a new neural network for solving linear and quadratic programming problems a new neural network for solving linear and quadratic programming problems is presented and is shown to be globally convergent the new neural network improves existing neural networks for solving these problems it avoids the parameter turning problem it is capable of achieving the exact solutions and it uses only simple hardware in which no analog multipliers for variables are required furthermore the network solves both the primal problems and their dual problems simultaneously

the representation dimension of a selfinjective algebra of euclidean type abstract we prove that the representation dimension of a selfinjective algebra of euclidean type is equal to three and give an explicit construction of the auslander generator of its module category

representation dimension and finitely generated cohomology abstract we consider selfinjective artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators for such an algebra we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules this provides a unified approach to computing lower bounds for the representation dimension of group algebras exterior algebras and artin complete intersections we also obtain new examples of classes of algebras with arbitrarily large representation dimension

dominant dimension and tilting modules we study which algebras have tilting modules that are both generated and cogenerated by projective injective modules crawley boevey and sauter have shown that auslander algebras have such tilting modules and for algebras of global dimension 2 auslander algebras are classified by the existence of such tilting modules in this paper we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2 independent of its global dimension in general such a tilting module is not necessarily cotilting here we show that the algebras which have a tilting cotilting module generated cogenerated by projective injective modules are precisely 1 minimal auslander gorenstein algebras when considering such a tilting module without the assumption that it is cotilting we study the global dimension of its endomorphism algebra and discuss a connection with the finitistic dimension conjecture furthermore as special cases we show that triangular matrix algebras obtained from auslander algebras and certain injective modules have such a tilting module we also give a description of which nakayama algebras have such a tilting module

representation type of hecke algebras of type a in this paper we provide a complete classification of the representation type for the blocks for the hecke algebra of type a stated in terms of combinatorical data the computation of the complexity of young modules is a key component in the proof of this classification result

representation dimension and quasi hereditary algebras we study auslander s representation dimension of artin algebras which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators we show the following statements 1 if an artin algebra a is stably hereditary then the representation dimension of a is at most 3 2 if two artin algebras are stably equivalent of morita type then they have the same representation dimension particularly if two self injective algebras are derived equivalent then they have the same representation dimension 3 any incidence algebra of a finite partially ordered set over a field has finite representation dimension moreover we use results on quasi hereditary algebras to show that 4 the auslander algebra of a nakayama algebra has finite representation dimension

a novel neural network for nonlinear convex programming in this paper we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method the main idea is to convert the convex programming problem into a variational inequality problem then a dynamical system and a convex energy function are constructed for resulting variational inequality problem it is shown that the proposed neural network is stable in the sense of lyapunov and can converge to an exact optimal solution of the original problem compared with the existing neural networks for solving the nonlinear convex programming problem the proposed neural network has no lipschitz condition no adjustable parameter and its structure is simple the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

the acyclic closure of an exact category and its triangulation abstract for any exact category a with splitting idempotents a maximal exact category t a containing a as a biresolving subcategory is constructed important types of exact categories including n tilting torsion classes categories of cohen macaulay modules over a cohen macaulay order or categories of gorenstein projectives are shown to be of the form t a the quotient category t a a in the sense of grothendieck always exists and carries a triangulated structure more generally it is proved that any biresolving subcategory a of an exact category m gives a triangulated localization m a for example the unbounded derived category d a of an exact category a is obtained directly with no passage through a homotopy category of complexes as applications some recent developments related to gorenstein projectivity non commutative crepant resolutions singularity categories and cohen macaulay representations are extended and improved in the new framework for example the concept of non commutative resolution of a noetherian frobenius category is extended to arbitrary exact categories which leads to an overarching connection with representation dimension of exact categories and n tilting

representation dimension via tilting theory abstract let be an artin algebra over a commutative artinian ring r and let t be a separating splitting tilting module with endomorphism end t the aim of this paper is to use tilting theory to study the representation dimension of the main result asserts that for an integer n 1 if is n gorenstein of finite cohen macaulay type then r e p d i m n 2 we conclude that if is a n gorenstein artin algebra of finite cohen macaulay type then r e p d i m n 2 this in particular shows that for any artin algebra r e p d i m g l d i m 2

finitistic dimension and igusa todorov algebras abstract the notion of igusa todorov algebras is introduced in connection with the little finitistic dimension conjecture and the conjecture is proved for those algebras such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds e g algebras with the representation dimension at most 3 algebras with radical cube zero monomial algebras and left serial algebras etc it is an open question whether all artin algebras are igusa todorov we provide some methods to construct many new classes of 2 igusa todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds in particular we show that the class of 2 igusa todorov algebras is closed under taking endomorphism algebras of projective modules hence if all quasi hereditary algebras are 2 igusa todorov then all artin algebras are 2 igusa todorov by v dlab c m ringel every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring proc amer math soc 107 1 1989 1 5 and have finite finitistic dimension

analysis and design of a k winners take all model with a single state variable and the heaviside step activation function this paper presents a k winners take all kwta neural network with a single state variable and a hard limiting activation function first following several kwta problem formulations related existing kwta networks are reviewed then the kwta model model with a single state variable and a heaviside step activation function is described and its global stability and finite time convergence are proven with derived upper and lower bounds in addition the initial state estimation and a discrete time version of the kwta model are discussed furthermore two selected applications to parallel sorting and rank order filtering based on the kwta model are discussed finally simulation results show the effectiveness and performance of the kwta model

on the representation dimension of tilted and laura algebras abstract we prove that the representation dimension of a tilted or of a strict laura algebra is at most three

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

the representation dimension of a selfinjective algebra of wild tilted type abstract we prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three and give an explicit construction of an auslander generator of its module category we also show that if a connected selfinjective algebra admits an acyclic generalised standard auslander reiten component then its representation dimension is equal to three

finitistic dimension conjecture and relative hereditary algebras abstract we introduce the notion of relative hereditary artin algebras as a generalization of algebras with representation dimension at most 3 we prove the following results 1 the relative hereditariness of an artin algebra is left right symmetric and is inherited by endomorphism algebras of projective modules 2 the finitistic dimensions of a relative hereditary algebra and its opposite algebra are finite as a consequence the finitistic projective dimension conjecture the finitistic injective dimension conjecture the gorenstein symmetry conjecture the wakamatsu tilting conjecture and the generalized nakayama conjecture hold for relative hereditary artin algebras and endomorphism algebras of projective modules over them in particular over algebras with representation dimension at most 3 we also show that the torsionless finiteness of an artin algebra is inherited by endomorphism algebras of projective modules and consequently give a partial answer to the question if the representation dimension of the endomorphism algebra of any projective module over an artin algebra a is bounded by the representation dimension of a

separable equivalence complexity and representation type abstract we generalise the notion of separable equivalence originally presented by linckelmann in 13 to an equivalence relation on additive categories we use this generalisation to show that from an initial equivalence between two algebras we may build equivalences between many related categories we also show that separable equivalence preserves the representation type of an algebra this generalises linckelmann s result in 13 where he showed this in the case of symmetric algebras we use these theorems to show that the group algebras of several small cyclic groups cannot be separably equivalent this gives several examples of algebras that have the same complexity but are not separably equivalent

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

tilting modules and representation dimensions in this paper we compare the representation dimensions of two algebras linked by certain tilting modules our main results can be stated as follows suppose t is a tilting module over a and b enda t then 1 if t is separating and splitting then rep dim a rep dim b 2 if t p 1s is an apr tilting module and the injective dimension of s is at most 2 then rep dim b rep dim a 1

radical embeddings and representation dimension given a representation finite algebra b and a subalgebra a of b such that the jacobson radicals of a and b coincide we prove that the representation dimension of a is at most three by a result of igusa and todorov this implies that the finitistic dimension of a is finite

cohomology of twisted tensor products abstract it is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies we look at twisted tensor products and investigate to which extent this is still true we give an explicit description of the ext algebra of the tensor product of two modules and under certain additional conditions describe an essential part of the hochschild cohomology ring of a twisted tensor product as an application we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the hochschild cohomology ring moreover both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra

on the finitistic dimension conjecture iii related to the pair eae a let a be an artin algebra and e an idempotent element in a in this paper we use co homological conditions on a to control the finitistic dimension of eae such a consideration is of particular interest for understanding the finitistic dimension conjecture let us denote the finitistic dimension and global dimension of a by fin dim a and gl dim a respectively suppose gl dim a 4 then fin dim eae if one of the following conditions holds 1 a aea has representation dimension at most 3 2 a 3 a is an a aea module 3 proj dim sa 3 for all simple a aea modules s this result can be considered as a first step to the question of whether gl dim a 4 implies fin dim eae moreover we show the following let a be an arbitrary artin algebra and e an idempotent element of a such that the syzygy dimension or the gorenstein dimension of the eae module ae is finite if fin dim a then fin dim eae

convex two level optimization in this paper a model for a two level planning problem is presented in the form of a static stackelberg game by assumption play is sequential and noncooperative however the leader can influence the actions of the followers through a set of coordination variables while the followers responses may partly determine the leader s payoff under certain convexity assumptions it is shown that the feasible region induced by the leader is continuous in the original problem variables this observation coupled with two corollary results are used as a basis for a hybrid algorithm which clings to the inducible region until a local optimum is found a branching scheme is then employed to located other segments of the region eventually terminating with the global optimum a number of examples are given to highlight the results while the algorithm s performance is tested in a comparison with two other procedures

an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions this paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties the method is an extension of our earlier method published in this journal zionts s wallenius j 1976 an interactive programming method for solving the multiple criteria problem management sci 22 6 652 663 various technical problems present in predecessor versions have been resolved in addition to presenting the supporting theory and algorithm we discuss certain options in implementation and summarize our practical experience with several versions of the method

on the representation dimension of finite dimensional algebras the importance of the representation finite artin algebras for the whole representation theory of artin algebras is well understood and there is much literature on the subject see 1 6 12 these are artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules moreover there is a bijective correspondence between the class of representation finite artin algebras and that of artin algebras with global dimension of at most 2 and with dominant dimension of at least 2 this result was established by auslander in 1 motivated by this correspondence auslander introduced the concept of representation dimension for artin algebras to study the connection of arbitrary artin algebras with representation finite artin algebras it is hoped that this notion gives a reasonable way of measuring how far an artin algebra is from being representation finite type 1 p 134 representation dimension is a morita invariant of artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators unfortunately little seems to be known about representation dimension so there are a lot of essential questions on this invariant dealing with the computation and estimation of the representation dimension of an artin algebra there are a few cases known auslander proved that the representation dimension of an artin algebra is two if and

multiplicative decision rules for multiobjective decision problems abstract properties of consistency of general decision rules for multiobjective decision problems are discussed decision rules based on the family of weighted geometric averages are introduced their properties are analysed and possible implementations are discussed

representation dimensions of triangular matrix algebras abstract let a be a finite dimensional hereditary algebra over an algebraically closed field k t 2 a a 0 a a be the triangular matrix algebra we prove that rep dim t 2 a is at most three if a is dynkin type and rep dim t 2 a is at most four if a is not dynkin type let a 1 a 0 da a be the duplicated algebra of a let t be a tilting a module and t t p be a tilting a 1 module we show that end a 1 t is representation finite if and only if the full subcategory x a y a f x a mod a y a 1 f t a add a of mod t 2 a is of finite type where is the auslander reiten translation and f t a is the torsion free class of mod a associated with t moreover we also prove that rep dim end a 1 t is at most three if a is dynkin type

homology of perfect complexes abstract it is proved that the sum of the loewy lengths of the homology modules of a finite free complex f over a local ring r is bounded below by a number depending only on r this result uncovers in the structure of modules of finite projective dimension obstructions to realizing r as a closed fiber of some flat local homomorphism other applications include as special cases uniform proofs of known results on free actions of elementary abelian groups and of tori on finite cw complexes the arguments use numerical invariants of objects in general triangulated categories introduced here and called levels they allow one to track through changes of triangulated categories homological invariants like projective dimension as well as structural invariants like loewy length an intermediate result sharpens with a new proof the new intersection theorem for commutative algebras over fields under additional hypotheses on the ring r stronger estimates are proved for loewy lengths of modules of finite projective dimension

a recurrent neural network for solving bilevel linear programming problem in this brief based on the method of penalty functions a recurrent neural network nn modeled by means of a differential inclusion is proposed for solving the bilevel linear programming problem blpp compared with the existing nns for blpp the model has the least number of state variables and simple structure using nonsmooth analysis the theory of differential inclusions and lyapunov like method the equilibrium point sequence of the proposed nns can approximately converge to an optimal solution of blpp under certain conditions finally the numerical simulations of a supply chain distribution model have shown excellent performance of the proposed recurrent nns

a new regularization scheme for mathematical programs with complementarity constraints we propose a new regularization scheme for mathematical programs with complementarity constraints mpcc by relaxing all the constraints of the complementarity system we show that under the mpcc linear independence constraint qualifications mpcc licq the lagrange multipliers exist for this regularization our method has strong convergence properties under mpcc linear independence constraint qualifications and some weak conditions of the strict complementarity in particular under mpcc licq it is shown that any accumulation point of the regularized stationary points is m stationary for the mpcc problem and if the asymptotically weak nondegeneracy condition holds at a stationary point of the regularized problem then it is strongly stationary an algorithm for solving the proposed regularization is presented and numerical experiments are reported some comparisons with other methods are discussed with illustrative examples

a two layer recurrent neural network for nonsmooth convex optimization problems in this paper a two layer recurrent neural network is proposed to solve the nonsmooth convex optimization problem subject to convex inequality and linear equality constraints compared with existing neural network models the proposed neural network has a low model complexity and avoids penalty parameters it is proved that from any initial point the state of the proposed neural network reaches the equality feasible region in finite time and stays there thereafter moreover the state is unique if the initial point lies in the equality feasible region the equilibrium point set of the proposed neural network is proved to be equivalent to the karush kuhn tucker optimality set of the original optimization problem it is further proved that the equilibrium point of the proposed neural network is stable in the sense of lyapunov moreover from any initial point the state is proved to be convergent to an equilibrium point of the proposed neural network finally as applications the proposed neural network is used to solve nonlinear convex programming with linear constraints and l 1 norm minimization problems

the representation dimension of hecke algebras and symmetric groups abstract we establish a lower bound for the representation dimension of all the classical hecke algebras of types a b and d for all the type a algebras and most of the algebras of types b and d we also establish upper bounds moreover we establish bounds for the representation dimension of group algebras of some symmetric groups

a neural network approach for solving nonlinear bilevel programming problem a neural network model is presented for solving nonlinear bilevel programming problem which is a np hard problem the proposed neural network is proved to be lyapunov stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem the asymptotic properties of the neural network are analyzed and the condition for asymptotic stability solution feasibility and solution optimality are derived the transient behavior of the neural network is simulated and the validity of the network is verified with numerical examples

gluing of idempotents radical embeddings and two classes of stable equivalences abstract stable equivalences are studied between any finite dimensional algebra a with a simple projective module and a simple injective module and an algebra b obtained from a by gluing the corresponding idempotents of a this extends results by martinez villa stable equivalences modulo projectives are compared to stable equivalences modulo semisimples and in either situation a characterization is given for a radical embedding to induce such a stable equivalence

a neurodynamic optimization approach to bilevel quadratic programming this paper presents a neurodynamic optimization approach to bilevel quadratic programming bqp based on the karush kuhn tucker kkt theorem the bqp problem is reduced to a one level mathematical program subject to complementarity constraints mpcc it is proved that the global solution of the mpcc is the minimal one of the optimal solutions to multiple convex optimization subproblems a recurrent neural network is developed for solving these convex optimization subproblems from any initial state the state of the proposed neural network is convergent to an equilibrium point of the neural network which is just the optimal solution of the convex optimization subproblem compared with existing recurrent neural networks for bqp the proposed neural network is guaranteed for delivering the exact optimal solutions to any convex bqp problems moreover it is proved that the proposed neural network for bilevel linear programming is convergent to an equilibrium point in finite time finally three numerical examples are elaborated to substantiate the efficacy of the proposed approach

on the representation dimension of some classes of algebras abstract we show that the representation dimension of the following classes of algebras is at most 3 a artin algebras a such that the functor hom a d a has finite length or dually hom a a has finite length these algebras coincide with the right left glued algebras as introduced in i assem f u coelho j pure appl algebra 96 3 1994 225 and b trivial extensions of iterated tilted algebras

an upper bound for the dimension of bounded derived categories abstract let be an artin algebra we give an upper bound for the dimension of the bounded derived category of the category mod of finitely generated right modules in terms of the projective and injective dimensions of certain class of simple right modules as well as the radical layer length of in addition we give an upper bound for the dimension of the singularity category of mod in terms of the radical layer length of

application of particle swarm optimization algorithm for solving bi level linear programming problem bi level linear programming is a technique for modeling decentralized decision it consists of the upper level and lower level objectives this paper attempts to develop an efficient method based on particle swarm optimization pso algorithm with swarm intelligence the performance of the proposed method is ascertained by comparing the results with genetic algorithm ga using four problems in the literature and an example of supply chain model the results illustrate that the pso algorithm outperforms ga in accuracy

gorenstein derived categories abstract gorenstein derived categories are defined and the relation with the usual derived categories is given the bounded gorenstein derived categories of gorenstein rings and of finite dimensional algebras are explicitly described via the homotopy categories of gorenstein projective modules and some applications are obtained gorenstein derived equivalences between cm finite gorenstein algebras are discussed

a one layer recurrent neural network with a discontinuous hard limiting activation function for quadratic programming in this paper a one layer recurrent neural network with a discontinuous hard limiting activation function is proposed for quadratic programming this neural network is capable of solving a large class of quadratic programming problems the state variables of the neural network are proven to be globally stable and the output variables are proven to be convergent to optimal solutions as long as the objective function is strictly convex on a set defined by the equality constraints in addition a sequential quadratic programming approach based on the proposed recurrent neural network is developed for general nonlinear programming simulation results on numerical examples and support vector machine svm learning show the effectiveness and performance of the neural network

the polynomial hierarchy and a simple model for competitive analysis the multi level linear programs of candler norton and townsley are a simple class of sequenced move games in which players are restricted in their moves only by common linear constraints and each seeks to optimize a fixed linear criterion function in his her own continuous variables and those of other players all data of the game and earlier moves are known to a player when he she is to move the one player case is just linear programming

convergence properties of a regularization scheme for mathematical programs with complementarity constraints we study the convergence behavior of a sequence of stationary points of a parametric nlp which regularizes a mathematical program with equilibrium constraints mpec in the form of complementarity conditions accumulation points are feasible points of the mpec they are c stationary if the mpec linear independence constraint qualification holds they are m stationary if in addition an approaching subsequence satisfies second order necessary conditions and they are b stationary if in addition an upper level strict complementarity condition holds these results complement recent results of fukushima and pang convergence of a smoothing continuation method for mathematical programs with equilibrium constraints in ill posed variational problems and regularization techniques springer verlag new york 1999 we further show that every local minimizer of the mpec which satisfies the linear independence upper level strict complementarity and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric nlp

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

on using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints we investigate the possibility of solving mathematical programs with complementarity constraints mpccs using algorithms and procedures of smooth nonlinear programming although mpccs do not satisfy a constraint qualification we establish sufficient conditions for their lagrange multiplier set to be nonempty mpccs that have nonempty lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter in this context the elastic mode transforms mpcc into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem which in turn makes it approachable by sequential quadratic programming sqp algorithms one such algorithm is shown to achieve local linear convergence once the problem is relaxed under stronger conditions we also prove superlinear convergence to the solution of an mpcc using an adaptive elastic mode approach for an sqp algorithm recently analyzed in an mpcc context in r fletcher s leyffer s sholtes and d ralph local convergence of sqp methods for mathematical programs with equilibrium constraints tech report na 210 university of dundee dundee uk 2002 our assumptions are more general since we do not use a critical assumption from that reference in addition we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen

a neural network for solving a convex quadratic bilevel programming problem a neural network is proposed for solving a convex quadratic bilevel programming problem based on lyapunov and lasalle theories we prove strictly an important theoretical result that for an arbitrary initial point the trajectory of the proposed network does converge to the equilibrium which corresponds to the optimal solution of a convex quadratic bilevel programming problem numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem

dimensions of triangulated categories via koszul objects lower bounds for the dimension of a triangulated category are provided these bounds are applied to stable derived categories of artin algebras and of commutative complete intersection local rings as a consequence one obtains bounds for the representation dimensions of certain artin algebras

the representation dimension of k x y x2 yn the representation dimension of an artin algebra was defined by m auslander in 1970 the precise value is not known in general and is very hard to compute even for small examples for group algebras it is known in the case of cyclic sylow subgroups for some group algebras in characteristic 2 of rank at least 3 the precise value of the representation dimension follows from recent work of r rouquier there is a gap for group algebras of rank 2 in this paper we show that for all n 0 and any field k the commutative algebras k x y x2 y2 n have representation dimension 3 for the proof we give an explicit inductive construction of a suitable generator cogenerator as a consequence we obtain that the group algebras in characteristic 2 of the groups c2 c2m have representation dimension 3 note that for m 3 these group algebras have wild representation type

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

invariant properties of representations under excellent extensions abstract in this paper we introduce the notion of weak excellent extensions of rings as a generalization of that of excellent extensions of rings let be a weak excellent extension of an artinian algebra we prove that if is of finite representation type resp cm finite cm free then so is furthermore if is an excellent extension of then the converse also holds true we also study when the representation dimension of an artinian algebra is invariant under excellent extensions

relative homology and maximal l orthogonal modules abstract let be an artin algebra and let m 1 and m 2 be generator cogenerators for mod such that the global dimension of their endomorphism ring is finite we characterize tilting modules of the form hom m 2 m 1 with respect to the relative theories induced by m 1 and m 2 on mod and apply our results to maximal l orthogonal modules

gorenstein algebras of finite cohen macaulay type abstract an artin algebra a is said to be cm finite if there are only finitely many isomorphism classes of indecomposable finitely generated gorenstein projective a modules inspired by auslander s idea on representation dimension we prove that for 2 n a is a cm finite n gorenstein algebra if and only if there is a resolving gorenstein projective a module e such that gl dim end a e op n

stepsize range and optimal value for taylor zhang discretization formula applied to zeroing neurodynamics illustrated via future equality constrained quadratic programming in this brief future equality constrained quadratic programming fecqp is studied via a zeroing neurodynamics method a continuous time zeroing neurodynamics ctzn model is presented by using taylor zhang discretization formula to discretize the ctzn model a taylor zhang discrete time zeroing neurodynamics tz dtzn model is presented to perform fecqp furthermore we focus on the critical parameter of the tz dtzn model i e stepsize by theoretical analyses we obtain an effective range of the stepsize which guarantees the stability of the tz dtzn model in addition we further discuss the optimal value of the stepsize which makes the tz dtzn model possess the optimal stability i e the best stability with the fastest convergence finally numerical experiments and application experiments for motion generation of a robot manipulator are conducted to verify the high precision of the tz dtzn model and the effective range and optimal value of the stepsize for fecqp

ideals in triangulated categories phantoms ghosts and skeleta we begin by showing that in a triangulated category specifying a projective class is equivalent to specifying an ideal i of morphisms with certain properties and that if i has these properties then so does each of its powers we show how a projective class leads to an adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence we use this to study various ideals in the stable homotopy category we examine phantom maps skeletal phantom maps superphantom maps and ghosts a ghost is a map which induces the zero map of homotopy groups we show that ghosts lead to a stable analogue of the lusternik schnirelmann category of a space and we calculate this stable analogue for low dimensional real projective spaces we also give a relation between ghosts and the hopf and kervaire invariant problems in the case ofa modules over ana ring spectrum the ghost spectral sequence is a universal coefficient spectral sequence from the phantom projective class we derive a generalized milnor sequence for filtered diagrams of finite spectra and from this it follows that the group of phantom maps fromxtoycan always be described as alim 1group the last two sections focus on algebraic examples in the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of kelly on the vanishing of composites of such maps we also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non trivially

an interactive weighted tchebycheff procedure for multiple objective programming the procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted tchebycheff metric using filtering techniques maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration the procedure has the advantage that it can converge to non extreme final solutions especially suitable for multiple objective linear programming the procedure is also applicable to integer and nonlinear multiple objective programs

powers of the phantom ideal the mono epi me exact structure on the morphisms of an exact category a e is introduced and used to prove ideal versions of salce s lemma christensen s ghost lemma and wakamatsu s lemma for an exact category salce s lemma establishes a bijective correspondence i i between the class of special precovering ideals of a e and that of its special preenveloping ideals me extensions of morphisms are used to define an extension i j of ideals christensen s lemma asserts that the class of special precovering respectively special preenveloping ideals is closed under products and extensions and that the bijective correspondence of salce s lemma satisfies ij j i and i j j i wakamatsu s lemma asserts that if a covering ideal i is closed under me extensions then it is a special precovering ideal as an application it is proved that if g is a finite group and is the ideal of phantom morphisms in the category k g mod then n 1 is the object ideal generated by projective modules where n is the nilpotency index of the jacobson radical j if r is a semiprimary ring with jn 0 then n is generated by projective modules for a right coherent ring r over which every cotorsion left r module has a coresolution of length n by pure injective modules n 1 is generated by flat modules

dynamical analysis of neural networks of subgradient system in this technical note we consider a class of neural network which is a generalization of neural network models considered in the optimization context under some mild assumptions this neural network can be translated into a negative subgradient dynamical system at first we study the existence and uniqueness of solution of this neural network then by nonsmooth ojasiewicz inequality we prove the convergence of the trajectories of this neural network in the end a constrained minimization problem is studied which can be associated with this neural network it is proved that the local constrained minimum of the cost function coincides with the stable equilibria point of this neural network

on the finitistic dimension conjecture ii related to finite global dimension abstract in this paper we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras on the one hand and the finitistic dimensions of their subalgebras on the other hand this is a continuation of the project in j pure appl algebra 193 2004 287 305 for an artin algebra a we denote by gl dim a fin dim a and rep dim a the global dimension finitistic dimension and representation dimension of a respectively the jacobson radical of a is denoted by rad a the main results in the paper are as follows let b be a subalgebra of an artin algebra a such that rad b is a left ideal in a then 1 if gl dim a 4 and rad a rad b a then fin dim b 2 if rep dim a 3 then fin dim b the results are applied to pullbacks of algebras over semi simple algebras moreover we have also the following dual statement 3 let b a be a surjective homomorphism between two algebras b and a suppose that the kernel of is contained in the socle of the right b module b b if gl dim a 4 or rep dim a 3 then fin dim b finally we provide a class of algebras with representation dimension at most three 4 if a is stably hereditary and rad b is an ideal in a then rep dim b 3

lower bounds for auslander s representation dimension the notion of representation dimension of a finite dimensional algebra has been introduced by auslander he has shown that an algebra is of finite representation type if and only if its representation dimension is at most two he has expected that the representation dimension of a representation infinite algebra should measure how far this algebra is from being representation finite in many cases and thanks to iyama also in general upper bounds for the representation dimension are known however there are non trivial lower bounds only for two similarly constructed series of algebras in this thesis new lower bounds for auslander s representation dimension are constructed and more general methods are developed first we consider group algebras of elementary abelian groups we show that their representation dimension is greater than the rank of the group this leads to a proof of a conjecture of benson concerning the loewy length of blocks of group algebras then a notion of dimension of a module category is introduced this notion enables us to find better lower bounds for the representation dimension in particular it allows us for the first time to determine the value of the representation dimension of non self injective algebras of representation dimension greater than three finally we give a general criterion for finding lower bounds for auslander s representation dimension suppose there is a family of modules such that sufficiently many members of the family have certain d extensions then the representation dimension of the algebra is at least d 2 we illustrate how this criterion is applied on some interesting families of algebras it turns out that the lower bound is often the precise value of the representation dimension

homological domino effects and the first finitistic dimension conjecture counterexamples to the first finitistic dimension conjecture are constructed in fact for any fieldk and any integerm 2 there exist finite dimensionalk algebras such that the little finitistic dimension of ism while the big finitistic dimension ism 1 our examples are monomial relation algebras within this class of algebras the big and little finitistic dimensions cannot differ by more than 1 the analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras

auslander type conditions and cotorsion pairs we study the properties of rings satisfying auslander type conditions if an artin algebra satisfies the auslander condition that is is an gorenstein artin algebra then we construct two kinds of subcategories which form functorially finite cotorsion pairs

the green correspondence for the representations of hecke algebras of type ᵣ we first prove the conjecture mentioned by leonard k jones in his thesis by applying this conjecture we obtain that the vertex of an indecomposable xf module is an i parabolic subgroup finally we establish the green correspondence for the representations of hecke algebras of type ar

recurrent neural network for non smooth convex optimization problems with application to the identification of genetic regulatory networks a recurrent neural network is proposed for solving the non smooth convex optimization problem with the convex inequality and linear equality constraints since the objective function and inequality constraints may not be smooth the clarke s generalized gradients of the objective function and inequality constraints are employed to describe the dynamics of the proposed neural network it is proved that the equilibrium point set of the proposed neural network is equivalent to the optimal solution of the original optimization problem by using the lagrangian saddle point theorem under weak conditions the proposed neural network is proved to be stable and the state of the neural network is convergent to one of its equilibrium points compared with the existing neural network models for non smooth optimization problems the proposed neural network can deal with a larger class of constraints and is not based on the penalty method finally the proposed neural network is used to solve the identification problem of genetic regulatory networks which can be transformed into a non smooth convex optimization problem the simulation results show the satisfactory identification accuracy which demonstrates the effectiveness and efficiency of the proposed approach

a new gradient based neural network for solving linear and quadratic programming problems a new gradient based neural network is constructed on the basis of the duality theory optimization theory convex analysis theory lyapunov stability theory and lasalle invariance principle to solve linear and quadratic programming problems in particular a new function f x y is introduced into the energy function e x y such that the function e x y is convex and differentiable and the resulting network is more efficient this network involves all the relevant necessary and sufficient optimality conditions for convex quadratic programming problems for linear programming and quadratic programming qp problems with unique and infinite number of solutions we have proven strictly that for any initial point every trajectory of the neural network converges to an optimal solution of the qp and its dual problem the proposed network is different from the existing networks which use the penalty method or lagrange method and the inequality constraints are properly handled the simulation results show that the proposed neural network is feasible and efficient

dimensions of triangulated categories with respect to subcategories abstract this paper introduces a concept of dimension of a triangulated category with respect to a fixed full subcategory for the bounded derived category of an abelian category upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given our methods not only recover some known results on the dimensions of derived categories in the sense of rouquier but also apply to various commutative and non commutative noetherian rings

representation dimension of exterior algebras we determine the representation dimension of exterior algebras this provides the first known examples of representation dimension 3 we deduce that the loewy length of the group algebra over f2 of a finite group is strictly bounded below by the 2 rank of the group a conjecture of benson a key tool is the use of the concept of dimension of a triangulated category

biserial algebras via subalgebras and the path algebra of d4 we give two new criteria for a basic algebra to be biserial the first one states that an algebra is biserial iff all subalgebras of the form eae where e is supported by at most 4 vertices are biserial the second one gives some condition on modules that must not exist for a biserial algebra these modules have properties similar to the module with dimension vector 1 1 1 1 for the path algebra of the quiver d4 both criteria generalize criteria for an algebra to be nakayama they rely on the description of a basic biserial algebra in terms of quiver and relations given by r vila freyer and w crawley boevey cbvf98

on the finitistic dimension conjecture of artin algebras abstract let a be an artin algebra and e be an idempotent element of a we prove that if a has representation dimension at most three then the finitistic dimension of eae is finite and deduce that if quasi hereditary algebras have representation dimensions at most three then the finitistic dimension conjecture holds

on subcategories closed under predecessors and the representation dimension abstract let be an artin algebra a and c be full subcategories of the category of finitely generated modules consisting of indecomposable modules and closed under predecessors and successors respectively in this paper we relate under various hypotheses the representation dimension of to those of the left support algebra of a and the right support algebra of c our results are then applied to the classes of laura algebras ada algebras and nakayama oriented pullbacks

quadratic bimodules and quadratic orders in some branches of mathematics especially the representation theory of artin algebras and orders it has been an important problem to study the geometric quotient gln mn m for a module m g m gmg 1 for g gln and m mn m an effective formulation of such problems is a bimodule problem 4 9 which is a study of the matrix category mat c m associated to an additive category c and a c c module m section 2 1 the bimodule problems have nice module theoretical interpretation by means of prinjective modules 22 29 30 they have already many useful applications in the representation theory of finite dimensional algebras artinian rings orders vector space categories and bocses see 6 7 9 16 19 25 28 29 in this paper we introduce the concept of quadratic extensions of rings section 1 2 which are closely related to many rings known in representation theory for example bass orders 10 17 backstrom orders 28 green orders 27 special biserial algebras 31 and clannish algebras 5 see sections 1 3 3 4 and 3 5 using quadratic extensions we introduce the concept of quadratic bimodules section 3 1 where we can easily formulate self reproducing systems 21 and clans 5 in terms of quadratic bimodules then we study their matrix category mat c m associated to a quadratic c c module m especially we give a description of all indecomposable objects in terms of walks sections 3 3 6 1 which can be regarded as a generalization of green walks 13 section 3 4 strings and bands 5 31 section 3 5 as an application we give a description of finitely generated indecomposable modules over quadratic orders section 1 3 2 in section 7 3 in forthcoming papers our results will be applied in construction of large new classes of orders of tame representation type some of the results of this paper were presented at the conference and workshop constanta 2000 and were announced in 19 without proof