the gl 2 c mckay correspondence in this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution as a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph also for any finite subgroup g of rm gl 2 mathbb c it means that the endomorphism ring of the special cm mathbb c x y g modules can be used to build the dual graph of the minimal resolution of mathbb c 2 g extending mckay s observation mckay proc symp pure math 37 183 186 1980 for finite subgroups of rm sl 2 mathbb c to all finite subgroups of rm gl 2 mathbb c

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

radical layers of representable functors i let 0 f a t4 b o c 0 be an almost split sequence then the induced sequence o g a g b g c ys 0 is exact where s gs c r c the maps a have degree 1 and y has degree 0 ii let i t i sot i be a left minimal almost split map where i is an indecomposable injective let k 03 be maximal such that im sot i c rk i if k co then the induced sequence 0 g sot i g i g i sot z is exact where the first map has degree k and the second has degree 1 if k co then the induced map g i t g z sot i is a monomorphism of degree 1 dual results hold for covariant functors x

quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations we present a practical and efficient means to compute the singular value decomposition svd of a real or complex quaternion matrix a based on bidiagonalization of a to a real or complex bidiagonal matrix b using quaternionic householder transformations computation of the svd of b using an existing subroutine library such as lapack provides the singular values of a the singular vectors of a are obtained trivially from the product of the householder transformations and the real or complex singular vectors of b we show in the paper that left and right quaternionic householder transformations are different because of the non commutative multiplication of quaternions and we present formulae for computing the householder vector and matrix in each case

algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory by means of complex representation and real representation of a quaternion matrix this paper studies the problem of diagonalization of a quaternion matrix gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory

computing global dimension of endomorphism rings via ladders abstract this paper deals with computing the global dimension of endomorphism rings of maximal cohen macaulay mcm modules over commutative rings several examples are computed in particular we determine the global spectra that is the sets of all possible finite global dimensions of endomorphism rings of mcm modules of the curve singularities of type a n for all n d n for n 13 and e 6 7 8 and compute the global dimensions of leuschke s normalization chains for all ade curves as announced in 12 moreover we determine the centre of an endomorphism ring of a mcm module over any curve singularity of finite mcm type in general we describe a method for the computation of the global dimension of an endomorphism ring end r m where r is a henselian local ring using add m approximations when m 0 is a mcm module over r and r is henselian local of krull dimension 2 with a canonical module and of finite mcm type we use auslander reiten theory and iyama s ladder method to explicitly construct these approximations

the classification of special cohen macaulay modules in this paper we completely classify all the special cohen macaulay cm modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities in every case we exhibit the specials explicitly in a combinatorial way our result relies on realizing the specials as those cm modules whose first ext group vanishes against the ring r thus reducing the problem to combinatorics on the ar quiver such possible ar quivers were classified by auslander and reiten we also give some general homological properties of the special cm modules and their corresponding reconstruction algebras

modified cholesky factorizations in interior point algorithms for linear programming we investigate a modified cholesky algorithm typical of those used in most interior point codes for linear programming cholesky based interior point codes are popular for three reasons their implementation requires only minimal changes to standard sparse cholesky algorithms allowing us to take full advantage of software written by specialists in that area they tend to be more efficient than competing approaches that use alternative factorizations and they perform robustly on most practical problems yielding good interior point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill conditioned we investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis illustrating our results with computational experiments finally we point out the potential limitations of this approach

jacobi method for quaternion matrix singular value decomposition the study of quaternion matrices has gained interest in many areas in recent years and the problem of diagonalizing such matrices has also attracted attention in this article we present an algorithm for computing the svd of a matrix with quaternion coefficients directly in quaternion arithmetic using a generalization of classical complex jacobi methods the extension of the jacobi transformation to the quaternion case is introduced for the diagonalization of a hermitian quaternion valued matrix based on this an implicit jacobi algorithm is proposed for computing the svd of a quaternion matrix the performance of the proposed algorithm is presented and compared with an already known algorithm using a complex equivalent of the quaternion matrix and shown to be superior in execution time and accuracy

quaternions and special relativity we reformulate special relativity by a quaternionic algebra on reals using real linear quaternions we show that previous difficulties concerning the appropriate transformations on the 3 1 space time may be overcome this implies that a complexified quaternionic version of special relativity is a choice and not a necessity

contravariantly finite resolving subcategories over commutative rings contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras in this paper we study contravariantly finite resolving subcategories over commutative rings the main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian gorenstein local ring in fact there exist only three such subcategories our method to obtain this classification also recovers as a by product the theorem of christensen piepmeyer striuli and takahashi concerning the relationship between the contravariant finiteness of the full subcategory of totally reflexive modules and the gorenstein property of the base ring

non commutative desingularization of determinantal varieties i we show that determinantal varieties defined by maximal minors of a generic matrix have a non commutative desingularization in that we construct a maximal cohen macaulay module over such a variety whose endomorphism ring is cohen macaulay and has finite global dimension in the case of the determinant of a square matrix this gives a non commutative crepant resolution

a fast structure preserving method for computing the singular value decomposition of quaternion matrices abstract in this paper we propose a fast structure preserving algorithm for computing the singular value decomposition of quaternion matrices the algorithm is based on the structure preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal jrs symplectic matrices the algorithm is efficient and numerically stable

global error bounds for convex conic problems this paper aims at deriving and proving some lipschitzian type error bounds for convex conic problems in a simple way first it is shown that if the recession directions satisfy slater s condition then a global lipschitzian type error bound holds alternatively if the feasible region is bounded then the ordinary slater condition guarantees a global lipschitzian type error bound these can be considered as generalizations of previously known results for inequality systems which also follow from general results by bauschke and borwein in siam review 38 1996 pp 367 426 and bauschke borwein and li in a 1997 report however the proofs in the current paper are considerably simpler some of the results are generalized to the intersection of multiple shifted cones with different shifts under slater s condition alone a global lipschitzian type error bound does not hold it is shown however that such an error bound holds for a specific conic region for linear systems we establish that the sharp constant involved in hoffman s error bound is nothing but the condition number for linear programming as used by vavasis and ye in math programming 74 1996 pp 79 120

a structure preserving algorithm for the quaternion cholesky decomposition in this paper we discuss the cholesky decomposition of the hermitian positive definite quaternion matrix for the first time the structure preserving gauss transformation is defined and then a novel structure preserving algorithm which is applied to its real representation matrix is proposed our algorithm needs only real number operations does not depend on the quaternion toolbox for matlab qtfm and has more portability although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or qtfm numerical experiments show that our algorithm runs faster

orlov spectra bounds and gaps the orlov spectrum is a new invariant of a triangulated category it was introduced by d orlov building on work of a bondal m van den bergh and r rouquier the supremum of the orlov spectrum of a triangulated category is called the ultimate dimension in this work we study orlov spectra of triangulated categories arising in mirror symmetry we introduce the notion of gaps and outline their geometric significance we provide the first large class of examples where the ultimate dimension is finite categories of singularities associated to isolated hypersurface singularities similarly given any nonzero object in the bounded derived category of coherent sheaves on a smooth calabi yau hypersurface we produce a generator by closing the object under a certain monodromy action and uniformly bound this generator s generation time in addition we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived fukaya category of a symplectic surface of genus greater than one

a norm bound for projections with complex weights abstract we consider projections weighted by a complex diagonal matrix these projections arise in the solution of alternating current ac electrical networks a norm bound for weighted projections in the case of an ill conditioned real positive definite weight matrix was established by dikin stewart todd and others we show that this bound generalizes to the complex case

some new properties of the equality constrained and weighted least squares problem abstract in this paper some new properties of the equality constrained and weighted least squares problem wlse min w 1 2 kx g 2 subject to lx h are obtained we derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors of this unconstrained ls problem we also present a new way to compute the minimum norm solution x wlse of the wlse problem by using the qr decomposition of the corresponding matrices and propose an algorithm to compute x wlse using the qr factorizations some numerical examples are provided to compare different methods for solving the wlse problem

a note on properties of condition numbers abstract the condition numbers described by stewart todd and others for a matrix a are given by the suprema of the norms of all the oblique projection operators respectively on the image space of a t and the null space of a this paper shows that these two numbers coincide

real structure preserving algorithms of householder based transformations for quaternion matrices in this paper we survey three different forms of householder based transformations for quaternion matrices in the literature and propose a new form of quaternion householder based transformation we propose real structure preserving algorithms of these householder based transformations which make the procedure computationally more flexible and efficient we compare the computation counts and assignment numbers of these algorithms we also compare the effectiveness of these real structure preserving algorithms applying to the quaternion qrd and the quaternion svd all these four real structure preserving algorithms are more efficient comparing to the algorithms which apply quaternion toolbox using quaternion arithmetics or algorithms which directly performs real householder transformations on the real representation of a quaternion matrix among these four real structure preserving algorithms the most efficient ones are based on quaternion householder reflection and new proposed householder based transformation

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

a mckay correspondence for reflection groups we construct a noncommutative desingularization of the discriminant of a finite reflection group g as a quotient of the skew group ring a s g if g is generated by order 2 reflections then this quotient identifies with the endomorphism ring of the reflection arrangement a g viewed as a module over the coordinate ring sg of the discriminant of g this yields in particular a correspondence between the nontrivial irreducible representations of g to certain maximal cohen macaulay modules over the coordinate ring sg these maximal cohen macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement a g viewed as a module over sg we identify some of the corresponding matrix factorizations namely the so called logarithmic co residues of the discriminant

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

on scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a diagonal matrix with positive diagonal elements the weighted pseudoinverse defined by x d x t dx 1 x t d and the associated oblique projection p d xx d arise in many applications in this paper we show that the norms of both matrices are bounded by numbers that are independent of d

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

maximal modifications and auslander reiten duality for non isolated singularities we first generalize classical auslander reiten duality for isolated singularities to cover singularities with a one dimensional singular locus we then define the notion of ct modules for non isolated singularities and we show that these are intimately related to noncommutative crepant resolutions nccrs when r has isolated singularities ct modules recover the classical notion of cluster tilting modules but in general the two concepts differ then wanting to generalize the notion of nccrs to cover partial resolutions of operatorname spec r in the main body of this paper we introduce a theory of modifying and maximal modifying modules under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three dimensional gorenstein rings are shown to be derived equivalent we then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory our mutation works in arbitrary dimension and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian