rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

graded maximal cohen macaulay modules over noncommutative graded gorenstein isolated singularities abstract in this paper we define a notion of noncommutative graded isolated singularity and study as gorenstein isolated singularities and the categories of graded maximal cohen macaulay modules over them in particular for an as gorenstein algebra a of dimension d 2 we show that a is a graded isolated singularity if and only if the stable category of graded maximal cohen macaulay modules over a has the serre functor using this result we also show the existence of cluster tilting objects in the categories of graded maximal cohen macaulay modules over veronese subalgebras of certain as regular algebras

on the trace of graded automorphisms abstract leta d 0 adbe a connected algebra with a graded algebra endomorphism the trace of is defined to be tr t d 0 tr ad td we prove that tr t is a rational function ifais either finitely generated commutative or right noetherian with finite global dimension or regular a version of molien s theorem follows in these three cases ifais a regular algebra or a frobenius algebra we prove a reciprocity for the trace we also partially generalize a theorem of watanabe on the gorenstein property to the noncommutative case

exponential b series the stiff case for the purpose of deriving the order conditions of exponential runge kutta and exponential rosenbrock methods we extend the well known concept of b series to exponential integrators as we are mainly interested in the stiff case taylor series expansions have their limitations our approach is based on the variation of constants formula which allows us to treat semilinear problems where the stiffness comes from the linear part by truncating the arising exponential b series to a certain order which requires regularity of the considered exact solution we are able to identify the sought after order conditions in particular we show how the stiff order conditions of arbitrary order can be obtained in a simple way from a set of recursively defined trees

tilting and cluster tilting for quotient singularities we shall show that the stable categories of graded cohen macaulay modules over quotient singularities have tilting objects in particular these categories are triangle equivalent to derived categories of finite dimensional algebras our method is based on higher dimensional auslander reiten theory which gives cluster tilting objects in the stable categories of ungraded cohen macaulay modules

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

rigidity of graded regular algebras we prove a graded version of alev polo s rigidity theorem the homogenization of the universal enveloping algebra of a semisimple lie algebra and the rees ring of the weyl algebras a n k cannot be isomorphic to their fixed subring under any finite group action we also show the same result for other classes of graded regular algebras including the sklyanin algebras

mesh deformation based on fully stressed design the method and 2 d examples mesh deformation in response to redefined boundary geometry is a frequently encountered task in shape optimization and analysis of fluid structure interaction we propose a simple and concise method for deforming meshes defined with three node triangular or four node tetrahedral elements the mesh deformation method is suitable for large boundary movement the approach requires two consecutive linear elastic finite element analyses of an isotropic continuum using a prescribed displacement at the mesh boundaries the first analysis is performed with homogeneous elastic property and the second with inhomogeneous elastic property the fully stressed design is employed with a vanishing poisson ratio and a proposed form of equivalent strain modified tresca equivalent strain to calculate from the strain result of the first analysis an element specific young s modulus for the second analysis the theoretical aspect of the proposed method its convenient numerical implementation using a typical linear elastic finite element code in conjunction with very minor extra coding for data processing and results for examples of large deformation of 2 d meshes are presented in this paper

connected graded gorenstein algebras with enough normal elements we generalize 12 1 1 and 1 2 to the following situation theorem 1 let a be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of a contains a homogeneous normal element of positive degree then 1 a is auslander gorenstein and cohen macaulay 2 a has a quasi frobenius quotient ring 3 every minimal prime ideal p is graded andgkdim a p d 4 if moreover a has finite global dimension then a is a domain and a maximal order in its quotient division ring to prove the above we need the following result which is a generalization of 3 2 46 ii theorem 2 let a be a connected graded noetherian as gorenstein algebra of injective dimension d then 1 the last term of the minimal injective resolution of aais isomorphic to a shift of a 2 for every noetherian graded a module m extd m a is finite dimensional over k

on the convergence of lawson methods for semilinear stiff problems since their introduction in 1967 lawson methods have achieved constant interest in the time discretization of evolution equations the methods were originally devised for the numerical solution of stiff differential equations meanwhile they constitute a well established class of exponential integrators which has turned out to be competitive for solving space discretizations of certain types of partial differential equations the popularity of lawson methods is in some contrast to the fact that they may have a bad convergence behaviour since they do not satisfy any of the stiff order conditions the aim of this paper is to explain this discrepancy it is shown that non stiff order conditions together with appropriate regularity assumptions imply high order convergence of lawson methods note however that the term regularity here includes the behaviour of the solution at the boundary for instance lawson methods will behave well in the case of periodic boundary conditions but they will show a dramatic order reduction for e g dirichlet boundary conditions the precise regularity assumptions required for high order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes in contrast to previous work the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem numerical examples for the schrodinger equation are included

high order exponential integrators for nonlinear schrodinger equations with application to rotating bose einstein condensates this article deals with the numerical integration in time of nonlinear schrodinger equations the main application is the numerical simulation of rotating bose einstein condensates the authors perform a change of unknown so that the rotation term disappears and they obtain as a result a nonautonomous nonlinear schrodinger equation they consider exponential integrators such as exponential runge kutta methods and lawson methods they provide an analysis of the order of convergence and some preservation properties of these methods in a simplified setting and they supplement their results with numerical experiments with realistic physical parameters moreover they compare these methods with the classical split step methods applied to the same problem

isogeometric shape optimization of photonic crystals via coons patches abstract in this paper we present an approach that extends isogeometric shape optimization from optimization of rectangular like nurbs patches to the optimization of topologically complex geometries we have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps salient features of this approach include the following 1 multi patch coons representation of design geometry the design geometry is represented as a collection of coons patches where the four boundaries of each patch are represented as nurbs curves the use of multiple patches is motivated by the need for representing topologically complex geometries the coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization 2 exact boundary conversion to the analysis geometry with guaranteed mesh injectivity the analysis geometry is a collection of nurbs patches that are converted from the multi patch coons representation with geometric exactness in patch boundaries the internal nurbs control points are embedded in the parametric domain of the coons patches with a built in mesh rectifier to ensure the injectivity of the resulting b spline geometry i e every point in the physical domain is mapped to one point in the parametric domain 3 analytical sensitivities sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities the nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding coons patches

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

hopf actions on filtered regular algebras we study finite dimensional hopf algebra actions on so called filtered artin schelter regular algebras of dimension n particularly on those of dimension 2 the first weyl algebra is an example of such an algebra with n 2 for instance results on the gorenstein condition and on the global dimension of the corresponding fixed subrings are also provided

down up algebras abstract the algebra generated by the down and up operators on a differential or uniform partially ordered set poset encodes essential enumerative and structural properties of the poset motivated by the algebras generated by the down and up operators on posets we introduce here a family of infinite dimensional associative algebras called down up algebras we show that down up algebras exhibit many of the important features of the universal enveloping algebrau s l2 of the lie algebra s l 2including a poincare birkhoff witt type basis and a well behaved representation theory we investigate the structure and representations of down up algebras and focus especially on verma modules highest weight representations and category o modules for them we calculate the exact expressions for all the weights since that information has proven to be particularly useful in determining structural results about posets

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

actions of small groups on two dimensional artin schelter regular algebras abstract in commutative invariant theory a classical result due to auslander says that if r k x 1 x n and g is a finite subgroup of au t gr r gl n k which contains no reflections then there is a natural graded isomorphism r g end r g r in this paper we show that a version of auslander s theorem holds if we replace r by an artin schelter regular algebra a of global dimension 2 and g by a finite subgroup of au t gr a which contains no quasi reflections this extends work of chan kirkman walton zhang as part of the proof we classify all such pairs a g up to conjugation of g by an element of au t gr a in all but one case we also write down explicit presentations for the invariant rings a g and show that they are isomorphic to factors of as regular algebras

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

noncommutative quasi resolutions abstract the notion of a noncommutative quasi resolution is introduced for a noncommutative noetherian algebra with singularities even for a non cohen macaulay algebra if a is a commutative normal gorenstein domain then a noncommutative quasi resolution of a naturally produces a noncommutative crepant resolution nccr of a in the sense of van den bergh and vice versa under some mild hypotheses we prove that i in dimension two all noncommutative quasi resolutions of a given noncommutative algebra are morita equivalent and ii in dimension three all noncommutative quasi resolutions of a given noncommutative algebra are derived equivalent these assertions generalize important results of van den bergh iyama reiten and iyama wemyss in the commutative and central finite cases

a mesh regularization scheme to update internal control points for isogeometric shape design optimization abstract this paper presents a variational method to update internal control points in isogeometric shape optimization the important properties of domain parameterization such as bijective mapping between parametric and physical domains uniform mesh and orthogonal mesh are enforced simultaneously the bijective mapping is achieved by minimizing a dirichlet energy functional to prevent the divergent behavior of the minimizing process due to the severely distorted initial mesh a constraint is introduced to enforce the positive jacobian of mapping from parametric to physical domains in spite of adding the constraint that might increase computational costs the proposed method is more efficient due to the convexity of dirichlet energy functional compared with the other unconstrained methods also it turns out that the proposed method is more effective to achieve the bijective mapping especially near a concave boundary the uniform parameterization of the domain is achieved by minimizing the dirichlet energy functional and the orthogonality of mesh is performed by minimizing a dimensionless functional the required design sensitivity of the employed functional and constraint is derived with respect to the position of internal control points the developed scheme of mesh regularization is effective to maintain the high quality of domain parameterization during the shape optimization process

noncommutative complete intersections abstract several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded k algebra these notions are justified by examples from noncommutative invariant theory

hopf dense galois extensions with applications abstract let h be a finite dimensional hopf algebra and let a be a left h module algebra motivated by the study of the isolated singularities of a h and the endomorphism ring end a h a we introduce the concept of hopf dense galois extensions in this paper hopf dense galois extensions yield certain equivalences between the quotient categories over a and a h a special class of hopf dense galois extensions consists of the so called densely group graded algebras which are weaker versions of strongly graded algebras a weaker version of dade s theorem holds for densely group graded algebras as applications we recover the classical equivalence of the noncommutative projective scheme over a noetherian n graded algebra a and its d th veronese subalgebra a d respectively hopf dense galois extensions are also applied to the study of noncommutative graded isolated singularities

fixed rings of generalized weyl algebras abstract we study actions by filtered automorphisms on classical generalized weyl algebras gwas in the case of a defining polynomial of degree two we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms is again a classical gwa extending a result of jordan and wells partial results are provided for the case of higher degree polynomials in addition we establish a version of auslander s theorem for finite cyclic groups of filtered automorphisms acting on classical gwas

isogeometric shape optimization of vibrating membranes we consider a model problem of isogeometric shape optimization of vibrating membranes whose shapes are allowed to vary freely the main obstacle we face is the need for robust and inexpensive extension of a b spline parametrization from the boundary of a domain onto its interior a task which has to be performed in every optimization iteration we experiment with two numerical methods one is based on the idea of constructing a quasi conformal mapping whereas the other is based on a spring based mesh model for carrying out this task which turn out to work sufficiently well in the present situation we perform a number of numerical experiments with our isogeometric shape optimization algorithm and present smooth optimized membrane shapes our conclusion is that isogeometric analysis fits well with shape optimization

constructing analysis suitable parameterization of computational domain from cad boundary by variational harmonic method in isogeometric analysis parameterization of computational domain has great effects as mesh generation in finite element analysis in this paper based on the concept of harmonic mapping from the computational domain to parametric domain a variational harmonic approach is proposed to construct analysis suitable parameterization of computational domain from cad boundary for 2d and 3d isogeometric applications different from the previous elliptic mesh generation method in finite element analysis the proposed method focuses on isogeometric version and converts the elliptic pde into a nonlinear optimization problem in which a regular term is integrated into the optimization formulation to achieve more uniform and orthogonal iso parametric structure near convex concave parts of the boundary several examples are presented to show the efficiency of the proposed method in 2d and 3d isogeometric analysis

homological properties of graded noetherian pi rings abstract let r be a connected graded noetherian pi ring if injdim r n r is auslander gorenstein and cohen macaulay with gelfand kirillov dimension equal to n if gldim r n r is a domain finitely generated as a module over its centre and a maximal order in its quotient division ring similar results hold if r is assumed to be local rather than connected graded alternatively suppose that r is a noetherian pi ring with gldim r r m 1 hd r m 2 for any two maximal ideals m i in the same clique then r is a direct sum of prime rings is integral over its centre and is auslander gorenstein if r is a prime ring then the centre z r of r is a krull domain and r equals its trace ring tr moreover hd r m height m for every maximal ideal m of r