse subcategories 16 we show that localization of triangulated categories is similarly defined and have a relation between localizations and cpaisse subcategories beilinson bernstein and deligne introduced the notion of t structure which is similar to torsion theory in abelian categories 2 we in particular consider a stable t structure which is an epaisse subcategory and deal with a correspondence between localizations of triangulated categories and stable r structures and then recollement in the sense of 2 is equivalent to bilocalization next we show that quotient and localization of abelian categories induce quotient and localization of its derived categories in section 1 we recall standard notations and terminologies of quotient and localization of abelian categories in section 2 we define localization of triangulated categories and consider a relation between localizations and stable r structures theorem 2 6 in section 3 we show that if a a c is a quotient of abelian categories then d a d a c is a quotient of triangulated categories where or b theorem 3 2 moreover

realisation functors in tilting theory derived equivalences and t structures are closely related we use realisation functors associated to t structures in triangulated categories to establish a derived morita theory for abelian categories with a projective generator or an injective cogenerator for this purpose we develop a theory of non compact or large tilting and cotilting objects that generalises the preceding notions in the literature within the scope of derived morita theory for rings we show that under some assumptions the realisation functor is a derived tensor product this fact allows us to approach a problem by rickard on the shape of derived equivalences finally we apply the techniques of this new derived morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t structures glueing to a tilting or a cotilting t structure as a further application we answer a question by xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras

recollements idempotent completions and t structures of triangulated categories abstract we first prove that the idempotent completion of a right or left recollement of triangulated categories is still a right or left recollement then show that the t structure on a triangulated category is compatible with taking idempotent completion finally an application of the main theorem is given which is focused on the boundedness and nondegeneration of the t structure induced by a recollement and its idempotent completion

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

tilting modules of finite projective dimension tilting modules are well known and are useful in the representation theory of artinian algebras 2 4 6 11 the purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results a unital left a module a t over a ring a is said to be a tilting module of projective dimension 0 is an integer if it satisfies the following three conditions 1 a t has a projective resolution

new geometric immersed interface multigrid solvers in l adams and z li siam j sci comput 24 2002 pp 463 479 a multigrid method was designed specifically for interface problems that have been discretized using the methods described therein and in z li and k ito siam j sci comput 23 2001 pp 339 361 for elliptic interface problems using the maximum principle preserving schemes in this paper we improve on this method by giving a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse grid matrices are m matrices we compare this new restrictor to injection and the transpose of interpolation we show that the number of v cycles is constant as the mesh size decreases and increases only slightly as the ratio of the discontinuous problem coefficient grows at the interface only when this new restrictor is used

an auslander buchweitz approximation approach to pre silting subcategories in triangulated categories abstract we apply the auslander buchweitz approximation triangles to study pre silting subcategories in a triangulated catego ry t an auslander reiten type correspondence between the class of silting subcategories of t and that of certain covariantly finite subcategories of t is established we introduce a relation on presilting subcategories of t which can be used to obtain another silting subcategory from a given one we also give a bazzoni s characterization for two presilting subcategories satisfying such a relation the results can be used to improve the auslander reiten type correspondence and bazzoni s characterization for small silting complexes established by wei

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

equivariant characteristic classes of singular complex algebraic varieties homology hirzebruch characteristic classes for singular varieties have been recently defined by brasselet schuurmann and yokura as an attempt to unify previously known characteristic class theories for singular spaces e g macpherson chern classes baum fulton macpherson todd classes and goresky macpherson l classes in this paper we define equivariant analogues of these classes for singular quasi projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology hirzebruch classes of global quotient varieties we also compute the new classes in the context of monodromy problems e g for varieties that fiber equivariantly in the complex topology over a connected algebraic manifold as another application we discuss atiyah meyer type formulae for twisted hirzebruch classes of global orbifolds 2012 wiley periodicals inc

generalized finite element systems for smooth differential forms and stokes problem we provide both a general framework for discretizing de rham sequences of differential forms of high regularity and some examples of finite element spaces that fit in the framework the general framework is an extension of the previously introduced notion of finite element systems and the examples include conforming mixed finite elements for stokes equation in dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes in dimension 3 we define one low order complex which may be branched into whitney forms at a chosen index stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension the finite element spaces all consist of composite polynomials the framework guarantees some nice properties of the spaces in particular the existence of commuting interpolators it also shows that some of the examples are minimal spaces

recollements and stratifying ideals abstract surjective homological epimorphisms with stratifying kernel can be used to construct recollements of derived module categories these stratifying recollements are derived from recollements of module categories can every recollement be put in this form up to equivalence a negative answer will be given after providing a characterisation of recollements equivalent to stratifying ones moreover criteria for a ring epimorphism to be stratifying will be presented as well as constructions of such epimorphisms

very high order compact finite difference schemes on non uniform grids for incompressible navier stokes equations this article presents a family of very high order non uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible navier stokes equations the high order compact schemes on non uniform grids developed in shukla and zhong r k shukla x zhong derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation j comput phys 204 2005 404 for linear model equations are extended to the full navier stokes equations in the vorticity and streamfunction formulation two methods for the solution of helmholtz and poisson equations using high order compact schemes on non uniform grids are developed the schemes are constructed so that they maintain a high order of accuracy not only in the interior but also at the boundary second order semi implicit temporal discretization is achieved through an implicit backward differentiation scheme for the linear viscous terms and an explicit adam bashforth scheme for the non linear convective terms the boundary values of vorticity are determined using an influence matrix technique the resulting discretized system with boundary closures of the same high order as the interior is shown to be stable when applied to the two dimensional incompressible navier stokes equations provided enough grid points are clustered at the boundary the resolution characteristics of the high order compact finite difference schemes are illustrated through their application to the one dimensional linear wave equation and the two dimensional driven cavity flow comparisons with the benchmark solutions for the two dimensional driven cavity flow thermal convection in a square box and flow past an impulsively started cylinder show that the high order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary

analysis of a new space time parallel multigrid algorithm for parabolic problems we present and analyze a new space time parallel multigrid method for parabolic equations the method is based on arbitrarily high order discontinuous galerkin discretizations in time and a finite element discretization in space the key ingredient of the new algorithm is a block jacobi smoother we present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters a precise criterion for semi coarsening in time or full coarsening and give an asymptotic two grid contraction factor estimate we then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties

solving poisson type equations with robin boundary conditions on piecewise smooth interfaces abstract we present two finite volume schemes to solve a class of poisson type equations subject to robin boundary conditions in irregular domains with piecewise smooth boundaries the first scheme results in a symmetric linear system and produces second order accurate numerical solutions with first order accurate gradients in the l norm for solutions with two bounded derivatives the second scheme is nonsymmetric but produces second order accurate numerical solutions as well as second order accurate gradients in the l norm for solutions with three bounded derivatives numerical examples are given in two and three spatial dimensions

statistical guarantees for the em algorithm from population to sample based analysis we develop a general framework for proving rigorous guarantees on the performance of the em algorithm and a variant known as gradient em our analysis is divided into two parts a treatment of these algorithms at the population level in the limit of infinite data followed by results that apply to updates based on a finite set of samples first we characterize the domain of attraction of any global maximizer of the population likelihood this characterization is based on a novel view of the em updates as a perturbed form of likelihood ascent or in parallel of the gradient em updates as a perturbed form of standard gradient ascent leveraging this characterization we then provide non asymptotic guarantees on the em and gradient em algorithms when applied to a finite set of samples we develop consequences of our general theory for three canonical examples of incompletedata problems mixture of gaussians mixture of regressions and linear regression with covariates missing completely at random in each case our theory guarantees that with a suitable initialization a relatively small number of em or gradient em steps will yield with high probability an estimate that is within statistical error of the mle we provide simulations to confirm this theoretically predicted behavior

second order finite difference ghost point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface abstract in this paper we propose a second order accurate numerical method to solve elliptic problems with discontinuous coefficients with general non homogeneous jumps in the solution and its gradient in 2d and 3d the method consists of a finite difference method on a cartesian grid in which complex geometries boundaries and interfaces are embedded and is second order accurate in the solution and the gradient itself in order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface two numerical values are assigned on grid points that are close to the interface a real value that represents the numerical solution on that grid point and a ghost value that represents the numerical solution extrapolated from the other side of the interface obtained by enforcing the assigned non homogeneous jump conditions on the solution and its flux the method is also extended to the case of matrix coefficient the linear system arising from the discretization is solved by an efficient multigrid approach unlike the 1d case grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient a proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor comparable with the one obtained by local fourier analysis for rectangular domains the method is robust enough to handle large jump in the coefficients order of accuracy monotonicity of the errors and good convergence factor are maintained by the scheme

finite element methods for the stokes problem on complicated domains abstract it is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process in case of complicated domains which appear in many applications such as ground water flows this requirement sometimes becomes a bottleneck the resolution condition links the computational complexity to the number and size of geometric details although the accuracy requirements possibly are moderate and would allow a locally coarse mesh width therefore even the coarsest available discretization can lead to a huge number of unknowns the composite mini element is a remedy to this dilemma because the degrees of freedom are not linked to the number of geometric details the basic concept for the stokes problem with uniform no slip boundary conditions has been introduced and analyzed in d peterseim s sauter the composite mini element coarse mesh computation of stokes flows on complicated domains sinum 46 6 2008 3181 3206 here we generalize the composite mini element to slip leak and neumann boundary conditions so that it becomes applicable to this much larger and more important problem class the main results are a the algorithmic concept remains unchanged and the new boundary conditions can be implemented as a weighted quadrature rule b the stability and convergence can be proved under very mild assumption on the domain geometries c the analysis is far from trivial and requires the development of substantially new tools compared to the simple case of uniform no slip boundary conditions

cohen macaulay differential graded modules and negative calabi yau configurations abstract in this paper we introduce the class of cohen macaulay cm dg differential graded modules over gorenstein dg algebras and study their basic properties we show that the category of cm dg modules forms a frobenius extriangulated category in the sense of nakaoka and palu and it admits almost split extensions we also study representation finite d self injective dg algebras a in detail for some positive integer d in particular we classify the auslander reiten ar quivers of cm a for a large class of d self injective dg algebras a in terms of d calabi yau cy configurations which are riedtmann s configurations for the case d 1 for any given d cy configuration c we show there exists a d self injective dg algebra a such that the ar quiver of cma is given by c for type a n by using a bijection between d cy configurations and certain purely combinatorial objects which we call maximal d brauer relations given by coelho simoes we construct such a through a brauer tree dg algebra

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

solving elliptic interface problems with jump conditions on cartesian grids abstract we present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient the source term the solution and its flux are discontinuous across an irregular interface the algorithm produces second order accurate solutions and first order accurate gradients in the l norm on cartesian grids the condition number is bounded regardless of the ratio of the diffusion constant and scales like that of the standard 5 point stencil approximation on a rectangular grid with no interface numerical examples are given in two and three spatial dimensions

higher apr tilting preserves n representation infiniteness abstract we show that m apr tilting preserves n representation infiniteness for 1 m n moreover we show that these tilting modules lift to tilting modules for the corresponding higher preprojective algebras which is n 1 cy algebras we also study the interplay of the two kinds of tilting modules

cohen macaulay approximation in fibred categories abstract we extend the auslander buchweitz axioms and prove cohen macaulay approximation results for fibred categories we show that these axioms apply for the fibred category of pairs consisting of a finite type flat family of cohen macaulay rings and modules in particular such a pair admits an approximation with a flat family of maximal cohen macaulay modules and a hull with a flat family of modules with finite injective dimension the existence of minimal approximations and hulls in the local flat case implies extension of upper semi continuous invariants as an example of mcm approximation we define a relative version of auslander s fundamental module

dg quotients of dg categories abstract keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with verdier s notion of quotient of a triangulated category modulo a triangulated subcategory this work is an attempt to further develop his theory

a space time galerkin least squares finite element formulation of the navier stokes equations for moving domain problems a space time galerkin least squares finite element formulation of the navier stokes equations is presented for the analysis of free surface flows moving spatial configurations and deforming fluid structure interfaces the variational equation is based on the time discontinuous galerkin method employing the physical entropy variables the space time elements are oriented in time to accommodate the spatial deformations if the elements are oriented along the particle paths the formulation is lagrangian and if they are fixed in time it is eulerian consequently this formulation is analogous to the arbitrary lagrangian eulerian ale technique a novel mesh rezoning strategy is presented to orient the elements in time and adapt the fluid mesh to the changing spatial configuration numerical results are presented to show the performance of the method

a dimensionally split cartesian cut cell method for the compressible navier stokes equations abstract we present a dimensionally split method for computing solutions to the compressible navier stokes equations on cartesian cut cell meshes the method is globally second order accurate in the l 1 norm fully conservative and allows the use of time steps determined by the regular grid spacing we provide a description of the three dimensional implementation of the method and evaluate its numerical performance by computing solutions to a number of test problems ranging from the nearly incompressible to the highly compressible flow regimes all the computed results show good agreement with reference results from theory experiment and previous numerical studies to the best of our knowledge this is the first presentation of a dimensionally split cut cell method for the compressible navier stokes equations in the literature

an efficient fluid solid coupling algorithm for single phase flows we present a simple and efficient fluid solid coupling method in two and three spatial dimensions in particular we consider the numerical approximation of the navier stokes equations on irregular domains and propose a novel approach for solving the hodge projection step on arbitrary shaped domains this method is straightforward to implement and leads to a symmetric positive definite linear system for both the projection step and for the implicit treatment of the viscosity we demonstrate the accuracy of our method in the l 1 and l norms and present its removing the errors associated with the conventional rasterization type discretizations we apply this method to the simulation of a flow past a cylinder in two spatial dimensions and show that our method can reproduce the known stable and unstable regimes as well as correct lift and drag forces we also apply this method to the simulation of a flow past a sphere in three spatial dimensions at low and moderate reynolds number to reproduce the known steady axisymmetric and non axisymmetric flow regimes we further apply this algorithm to the coupling of flows with moving rigid bodies

t structures and cotilting modules over commutative noetherian rings for a commutative noetherian ring r we establish a bijection between the resolving subcategories consisting of finitely generated r modules of finite projective dimension and the compactly generated t structures in the unbounded derived category mathcal d r that contain r 1 in their heart under this bijection the t structures mathcal u mathcal v such that the aisle mathcal u consists of objects with homology concentrated in degrees n correspond to the n cotilting classes in mathrm mod text r as a consequence of these results we prove that the little finitistic dimension findim r of r equals an integer n if and only if the direct sum bigoplus k 0 n e k r of the first n 1 terms in a minimal injective coresolution 0 rightarrow r rightarrow e 0 r rightarrow e 1 r rightarrow cdots of r is an injective cogenerator of mathrm mod text r

silting reduction and calabi yau reduction of triangulated categories it is shown that the silting reduction t thickp of a triangulated category t with respect to a presilting subcategory p can be realised as a certain subfactor category of t and that there is a one to one correspondence between the set of pre silting subcategories of t containing p and the set of pre silting subcategories of t thickp this is analogous to a result for calabi yau reduction this result is applied to show that the amiot guo keller construction of d calabi yau triangulated categories with d cluster tilting objects takes silting reduction to calabi yau reduction and conversely calabi yau reduction lifts to silting reduction