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

new variable separation solutions of two dimensional burgers system abstract in this paper with the help of g g expansion method we obtain some exact solutions of the riccati equation based on the riccati equation and its exact solutions we find some variable separation solutions with two arbitrary functions of two dimensional burgers system as some special examples these exact solutions can be reduced to variable separation solutions in kink solution soliton solution periodic solution and rational function solution forms

application of the exp function method for nonlinear differential difference equations in this paper we established abundant travelling wave solutions for some nonlinear differential difference equations it is shown that the exp function method with the help of symbolic computation provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics

realizing stable categories as derived categories abstract in this paper we discuss a relationship between representation theory of graded self injective algebras and that of algebras of finite global dimension for a positively graded self injective algebra a such that a 0 has finite global dimension we construct two types of triangle equivalences first we show that there exists a triangle equivalence between the stable category of z graded a modules and the derived category of a certain algebra of finite global dimension secondly we show that if a has gorenstein parameter l then there exists a triangle equivalence between the stable category of z l z graded a modules and a derived orbit category of which is a triangulated hull of the orbit category of the derived category

simplicial structures in higher auslander reiten theory abstract we develop a novel combinatorial perspective on the higher auslander algebras of type a a family of algebras arising in the context of iyama s higher auslander reiten theory this approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to eilenberg maclane spaces and higher dimensional versions of waldhausen s s construction in algebraic k theory as an application of our techniques we provide a generalisation of the higher reflection functors of iyama and oppermann to representations with values in stable categories the resulting combinatorial framework of slice mutation can be regarded as a higher dimensional variant of the abstract representation theory of type a quivers developed by groth and s ovicek our simplicial point of view then naturally leads to an interplay between slice mutation horn filling conditions and the higher segal conditions of dyckerhoff and kapranov in this context we provide a classification of higher segal objects with values in any abelian category or stable category

preprojective algebras and c sortable words let q be an acyclic quiver and lambda be the complete preprojective algebra of q over an algebraically closed field k to any element w in the coxeter group of q buan iyama reiten and scott have introduced and studied in cite bua2 a finite dimensional algebra lambda w lambda i w in this paper we look at filtrations of lambda w associated to any reduced expression mathbf w of w we are especially interested in the case where the word mathbf w is c sortable where c is a coxeter element in this situation the consecutive quotients of this filtration can be related to tilting kq modules with finite torsionfree class

on faithfully balanced modules f cotilting and f auslander algebras abstract the aim of this paper is to give a unification of the theory of co tilting modules and higher auslander correspondence by using faithfully balanced modules i e faithful modules with the double centralizer property in the language of relative homological algebra following auslander solberg a relative version of faithfully balancedness is given which will be called 1 f faithful the relative versions of higher auslander correspondence and brenner butler theorem are also developed

the existence of maximal n orthogonal subcategories abstract for an n 1 auslander algebra with global dimension n we give some necessary conditions for admitting a maximal n 1 orthogonal subcategory in terms of the properties of simple modules with projective dimension n 1 or n for an almost hereditary algebra with global dimension 2 we prove that admits a maximal 1 orthogonal subcategory if and only if for any non projective indecomposable module m m is injective is equivalent to that the reduced grade of m is equal to 2 we give a connection between the gorenstein symmetric conjecture and the existence of maximal n orthogonal subcategories of t for a cotilting module t for a gorenstein algebra we prove that all non projective direct summands of a maximal n orthogonal module are n periodic in addition we study the relation between the complexity of modules and the existence of maximal n orthogonal subcategories for the tensor product of two finite dimensional algebras

joint uncertainty propagation in linear structural dynamics using stochastic reduced basis methods uncertainties in the properties of joints produce uncertainties in the dynamic response of built up structures line joints such as glued or continuously welded joints have spatially distributed uncertainty and can be modeled by a discretized random field techniques such as monte carlo simulation can be applied to estimate the output statistics but computational cost can be prohibitive this paper addresses how uncertainties in joints might be included straightforwardly in a finite element model with particular reference to approaches based on fixed interface craig bampton component mode synthesis and a stochastic reduced basis method with two variants these methods are used to determine the output statistics of a structure unlike perturbation based methods good accuracy can be achieved even when the coefficients of variation of the input random variables are not small undamped as well as proportionally damped components are considered efficient implementations are proposed based on an exact matrix identity that leads to a significantly lower computational cost if the number of joint degrees of freedom is sufficiently small compared with the structure s overall number of degrees of freedom a numerical example is presented the proposed formulation is an efficient and effective implementation of a stochastic reduced basis projection scheme it is seen that the method can be up to orders of magnitude faster than direct monte carlo simulation while providing results of comparable accuracy furthermore the proposed implementation is more efficient when fewer joints are affected by uncertainty

exact three wave solutions for the 3 1 dimensional jimbo miwa equation exact three wave solutions including periodic cross kink wave solutions doubly periodic solitary wave solutions and breather type of two solitary wave solutions for the 3 1 dimensional jimbo miwa equation are obtained using the generalized three wave method it is shown that the generalized three wave method with the help of symbolic computation provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics

auslander gorenstein algebras and precluster tilting abstract we generalize the notions of n cluster tilting subcategories and selfinjective algebras into n precluster tilting subcategories and n selfinjective algebras where we show that a subcategory naturally associated to n precluster tilting subcategories has a higher auslander reiten theory furthermore we give a bijection between n precluster tilting subcategories and n minimal auslander gorenstein algebras which is a higher dimensional analog of auslander solberg correspondence 8 as well as a gorenstein analog of n auslander correspondence 22 the auslander reiten theory associated to an n precluster tilting subcategory is used to classify the n minimal auslander gorenstein algebras into four disjoint classes our method is based on relative homological algebra due to auslander solberg

on n translation algebras abstract motivated by iyama s higher representation theory we introduce n translation quivers and n translation algebras the classical z q construction of the translation quiver is generalized to construct an n 1 translation quiver from an n translation quiver using trivial extension and smash product we find a sufficient and necessary condition for the quadratic dual of n translation algebras to have n almost split sequences in the category of its projective modules

multiple front solutions for the burgers equation and the coupled burgers equations abstract in this work multiple front solutions for the burgers equation and the coupled burgers equations are examined the tanh coth method and the cole hopf transformation are used the work highlights the power of the proposed schemes and the structures of the obtained multiple front solutions

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

higher nakayama algebras i construction we introduce higher dimensional analogues of the nakayama algebras from the viewpoint of iyama s higher auslander reiten theory more precisely for each nakayama algebra a and each positive intege

extended trial equation method to generalized nonlinear partial differential equations in this article we give the extended trial equation method for solving nonlinear partial differential equations with higher order nonlinearity by use of this method the exact traveling wave solutions including soliton solution singular soliton solutions rational function solution and elliptic integral function solution to one dimensional general improved kdv gikdv equation and r m n equation are obtained also a more general trial equation method is proposed

almost split sequences in dimension two a does not split b if h x r c is a morphism in c which is not a splittable epimorphism in c then there is a t x c such that gt h c if j ci y is a morphism in c which is not a splittable monomorphism in c then there is an s c y in c such that j sf almost split sequences were first introduced by us in the case a is the category of finitely generated modules over an artin algebra i and c a see 6 71 the reader is referred to the expository articles 11 121 for an account of the role almost split sequences have played in the theory of the representation theory of artin algebras that almost split sequences exist for various subcategories of module categories over a much wider class of rings than artin algebras was first shown in 2 31 where various existence theorems for almost split sequences were given these results seemed to indicate that there might be connections between the structure and existence of almost split sequences and algebraic geometry recent developments show that this is indeed the case in 4 it is shown that the structure of the almost split sequences in the category of reflexive modules of a complete rational double point over the complex numbers determines the desingularization graph of the singularity also it has been shown in s that a complete cohen macaulay local ring s is an isolated singularity if and only if the category of cohen macaulay

stochastic reduced basis methods stochastic reduced basis methods for solving large scale linear random algebraic systems of equations such as those obtained by discretizing linear stochastic partial differential equations in space time and the random dimension are introduced the fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients or random functions we present a theoretical justification for employing basis vectors spanning the preconditioned stochastic krylov subspace to approximate the response process subsequently variants of the bubnov galerkin scheme are employed to compute the undetermined coefficients which allow explicit expressions for the response quantities to be derived we also examine some theoretical properties of the projection scheme and procedures for computing the response statistics numerical studies are presented for static and dynamic analysis of stochastic structural systems we demonstrate that significant improvements over the neumann expansion scheme as well as other relevant techniques in the literature can be achieved

n representation infinite algebras abstract from the viewpoint of higher dimensional auslander reiten theory we introduce a new class of finite dimensional algebras of global dimension n which we call n representation infinite they are a certain analog of representation infinite hereditary algebras and we study three important classes of modules n preprojective n preinjective and n regular modules we observe that their homological behaviour is quite interesting for instance they provide first examples of algebras having infinite ext 1 orthogonal families of modules moreover we give general constructions of n representation infinite algebras applying minamoto s theory on fano algebras in non commutative algebraic geometry we describe the category of n regular modules in terms of the corresponding preprojective algebra then we introduce n representation tame algebras and show that the category of n regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra this generalizes the classical description of regular modules over tame hereditary algebras as an application we show that the representation dimension of an n representation tame algebra is at least n 2

model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics summary a priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework the proper generalized decomposition method is used to construct a quasi optimal separated representation of the random solution at some frequency samples at each frequency an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions an extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters this strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties this reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest because the dynamic response is highly nonlinear with respect to the input random parameters a second level of separation of variables is introduced for the representation of functions of multiple random parameters thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension copyright 2011 john wiley sons ltd

reduced basis approximations of structural displacements for optimaldesign a new solution procedure for improved approximate reanalyses of structures using results of a single precise analysis is presented the proposed procedure is based on combining the computed terms of a series expansion used as high quality basis vectors with coefficients of a reduced basis expression the latter coefficients can readily be determined for each trial design by various criteria the results are measured by the errors in satisfying the analysis equations and are compared in terms of the quality of the approximations the proposed approach is suitable for various types of design variables and can be used with a general finite element model a general reanalysis procedure is introduced and the physical significance of first order approximations is demonstrated several numerical examples illustrate the effectiveness of the solution process it is shown that high quality approximations of displacements can be obtained with a small computational effort for very large changes in the design variables

exp function method for solving nonlinear evolution equations with higher order nonlinearity abstract in this paper the exp function method is used to obtain generalized solitary solutions of the generalized drinfel d sokolov wilson dsw system and the generalized 2 1 dimensional burgers type equation then some of the solitary solutions are converted to periodic solutions or hyperbolic function solutions by a simple transformation the results show that the exp function method is a powerful and convenient mathematical tool for solving nonlinear evolution equations with higher order nonlinearity

new variable separation solutions of the 2 1 dimensional asymmetric nizhnik novikov veselov system abstract by means of the exp function method we obtain some new exact solutions of the riccati equation based on the riccati equation and its new exact solutions we find new and more general variable separation solutions with two arbitrary functions of the 2 1 dimensional asymmetric nizhnik novikov veselov annv system as some special examples these new variable separation solutions can degenerate into some known solutions reported in open literature

a note on abelian quotient categories abstract let c be a triangulated category with a serre functor s and x a non zero contravariantly finite rigid subcategory of c then x is cluster tilting if and only if the quotient category c x is abelian and s x x 2 as an application this result generalizes the work by beligiannis

exact solutions of the drinfel d sokolov wilson equation using the exp function method abstract the generalized solitary solutions of the classical drinfel d sokolov wilson equation dswe are obtained using the exp function method then some of these solutions are easily converted into kink shaped solutions and blow up solutions by a simple transformation

the g g expansion method and travelling wave solutions for a higher order nonlinear schrodinger equation abstract the higher order nonlinear schrodinger equation is reduced to a nonlinear ordinary differential equation ode by using a simple transformation various solutions of the nonlinear ode are obtained by the g g expansion method proposed recently with the aid of solutions of the nonlinear ode more explicit travelling wave solutions involving two arbitrary parameters of the higher order nonlinear schrodinger equation are found out the travelling waves are expressed by the hyperbolic functions trigonometric functions and rational functions when the parameters are taken as special values the solitary waves are also derived from the travelling waves

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

τ2 stable tilting complexes over weighted projective lines abstract let x be a weighted projective line and coh x the associated category of coherent sheaves we classify the tilting complexes t in d b coh x such that 2 t t where is the auslander reiten translation in d b coh x as an application of this result we classify the 2 representation finite algebras which are derived equivalent to a canonical algebra this complements iyama oppermann s classification of the iterated tilted 2 representation finite algebras by passing to 3 preprojective algebras we obtain a classification of the selfinjective cluster tilted algebras of canonical type this complements ringel s classification of the selfinjective cluster tilted algebras

d representation finite self injective algebras abstract in this paper we initiate the study of higher dimensional auslander reiten theory of self injective algebras we give a systematic construction of weakly d representation finite self injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the serre functor the condition holds in particular for all fractionally calabi yau algebras of global dimension at most d this generalizes riedtmann s classical construction of representation finite self injective algebras our method is based on an adaptation of gabriel s covering theory for k linear categories to the setting of higher dimensional auslander reiten theory applications include n fold trivial extensions and classical and higher preprojective algebras which are shown to be d representation finite in many cases we also get a complete classification of all d representation finite self injective nakayama algebras for arbitrary d

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

soliton solutions of the korteweg de vries and the kadomtsev petviashvili equations the wronskian technique the well known soliton solutions of the kadomtsev petviashvili equations are written in terms of determinants of wronskian form by using this compact representation together with the hirota bilinear form of the equations it is demonstrated by elementary algebraic methods that the n soliton solution satisfies the evolution equation and the n and n 1 soliton solutions satisfy the associated backlund transformation the relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the n soliton solution is also given

d abelian quotients of d 2 angulated categories abstract let t be a triangulated category if t is a cluster tilting object and i add t is the ideal of morphisms factoring through an object of add t then the quotient category t i is abelian this is an important result of cluster theory due to keller reiten and konig zhu more general conditions which imply that t i is abelian were determined by grimeland and the first author now let t be a suitable d 2 angulated category for an integer d 1 if t is a cluster tilting object in the sense of oppermann thomas and i add t is the ideal of morphisms factoring through an object of add t then we show that t i is d abelian the notions of d 2 angulated and d abelian categories are due to geiss keller oppermann and jasso they are higher homological generalisations of triangulated and abelian categories which are recovered in the special case d 1 we actually show that if end t t is the endomorphism algebra of t then t i is equivalent to a d cluster tilting subcategory of mod in the sense of iyama this implies that t i is d abelian moreover we show that is a d gorenstein algebra more general conditions which imply that t i is d abelian will also be determined generalising the triangulated results of grimeland and the first author

skew group algebras of jacobian algebras for a quiver with potential q w with an action of a finite cyclic group g we study the skew group algebra lambda g of the jacobian algebra lambda p q w by a result of reiten and riedtmann

n angulated categories we dene n angulated categories by modifying the axioms of triangulated categories in a natural way we show that heller s parametrization of pre triangulations extends to pre n angulations we obtain a large class of examples of n angulated cate gories by considering n 2 cluster tilting subcategories of triangulated categories which are stable under the n 2 nd power of the suspension functor as an application we show how n angulated calabi yau categories yield triangulated calabi yau categories of higher calabi yau dimension finally we sketch a link to algebraic geometry and string theory

n representation finite algebras and n apr tilting we introduce the notion of n representation finiteness generalizing representationfinite hereditary algebras we establish the procedure of n apr tilting and show that it preserves n representation finiteness we give some combinatorial description of this procedure and use this to completely describe a class of n representation finite algebras called type a

pure semisimple n cluster tilting subcategories abstract from the viewpoint of higher homological algebra we introduce pure semisimple n abelian categories which are analogs of pure semisimple abelian categories let be an artin algebra and m be an n cluster tilting subcategory of mod we show that m is pure semisimple if and only if each module in m is a direct sum of finitely generated modules let m be an n cluster tilting subcategory of mod we show that add m is an n cluster tilting subcategory of mod if and only if m has an additive generator if and only if mod m is locally finite this generalizes auslander s classical results on pure semisimplicity of artin algebras