a new class of truly consistent splitting schemes for incompressible flows a new class of splitting schemes for incompressible flows is introduced the new schemes are based on a weak form of the pressure poisson equation and at each time step only require to solve a set of helmholtz type equations for the velocity and a poisson equation in the weak form for the pressure just as pressure correction and velocity correction schemes however unlike pressure correction and velocity correction schemes the new splitting schemes are free of splitting errors and deliver full accuracy on the vorticity and the pressure

an arbitrary order runge kutta discontinuous galerkin approach to reinitialization for banded conservative level sets abstract we present a runge kutta discontinuous galerkin method for solving conservative reinitialization in the context of the conservative level set method 18 19 this represents an extension of the method recently proposed by owkes and desjardins 21 by solving the level set equations on the refined level set grid 9 and projecting all spatially dependent variables into the full basis used by the discontinuous galerkin discretization by doing so we achieve the full k 1 order convergence rate in the l 1 norm of the level set field predicted for rkdg methods given kth degree basis functions when the level set profile thickness is held constant with grid refinement shape and volume errors for the 0 5 contour of the level set on the other hand are found to converge between first and second order we show a variety of test results including the method of manufactured solutions reinitialization of a circle and sphere zalesak s disk and deforming columns and spheres all showing substantial improvements over the high order finite difference traditional level set method studied for example by herrmann 9 we also demonstrate the need for kth order accurate normal vectors as lower order normals are found to degrade the convergence rate of the method

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 extension of the discontinuous galerkin method for the singular poisson equation we present a numerical method for solving a singular poisson equation which solution contains jumps and kinks due to a singular right hand side equations of this type may arise e g within the pressure computation of incompressible multiphase flows the method is an extension to the well known discontinuous galerkin dg method being able to represent the jumps and kinks with subcell accuracy in the proposed method an ansatz function which already fulfills the jump condition is subtracted from the original problem thereby reducing it to a standard poisson equation without a jump invoking a technique that we refer to as patching the construction of the ansatz function can be limited to a very narrow domain around the jump position thus making the construction numerically cheap and easy under optimal conditions the method shows a convergence order of p 1 for dg polynomial degree p still in the worst case a convergence order of approximately 2 4 is preserved for dg polynomial degree of 2

a high order elliptic pde based level set reinitialisation method using a discontinuous galerkin discretisation abstract in this paper an efficient high order accurate level set reinitialisation method is proposed based on the elliptic reinitialisation method basting and kuzmin 2013 1 which is discretised spatially using the discontinuous galerkin dg symmetric interior penalty method sipg in order to achieve this a number of improvements have been made to the elliptic reinitialisation method including reformulation of the underlying minimisation problem driving the solution adoption of a lagrange multiplier approach for enforcing a dirichlet boundary condition on the implicit level set interface and adoption of a narrow band approach numerical examples confirm the high order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the sipg method that is h p 1 and h p in the l 2 and dg norms respectively furthermore the degree to which the level set function satisfies the eikonal equation improves proportionally to h p and the often ignored homogeneous dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h 2 for all polynomial orders

a gradient augmented level set method for unstructured grids we present and test a gradient augmented level set method for unstructured grids which has been designed to allow for easy integration in parallel flow solvers the means to this end is a hermite formulation where the gradient is stored and advected as an independent variable along with the level set function the method uses narrow band storage in conjunction with a fast marching method which is also designed to take advantage of the gradient augmentation we demonstrate through standard passive advection test cases that the proposed method is able to compute interface motion with performance comparable to popular high performing methods on structured grids

a convex analysis framework for blind separation of non negative sources this paper presents a new framework for blind source separation bss of non negative source signals the proposed framework referred herein to as convex analysis of mixtures of non negative sources camns is deterministic requiring no source independence assumption the entrenched premise in many existing usually statistical bss frameworks the development is based on a special assumption called local dominance it is a good assumption for source signals exhibiting sparsity or high contrast and thus is considered realistic to many real world problems such as multichannel biomedical imaging under local dominance and several standard assumptions we apply convex analysis to establish a new bss criterion which states that the source signals can be perfectly identified in a blind fashion by finding the extreme points of an observation constructed polyhedral set methods for fulfilling the camns criterion are also derived using either linear programming or simplex geometry simulation results on several data sets are presented to demonstrate the efficacy of the proposed method over several other reported bss methods

classification of abelian hereditary directed categories satisfying serre duality in an ongoing project to classify all hereditary abelian categories we provide a classification of ext finite directed hereditary abelian categories satisfying serre duality up to derived equivalence in order to prove the classification we will study the shapes of auslander reiten components extensively and use appropriate generalizations of tilting objects and coordinates namely partial tilting sets and probing of objects by quasi simples

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

a parabolic level set reinitialisation method using a discontinuous galerkin discretisation level set reinitialisation is a part of the level set methodology which allows one to generate at any point during level set evolution a level set function which is a signed distance function to its own zero isocontour whilst not in general a required condition maintaining the level set function as a signed distance function is often desirable as it removes a known source of numerical instability this paper presents a novel level set reinitialisation method based on the solution of a nonlinear parabolic pde the pde is discretised using a symmetric interior penalty discontinuous galerkin method in space and an implicit euler method in time also explored are explicit and semi implicit time discretisations however numerical experiments demonstrate that such methods suffer from severe time step restrictions leading to prohibitively large numbers of iterations required to achieve convergence the proposed method is shown to be high order accurate through a number of numerical examples more specifically the presented experimental orders of convergence align with the well established optimal convergence rates for the symmetric interior penalty method that is the error in the norm decreases proportionally to and the error in the dg norm decreases proportionally to

some improvements of the fast marching method the fast marching method published by sethian proc natl acad sci usa 93 1996 pp 1591 1595 is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic it has been used in a wide variety of applications such as robotic path planning r kimmel and j sethian fast marching methods for computing distance maps and shortest paths tech report 669 cpam university of california berkeley 1996 crack propagation m stolarska et al modelling crack growth by level sets in the extended finite element method comput methods appl mech engrg to appear internat j numer methods engrg 51 2001 pp 943 960 n sukumar d l chopp and b moran extended finite element method and fast marching method for three dimensional fatigue crack propagation j comput phys submitted seismology j sethian and a popovici geophysics 64 1999 pp 516 523 photolithography j sethian fast marching level set methods for three dimensional photolithography development in proceedings of the spie 1996 international symposium on microlithography santa clara ca 1996 and medical imaging r malladi and j sethian proc natl acad sci usa 93 1996 pp 9389 9392 it has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and or an extended velocity function in this paper we improve upon the second order fast marching method of sethian siam rev 41 1999 pp 199 235 by constructing a second order approximation of the interface generated from local data on the mesh the data is interpolated on a single box of the mesh using a bicubic approximation the distance to the front is then calculated by using a variant of newton s method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node the result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method

a convex analysis based minimum volume enclosing simplex algorithm for hyperspectral unmixing hyperspectral unmixing aims at identifying the hidden spectral signatures or endmembers and their corresponding proportions or abundances from an observed hyperspectral scene many existing hyperspectral unmixing algorithms were developed under a commonly used assumption that pure pixels exist however the pure pixel assumption may be seriously violated for highly mixed data based on intuitive grounds craig reported an unmixing criterion without requiring the pure pixel assumption which estimates the endmembers by vertices of a minimum volume simplex enclosing all the observed pixels in this paper we incorporate convex analysis and craig s criterion to develop a minimum volume enclosing simplex mves formulation for hyperspectral unmixing a cyclic minimization algorithm for approximating the mves problem is developed using linear programs lps which can be practically implemented by readily available lp solvers we also provide a non heuristic guarantee of our mves problem formulation where the existence of pure pixels is proved to be a sufficient condition for mves to perfectly identify the true endmembers some monte carlo simulations and real data experiments are presented to demonstrate the efficacy of the proposed mves algorithm over several existing hyperspectral unmixing methods

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway

on a cluster category of infinite dynkin type and the relation to triangulations of the infinity gon let k be a field and let d be a k linear algebraic triangulated category with split idempotents let be the suspension functor of d and let s be a 2 spherical object of d that is the morphism space d s i s is k for i 0 and i 2 and vanishes otherwise assume that s classically generates d that is each object of d can be built from s using de suspensions direct sums direct summands and distinguished triangles it was proved in 15 thm 2 1 that d is uniquely determined by these properties as we will explain d is a good candidate for a cluster category of dynkin type a for instance we show that there is a bijection between the cluster tilting subcategories of d and certain triangulations of the gon we use this to give an example of a subcategory a which is weak cluster tilting that is satisfies a 1a a but fails to be functorially finite perpendicular

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

joint bayesian endmember extraction and linear unmixing for hyperspectral imagery this paper studies a fully bayesian algorithm for endmember extraction and abundance estimation for hyperspectral imagery each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra following the linear mixing model the estimation of the unknown endmember spectra is conducted in a unified manner by generating the posterior distribution of abundances and endmember parameters under a hierarchical bayesian model this model assumes conjugate prior distributions for these parameters accounts for nonnegativity and full additivity constraints and exploits the fact that the endmember proportions lie on a lower dimensional simplex a gibbs sampler is proposed to overcome the complexity of evaluating the resulting posterior distribution this sampler generates samples distributed according to the posterior distribution and estimates the unknown parameters using these generated samples the accuracy of the joint bayesian estimator is illustrated by simulations conducted on synthetic and real aviris images

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

some new axioms for triangulated categories and this mapping cone should by rights be a triangle it is for instance true that any homological functor applied to gives a long exact sequence unfortunately the world of triangulated categories is a bad one and need not be a triangle it is however true that givenf and g there exists an h for which is a triangle see theorem 1 8 not all morphisms of triangles are equal some are better than others it turns out that theorem 1 8 is equivalent to the octahedral axiom in the first two sections of this article we study two possible notions of good morphisms between triangles and we quickly decide that neither is satisfactory the morphisms do not compose well the composite of good morphisms need not be good worse still the non category of good morphisms is non additive the sum of two good morphisms need not be good the problem goes right back to the definition of a triangulated category

numerical treatment of the navier stokes equations with slip boundary condition in this article we present efficient numerical methods for the navier stokes equations with slip boundary conditions a first method is based on a saddle point formulation of the slip boundary condition a congruent gradient cg method is applied to the schur complement operator in order to solve the problem we present two preconditioners for the cg method which result in convergence rates independent of the mesh size for a second method the slip boundary condition is enforced pointwise for nodal values of the velocity at boundary nodes

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

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

derived equivalences for hereditary artin algebras we study the role of the serre functor in the theory of derived equivalences let a a be an abelian category and let u v u v be a t structure on the bounded derived category d b a d b a with heart h h we investigate when the natural embedding h d b a h d b a can be extended to a triangle equivalence d b h d b a d b h d b a our focus of study is the case where a a is the category of finite dimensional modules over a finite dimensional hereditary algebra in this case we prove that such an extension exists if and only if the t structure is bounded and the aisle u u of the t structure is closed under the serre functor

exchange graphs and ext quivers abstract we study the oriented exchange graph eg n q of reachable hearts in the finite dimensional derived category d n q of the cy n ginzburg algebra n q associated to an acyclic quiver q we show that any such heart is induced from some heart in the bounded derived category d q via some lagrangian immersion l d q d n q we build on this to show that the quotient of eg n q by the seidel thomas braid group is the exchange graph ceg n 1 q of cluster tilting sets in the higher cluster category c n 1 q as an application we interpret buan thomas coloured quiver for a cluster tilting set in terms of the ext quiver of any corresponding heart in d n q

a high order and interface preserving discontinuous galerkin method for level set reinitialization abstract a high order numerical method for interface preserving level set reinitialization is presented in this paper in the interface cells the gradient of the level set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved in the non interface cells we compute the gradient of the level set function by solving a hamilton jacobi equation as a conservation law system using the discontinuous galerkin method following the work by hu and shu siam j sci comput 21 1999 660 690 the missing constant is then recovered by the continuity of the level set function while taking into account the characteristics to handle highly distorted initial conditions we develop a hybrid numerical flux that combines the lax friedrichs flux and the penalty flux our method is stable for non trivial test cases and handles singularities away from the interface very well when derivative singularities are present on the interface a second derivative limiter is designed to suppress the oscillations at least n 1 th order accuracy in the interface cells and nth order in the whole domain are observed for smooth solutions when nth degree polynomials are used two dimensional test cases are presented to demonstrate superior properties such as accuracy long term stability interface preserving capability and easy treatment of contact lines we also show some preliminary results on the pinch off process of a pendant drop where topological changes of the fluid interface are involved our method is readily extendable to three dimensions and adaptive meshes

the algebras with discrete derived category abstract we classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non isomorphic indecomposable complexes

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

a fast hyperplane based minimum volume enclosing simplex algorithm for blind hyperspectral unmixing hyperspectral unmixing hu is a crucial signal processing procedure to identify the underlying materials or endmembers and their corresponding proportions or abundances from an observed hyperspectral scene a well known blind hu criterion advocated by craig during the early 1990s considers the vertices of the minimum volume enclosing simplex of the data cloud as good endmember estimates and it has been empirically and theoretically found effective even in the scenario of no pure pixels however such kinds of algorithms may suffer from heavy simplex volume computations in numerical optimization etc in this paper without involving any simplex volume computations by exploiting a convex geometry fact that a simplest simplex of n vertices can be defined by n associated hyperplanes we propose a fast blind hu algorithm for which each of the n hyperplanes associated with the craig s simplex of n vertices is constructed from n 1 affinely independent data pixels together with an endmember identifiability analysis for its performance support without resorting to numerical optimization the devised algorithm searches for the n n 1 active data pixels via simple linear algebraic computations accounting for its computational efficiency monte carlo simulations and real data experiments are provided to demonstrate its superior efficacy over some benchmark craig criterion based algorithms in both computational efficiency and estimation accuracy

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

a level set discontinuous galerkin method for free surface flows we present a discontinuous galerkin method on a fully unstructured grid for the modeling of unsteady incompressible fluid flows with free surfaces the surface is modeled by embedding and represented by a level set we discuss the discretization of the flow equations and the level set equation as well a various ways of advancing the equations in time using velocity projection techniques the efficacy of the method for the representation of the level set and its reinitialization is discussed and several numerical tests confirm the robustness and versatility of the proposed scheme

some adjoints in homotopy categories let r be a ring in a previous paper 11 we found a new description for the category k r proj it is equivalent to the verdier quotient k r flat s for some suitable s c k r flat in this article we show that the quotient map from k r flat to k r flat s always has a right adjoint this gives a new fully faithful embedding of k r proj into k r flat its virtue is that it generalizes to nonaffine schemes

a nodal discontinuous galerkin finite element method for nonlinear elastic wave propagation a nodal discontinuous galerkin finite element method dg fem to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented this dg fem method combines the geometrical flexibility of the finite element method and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method required for nonlinear elastodynamics simulations in order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications the equations of nonlinear elastodynamics have been written in a conservative form the adopted formalism allows the introduction of different kinds of elastic nonlinearities such as the classical quadratic and cubic nonlinearities or the quadratic hysteretic nonlinearities absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate when needed semi infinite or infinite media the developed dg fem scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations lamb s problem and plane wave nonlinear propagation

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

idempotent filtering in spectral and spectral element methods the comparison of numerical results for implicit explicit and fully explicit runge kutta time integration methods for a nozzle flow problem shows that filtering can significantly degrade the accuracy of the numerical solution for long time integration problems we demonstrate analytically and numerically that filtering in time errors become additive for u n x t k dt u n x t u n x t when nonidempotent filters are used and suggest the development and implementation of idempotent filters

tracking discontinuities in hyperbolic conservation laws with spectral accuracy it is well known that the spectral solutions of conservation laws have the attractive distinguishing property of infinite order convergence also called spectral accuracy when they are smooth e g c canuto m y hussaini a quarteroni t a zang spectral methods for fluid dynamics springer verlag heidelberg 1988 j p boyd chebyshev and fourier spectral methods second ed dover new york 2001 c canuto m y hussaini a quarteroni t a zang spectral methods fundamentals in single domains springer verlag berlin heidelberg 2006 if a discontinuity or a shock is present in the solution this advantage is lost there have been attempts to recover exponential convergence in such cases with rather limited success the aim of this paper is to propose a discontinuous spectral element method coupled with a level set procedure which tracks discontinuities in the solution of nonlinear hyperbolic conservation laws with spectral convergence in space spectral convergence is demonstrated in the case of the inviscid burgers equation and the one dimensional euler equations

some examples of t structures for finite dimensional algebras abstract we describe the heart of the canonical t structure on the perfect derived category of a strictly positive graded algebra as the module category over the quadratic dual applying this result we obtain examples showing new phenomena on t structures on derived categories of finite dimensional algebras

stable spectral methods for conservation laws on triangles with unstructured grids this paper presents an asymptotically stable scheme for the spectral approximation of linear conservation laws defined on a triangle lagrange interpolation on a general two dimensional nodal set is employed and by imposing the boundary conditions weakly through a penalty term the scheme is proven stable in l 2 this result is established for a general unstructured grid in the triangle a special case for which the nodes along the edges of the triangle are chosen as the legendre gauss lobatto quadrature points is discussed in detail the eigenvalue spectrum of the approximation to the advective operator is computed and is shown to result in an o n 2 restriction on the time step when considering explicit time stepping c 1999 elsevier science s a all rights reserved

discrete derived categories i homomorphisms autoequivalences and t structures discrete derived categories were studied initially by vossieck j algebra 243 168 176 2001 and later by bobinski et al cent eur j math 2 19 49 2004 in this article we describe the homomorphism hammocks and autoequivalences on these categories we classify silting objects and bounded t structures

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

algorithms for a minimum volume enclosing simplex in three dimensions we develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in cal r 3 if the convex hull of the points has n vertices then our algorithm takes theta n 4 time combining our exact but slow algorithm with a simple but crude approximation technique we also develop an varepsilon approximation algorithm the algorithm computes in o n 1 varepsilon 6 time a simplex whose volume is within 1 varepsilon factor of the optimal for any varepsilon 0

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions