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

on trees with perfect matchings a tree is said to have a perfect matching if it has a spanning forest whose components are paths on two vertices only in this paper we develop upper bounds on the algebraic connectivity of such trees and we consider other eigenvalue properties of its laplacian matrix furthermore for trees with perfect matchings we refine a result due to kirkland neumann and shader concerning the connection between the maximal diagonal entry of the group inverse of the laplacian matrix of a general tree and the pendant vertices of the tree and use this refinement to narrow down the set of the pendant vertices of a tree with a perfect matching which can correspond to the maximal diagonal entry in the group inverse of its laplacian matrix

diffusive logistic equations with indefinite weights population models in disrupted environments the dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t d u m x cu u in 0 where u represents the population density c d 0 are constants describing the limiting effects of crowding and the diffusion rate of the population respectively and m x describes the local growth rate of the population if the environment is bounded and is surrounded by uninhabitable regions then u 0 on 0 the growth rate m x is positive on favourablehabitats and negative on unfavourable ones the object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled the models are shown to possess a unique stable positive steady state implying persistence for the population provided l d where is the principle positive eigenvalue for the problem m x in 0 on analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat in the limit as the diffusion rate d 0 the solutions tend toward the positive part of m x c and if m is discontinuous develop interior transition layers the analysis uses bifurcation and continuation methods the variational characterisation of eigenvalues upper and lower solution techniques and singular perturbation theory

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

spectral sequences for the cohomology rings of a smash product abstract stefan and guichardet have provided lyndon hochschild serre type spectral sequences which converge to the hochschild cohomology and ext groups of a smash product we show that these spectral sequences carry natural multiplicative structures and that these multiplicative structures can be used to calculate the cup product on hochschild cohomology and the yoneda product on an ext algebra

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

gorenstein subrings of invariants under hopf algebra actions abstract this paper concerns conditions on the action of a finite dimensional semisimple hopf algebra on an artin schelter regular algebra that force the subring of invariants to satisfy the artin schelter gorenstein condition classical results such as watanabe s theorem and stanley s theorem are extended from the case of a group action to the context of a hopf algebra action a hopf algebra version of the homological determinant is introduced and it becomes an important tool in the generalization from group actions to hopf algebra actions

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

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

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

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

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

the nehari manifold for elliptic equation involving the square root of the laplacian abstract in this paper we introduce the nehari manifold for elliptic problems involving the square root of the laplacian we use it to establish the existence of solutions and multiple solutions for some nonlinear elliptic problem with sign changing weight we also establish some bifurcation results and non existence results

on some critical problems for the fractional laplacian operator abstract we study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional laplacian 2 u u q u n n u 0 in u 0 on where r n is a smooth bounded domain n 1 0 0 q n n 0 min n 2 for suitable conditions on depending on q we prove in the case q 1 there exist at least two solutions for every 0 and some 0 at least one if no solution if for q 1 we show existence of at least one solution for 0 1 and nonexistence for 1 when q 1 the existence is shown for every 0 also we prove that the solutions are bounded and regular

actions of quantum linear spaces on quantum algebras abstract we study actions of bosonizations of quantum linear spaces on quantum algebras under mild conditions we classify actions on quantum affine spaces and quantum matrix algebras in the former case it is shown that all actions of generalized taft algebras are trivial extensions of actions on quantum planes in both cases we achieve bounds on the rank of the bosonization acting on the algebra

the discriminant criterion and automorphism groups of quantized algebras abstract we compute the automorphism groups of some quantized algebras including tensor products of quantum weyl algebras and some skew polynomial rings

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

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

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

existence theorems for dualizing complexes over non commutative graded and filtered rings abstract in this note we prove existence theorems for dualizing complexes over graded and filtered rings thereby generalizing some results by zhang yekutieli and jorgensen

bounds on the largest eigenvalues of trees with a given size of matching very little is known about upper bound for the largest eigenvalue of a tree with a given size of matching in this paper we find some upper bounds for the largest eigenvalue of a tree in terms of the number of vertices and the size of matchings which improve some known results

mckay correspondence for semisimple hopf actions on regular graded algebras i abstract in establishing a more general version of the mckay correspondence we prove auslander s theorem for actions of semisimple hopf algebras h on noncommutative artin schelter regular algebras a of global dimension two where a is a graded h module algebra and the hopf action on a is inner faithful with trivial homological determinant we also show that each fixed ring a h under such an action arises as an analogue of a coordinate ring of a kleinian singularity

stable categories of graded maximal cohen macaulay modules over noncommutative quotient singularities abstract tilting objects play a key role in the study of triangulated categories a famous result due to iyama and takahashi asserts that the stable categories of graded maximal cohen macaulay modules over quotient singularities have tilting objects this paper proves a noncommutative generalization of iyama and takahashi s theorem using noncommutative algebraic geometry namely if s is a noetherian as regular koszul algebra and g is a finite group acting on s such that s g is a gorenstein isolated singularity then the stable category cm z s g of graded maximal cohen macaulay modules has a tilting object in particular the category cm z s g is triangle equivalent to the derived category of a finite dimensional algebra

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

ordering trees with algebraic connectivity and diameter abstract this paper is motivated by abreu s paper which proposed finding a total ordering of trees by their algebraic connectivity is still an open problem we first discuss how to find a total ordering of trees with diameter 4 then these results are used to determine first several orderings of trees by algebraic connectivity in nonincreasing order in last we present a class of trees with large diameter whose algebraic connectivity is larger than that of a tree with diameter 3

global structure of positive solutions for nonlocal boundary value problems involving integral conditions abstract in this paper we consider the nonlinear eigenvalue problems u h t f u 0 0 t 1 u 0 0 u 1 0 1 u s d a s where 0 1 u s d a s is a stieltjes integral with a nondecreasing and a t is not a constant on 0 1 h c 0 1 0 and h t 0 on any subinterval of 0 1 f c 0 0 and f s 0 for s 0 and f 0 f 0 f 0 lim s 0 f s s f lim s f s s we investigate the global structure of positive solutions by using global bifurcation techniques

properties of the fixed ring of a preprojective algebra abstract for a finite group acting on a polynomial ring the chevalley shephard todd theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo reflections in recent years progress was made in work of kirkman kuzmanovich zhang and others to extend this result to artin schelter regular algebras by expanding pseudo reflections to quasi reflections naturally the question arises if the theory generalizes further to non connected noncommutative algebras our objects of study will be preprojective algebras which are certain factor algebras of path algebras corresponding to extended dynkin diagrams of type a d or e this work answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible on our way we will point out additional difficulties in establishing quasi reflections using the trace and reveal situations which do not occur for artin schelter regular algebras

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

hopf algebra actions abstract finite groups acting on rings by automorphisms and group graded rings are instances of hopf algebras h acting on h module algebras a we study such actions let a h a a h a e h a all h h the ring of h invariants and form the smash product a h we study the ring extensions ah a a h we prove a maschke type theorem for a h modules we form an associated morita context ah a a a h and use these to get connections between the various rings

on the second largest laplacian eigenvalues of graphs the second largest laplacian eigenvalue of a graph is the second largest eigenvalue of the associated laplacian matrix in this paper we study extremal graphs for the extremal values of the second largest laplacian eigenvalue and the laplacian separator of a connected graph respectively all simple connected graphs with second largest laplacian eigenvalue at most 3 are characterized it is also shown that graphs with second largest laplacian eigenvalue at most 3 are determined by their laplacian spectrum moreover the graphs with maximum and the second maximum laplacian separators among all connected graphs are determined

skew group algebras of calabi yau algebras abstract the calabi yau property of skew group algebras is discussed it is shown that the skew group algebra a g of a koszul calabi yau algebra a with a finite subgroup g of automorphisms of a is calabi yau if and only if g is a finite subgroup of the special linear group sl a which is defined by means of the homological determinant using the a algebra structure on the yoneda algebra some results in bocklandt et al 2010 bsw are generalized say every connected graded p koszul calabi yau algebra is derived from a superpotential the superpotential for the skew group algebra a g is also constructed

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

on the spectra of certain rooted trees abstract let t be an unweighted tree with vertex root v which is the union of two trees t 1 v 1 e 1 t 2 v 2 e 2 such that v 1 v 2 v and t 1 and t 2 have the property that the vertices in each of their levels have equal degree we characterize completely the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees moreover we give some results about the multiplicity of the eigenvalues applications to some particular trees are developed

irreducible representations of the 4 dimensional sklyanin algebra at points of infinite order abstract in 1982 sklyanin funct anal appl 16 1982 27 34 defined a family of graded algebras a e depending on an elliptic curve e and a point e which is not 4 torsion basic properties of these algebras were established in smith and stafford compositio math 83 1992 259 289 and a study of their representation theory was begun in levasseur and smith bull soc math france121 1993 35 90 the present paper classifies the finite dimensional simple a modules when is a point of infinite order sklyanin funct anal appl 17 1983 273 284 defines for each k n a representation of a in a certain k dimensional subspace of theta functions of order 2 k 1 we prove that these are irreducible representations and that any other simple module is obtained by twisting one of these by an automorphism of a the automorphism group of a is explicitly computed the method of proof relies on results in levasseur and smith in particular it is proved that every finite dimensional simple module is a quotient of a line module an important part of the analysis is a determination of the 1 critical a modules and the fact that such a module is equivalent to a quotient of a line module by a shifted line module

spectrum and constant sign solutions for a fractional laplace problem with sign changing weight abstract in this paper we study the spectrum of the fractional laplace operator with sign changing weight and show that there exist two simple isolated principal eigenvalues 1 and 1 by use of the obtained spectrum results we study the existence multiplicity and nonexistence of constant sign solutions to corresponding nonlinear problems according to the asymptotic behavior of nonlinear term f at 0 and whether f possesses zeros in r 0 as far as we know the spectral results presented here are new and the rest theorems partially generalize the corresponding ones in the literature

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

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

on the two conjectures of graffiti we study the laplacian eigenvalues of trees on n vertices with independence number and describe all extremal graphs that attain the maximal laplacian spectral radius and algebraic connectivity then the results are used to confirm two conjectures of graffiti wow conjectures 584 and 636 on the relationship between the laplacian eigenvalues and the independence number of a graph

gorensteinness of invariant subrings of quantum algebras abstract we prove auslander gorenstein and gkdim macaulay properties for certain invariant subrings of some quantum algebras the weyl algebras and the universal enveloping algebras of finite dimensional lie algebras

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

ideal classes of three dimensional sklyanin algebras abstract in this paper we classify graded reflexive ideals up to isomorphism and shift in certain three dimensional artin schelter regular algebras this classification is similar to the classification of right ideals in the first weyl algebra a problem that was completely settled recently the situation we consider is substantially more complicated however