subdivisions and local h vectors in part i a general theory of f vectors of simplicial subdivisions or triangulations of simplicial complexes is developed based on the concept of lo cal h vector as an application we prove that the h vector of a cohen macaulay complex increases under quasi geometric subdivision thus establishing a spe cial case of a conjecture of kalai and this author techniques include com mutative algebra homological algebra and the intersection homology of toric varieties in part ii we extend the work of part i to more general situations first a formal generalization of subdivision is given based on incidence alge bras special cases are then developed in particular one based on subdivisions of eulerian posets and involving generalized h vectors other cases deal with kazhdan lusztig polynomials ehrhart polynomials and a q analogue of eule rian posets many applications and examples are given throughout department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address rstan math mit edu this content downloaded from 157 55 39 210 on sat 30 jul 2016 05 59 16 utc all use subject to http about jstor org terms

handling solid fluid interfaces for viscous flows explicit jump approximation vs ghost cell approaches the ghost cell approaches gca for handling stationary solid boundaries regular or irregular are first investigated theoretically and numerically for the diffusion equation with dirichlet boundary conditions the main conclusion of this part of investigation is that the approximation for the diffusion term has to be second order accurate everywhere in order for the numerical solution to be rigorously second order accurate violating this principle the linear and quadratic gcas have the following shortcomings 1 restrictive constraints on grid size when the viscosity is small 2 susceptibleness to instability of a time explicit formulation for strongly transient flows 3 convergence deterioration to zeroth or first order for solutions with high frequency modes therefore the widely used linear extrapolation for enforcing no slip boundary conditions should be avoided even for regular solid boundaries as a remedy a simple method based on explicit jump approximation eja is proposed eja hinges on the idea that a velocity derivative jump at the boundary reduces to the value of the velocity derivative at the fluid side because the velocity of the stationary boundary is zero although the time marching linear system of eja is not symmetric it is strictly diagonal dominant with positive diagonal entries numerical results show that over a large range of viscosity and grid sizes eja performs much better than gcas in terms of stability and accuracy furthermore the second order convergence of eja does not depend on viscosity and the spectrum of the solution as those of gcas do this paper is written with enough details so that one can reproduce the numerical results

some koszul rings from geometry abstract we give examples of koszul rings that arise naturally in algebraic geometry in the first part we prove a general result on koszul property associated to an ample line bundle on a projective variety specifically we show how koszul property of multiples of a base point free ample line bundle depends on its castelnuovo mumford regularity in the second part we give examples of koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type

equivariant constructible koszul duality for dual toric varieties abstract for affine toric varieties x and x defined by dual cones we define an equivalence of categories between mixed versions of the equivariant derived category d t b x and the derived category of sheaves on x which are locally constant with unipotent monodromy on each orbit this equivalence satisfies the koszul duality formalism of beilinson ginzburg and soergel

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

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

boosting the accuracy of finite difference schemes via optimal time step selection and non iterative defect correction abstract in this article we present a unified analysis of the simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations pdes by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non iterative manner the power of the technique which is applicable to time dependent semilinear scalar pdes where the leading order spatial derivative has a constant coefficient is its ability to increase the accuracy of formally low order finite difference schemes without major modification to the basic numerical algorithm through straightforward numerical analysis arguments we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method we demonstrate the utility of optimal time step ots selection combined with non iterative defect correction nidc on several different types of finite difference schemes for a wide array of classical linear and semilinear pdes in one and more space dimensions on both regular and irregular domains

whittaker coinvariants for gl m n abstract let w m n be the finite w algebra attached to the principal nilpotent orbit in the general linear lie superalgebra gl m n c in this paper we study the whittaker coinvariants functor which is an exact functor from category o for gl m n c to a certain category of finite dimensional modules over w m n we show that this functor has properties similar to soergel s functor v in the setting of category o for a semisimple lie algebra we also use it to compute the center of w m n explicitly and deduce consequences for the classification of blocks of o up to morita derived equivalence

graded decomposition numbers for the blob algebra abstract we give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity

koszul duality for toric varieties we show that certain categories of perverse sheaves on affine toric varieties x and x v defined by dual cones are koszul dual in the sense of beilinson ginzburg and soergel 1996 the functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties

highest weight sl_2 categorifications ii structure theory this paper continues the study of highest weight categorical sl2 actions initi ated in part i we start by refining the definition given there and showing that all examples considered in part i are also highest weight categorifications in the refined sense then we prove that any highest weight sl2 categorification can be filtered in such a way that the suc cessive quotients are so called basic highest weight sl2 categorifications for a basic highest weight categorification we determine minimal projective resolutions of standard objects we use this in particular to examine the structure of tilting objects in basic categorifications and to show that the ringel duality is given by the rickard complex we apply some of these structural results to categories o for cyclotomic rational cherednik algebras we finish by discussing open problems

higher order cartesian grid based finite difference schemes for elliptic equations on irregular domains second and fourth order cartesian grid based finite difference methods are proposed for elliptic and parabolic partial differential equations and associated eigenvalue problems on irregular domains with general boundary conditions our methods are based on the continuation of a solution idea using multivariable taylor s expansion of the solution about selected boundary points and the core ideas of the immersed interface method the methods offer systematic treatment of the general boundary conditions in two and three dimensional domains and are directly applied to semi discretize heat equations on irregular domains convergence analysis and numerical examples are presented the validity and effectiveness of the proposed methods are demonstrated through our numerical results including computations of the eigenvalues of the associated eigenvalue problem

a least square extrapolation method for improving solution accuracy of pde computations richardson extrapolation re is based on a very simple and elegant mathematical idea that has been successful in several areas of numerical analysis such as quadrature or time integration of odes in theory re can be used also on pde approximations when the convergence order of a discrete solution is clearly known but in practice the order of a numerical method often depends on space location and is not accurately satisfied on different levels of grids used in the extrapolation formula we propose in this paper a more robust and numerically efficient method based on the idea of finding automatically the order of a method as the solution of a least square minimization problem on the residual we introduce a two level and three level least square extrapolation method that works on nonmatching embedded grid solutions via spline interpolation our least square extrapolation method is a post processing of data produced by existing pde codes that is easy to implement and can be a better tool than re for code verification it can be also used to make a cascade of computation more numerically efficient we can establish a consistent linear combination of coarser grid solutions to produce a better approximation of the pde solution at a much lower cost than direct computation on a finer grid to illustrate the performance of the method examples including two dimensional turning point problem with sharp transition layer and the navier stokes flow inside a lid driven cavity are adopted

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

a note on tests for exponentiality by deshpande summary recently deshpande 1983 has proposed a class of simple tests for testing exponentiality against the increasing failure rate average class of nonexponential probability distributions these tests have a number of desirable properties however they suffer from some omissions the object of this note is to provide the missing arguments so that the paper can be more useful

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

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

perverse sheaves and actions where t x denotes the action of t on x the set xw is known to be a locally closed c stable algebraic subvariety of x isomorphic to an affine space the pieces xw form a cell decomposition x uwewxw w e w the socalled bialynicki birula decomposition we assume this decomposition to be an algebraic stratification of x the closure of a cell may not be a union of cells in general let zw be the intersection cohomology complex on the closure of a cell xw extended by 0 to the whole of x the hypercohomology h 2w has a natural structure of a graded h x c module further for any x y e w there is a natural morphism of graded spaces

a new fourth order compact scheme for the navierstokes equations in irregular domains we present a high order finite difference scheme for navierstokes equations in irregular domains the scheme is an extension of a fourth order scheme for navierstokes equations in streamfunction formulation on a rectangular domain ben artzi etal 2010 the discretization offered here contains two types of interior points the first is regular interior points where all eight neighboring points of a grid point are inside the domain and not too close to the boundary the second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary in the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes using discretizations of pure derivatives in the x y and along the diagonals of the element

a numerical study of electro migration voiding by evolving level set functions on a fixed cartesian grid a numerical method for studying migration of voids driven by surface diffusion and electric current in a metal conducting line is developed the mathematical model involves moving boundaries governed by a fourth order nonlinear partial differential equation which contains a nonlocal term corresponding to the electrical field and a nonlinear term corresponding to the curvature numerical challenges include efficient computation of the electrical field with sufficient accuracy to afford fourth order differentiation along the void boundary and to capture singularities arising in topological changes we use the modified immersed interface method with a fixed cartesian grid to solve for the electrical field and the fast local level set method to update the position of moving voids numerical examples are performed to demonstrate the physical mechanisms by which voids interact under electromigration

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

a class of tests for exponentiality against increasing failure rate average alternatives a class of tests is proposed for testing exponentiality against the increasing failure rate average class of nonexponential probability distributions some of these tests are consistent as well for the larger class new better than used compared to its competitors these statistics are remarkably simple they are u statistics and hence asymptotically normally distributed the asymptotic relative efficiency with respect to the hollanderproschan statistic and the cumulative total time on test statistic are reasonably high

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

mirror duality and string theoretic hodge numbers we prove in full generality the mirror duality conjecture for string theoretic hodge numbers of calabi yau complete intersections in gorenstein toric fano varieties the proof is based on properties of intersection cohomology

two block springer fibers of types c and d a diagrammatic approach to springer theory we explain an elementary topological construction of the springer representation on the homology of topological springer fibers of types c and d in the case of nilpotent endomorphisms with two jordan blocks the weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types b and d we determine the decomposition of the representations into irreducibles and relate our construction to classical springer theory as an application we obtain presentations of the cohomology rings of all two block springer fibers of types c and d moreover we deduce explicit isomorphisms between the kazhdan lusztig cell modules attached to the induced trivial module and the irreducible specht modules in types c and d

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

on distribution free goodness of fit testing of exponentiality there is a need to test the hypothesis of exponentiality against a wide variety of alternative hypotheses across many areas of economics and finance local or contiguous alternatives are the closest alternatives against which it is still possible to have some power hence goodness of fit tests should have some power against all or a huge majority of local alternatives such tests are often based on nonlinear statistics with a complicated asymptotic null distribution thus a second desirable property of a goodness of fit test is that its statistic will be asymptotically distribution free we suggest a whole class of goodness of fit tests with both of these properties by constructing a new version of empirical process that weakly converges to a standard brownian motion under the hypothesis of exponentiality all statistics based on this process will asymptotically behave as statistics from a standard brownian motion and so will be asymptotically distribution free we show the form of transformation is especially simple in the case of exponentiality surprisingly there are only two asymptotically distribution free versions of empirical process for this problem and only this one has a convenient limit distribution many tests of exponentiality have been suggested based on asymptotically linear functionals from the empirical process we illustrate none of these can be used as goodness of fit tests contrary to some previous recommendations of considerable interest is that a selection of well known statistics all lead to the same test asymptotically with negligible asymptotic power against a great majority of local alternatives finally we present an extension of our approach that solves the problem of multiple testing both for exponentiality and for other more general hypotheses

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

modeling water transport across elastic boundaries using an explicit jump method a mathematical model is presented to simulate water and solute transport in a highly viscous fluid with a water permeable elastic immersed membrane in this model fluid motion is described by stokes flow whereas water fluxes across the membrane are driven by transmural pressure and solute concentration differences the elastic forces arising from the membrane being distorted from its relaxed configuration and the transmembrane water fluxes introduce into model solutions discontinuities across the membrane such discontinuities are faithfully captured using a second order explicit jump method a mayo siam j numer anal 21 1984 pp 285 299 in which jumps in the solution and its derivatives are incorporated into a finite difference scheme numerical results suggest that the method exhibits desirable volume accuracy and mass conservation

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

an efficient numerical method for the two fluid stokes equations with a moving immersed boundary we consider the immersed boundary problem in which the boundary separates two very viscous fluids with differing viscosities the moving elastic boundary may exert a force on the local fluid the model solution is obtained using the immersed interface method which computes second order accurate approximations by incorporating known jumps in the solution or its derivatives into a finite difference method these jump conditions become coupled when the fluid viscosity has a jump across the boundary and this coupling renders the application of the immersed interface method challenging we present a method that first uses boundary integral equations to reduce the two fluid stokes problem to the single fluid case and then solves the single fluid problem using the immersed interface method using this method we assess through two numerical examples how the fluid dynamics are affected by differing viscosities in the two fluid regions we also propose an implicit algorithm and a fractional step algorithm for advancing the boundary position because both algorithms make use of the integral form of the solution neither one requires the solution of a large system of coupled nonlinear equations as is traditionally the case numerical results suggest that for sufficiently stiff problems the fractional time stepping algorithm is the most efficient in the sense that it allows the largest time interval between subsequent updates of global model solutions

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

a cartesian grid embedded boundary method for the heat equation on irregular domains we present an algorithm for solving the heat equation on irregular time dependent domains it is based on the cartesian grid embedded boundary algorithm of johansen and colella j comput phys 147 2 60 85 for discretizing poisson s equation combined with a second order accurate discretization of the time derivative this leads to a method that is second order accurate in space and time for the case where the boundary is moving we convert the moving boundary problem to a sequence of fixed boundary problems combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves we find that in the moving boundary case the use of crank nicolson time discretization is unstable requiring us to use the l sub 0 stable implicit runge kutta method of twizell gumel and arigu

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

three dimensional elliptic solvers for interface problems and applications second order accurate elliptic solvers using cartesian grids are presented for three dimensional interface problems in which the coefficients the source term the solution and its normal flux may be discontinuous across an interface one of our methods is designed for general interface problems with variable but discontinuous coefficient the scheme preserves the discrete maximum principle using constrained optimization techniques an algebraic multigrid solver is applied to solve the discrete system the second method is designed for interface problems with piecewise constant coefficient the method is based on the fast immersed interface method and a fast 3d poisson solver the second method has been modified to solve helmholtz poisson equations on irregular domains an application of our method to an inverse interface problem of shape identification is also presented in this application the level set method is applied to find the unknown surface iteratively

hard lefschetz theorem for nonrational polytopes the hard lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope one can construct the intersection cohomology combinatorially from the polytope hence it is well defined even for nonrational polytopes when there is no variety associated to it we prove the hard lefschetz theorem for the intersection cohomology of a general polytope

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd