sklyanin algebras and hilbert schemes of points we construct projective moduli spaces for torsion free sheaves on noncommutative projective planes these moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual commutative projective plane p2 the generic noncommutative plane corresponds to the sklyanin algebra s skl e constructed from an automorphism of infinite order on an elliptic curve e p2 in this case the fine moduli space of line bundles over s with first chern class zero and euler characteristic 1 n provides a symplectic variety that is a deformation of the hilbert scheme of n points on p2 e

the analysis of contingency tables by graphical models summary some aspects of the analysis of multidimensional contingency tables in practice are considered the class of graphical models defined by darroch lauritzen speed 1980 is described strategies for model selection based on this class are considered and an example is given

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

valuation based systems for bayesian decision analysis this paper proposes a new method for representing and solving bayesian decision problems the representation is called a valuation based system and has some similarities to influence diagrams however unlike influence diagrams which emphasize conditional independence among random variables valuation based systems emphasize factorizations of joint probability distributions also whereas influence diagram representation allows only conditional probabilities valuation based system representation allows all probabilities the solution method is a hybrid of local computational methods for the computation of marginals of joint probability distributions and the local computational methods for discrete optimization problems we briefly compare our representation and solution methods to those of influence diagrams

growth of graded twisted calabi yau algebras abstract we initiate a study of the growth and matrix valued hilbert series of n graded twisted calabi yau algebras that are homomorphic images of path algebras of weighted quivers generalizing techniques previously used to investigate artin schelter regular algebras and graded calabi yau algebras several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras we give particular attention to twisted calabi yau algebras of dimension d 3 giving precise descriptions of their matrix valued hilbert series and partial results describing which underlying quivers yield algebras of finite gk dimension for d 2 we show that these are algebras with mesh relations for d 3 we show that the resulting algebras are a kind of derivation quotient algebra arising from an element that is similar to a twisted superpotential

repackable multi log2n networks multi log 2 n networks have a wide range of applications in telecommunications in this paper we present a new type of nonblocking multi log 2 n networks called repackable nonblocking wherein rearrangement activities are required only during the departures but not upon arrivals furthermore the frequency of rearrangement activities is so low that under many circumstances they perform like strictly nonblocking networks but with a much lower hardware complexity

sequential valuation networks for asymmetric decision problems abstract this paper deals with representation and solution of asymmetric decision problems we describe a new representation called sequential valuation networks that is a hybrid of covaliu and oliver s sequential decision diagrams and shenoy s valuation networks the solution algorithm is based on the idea of decomposing a large asymmetric problem into smaller sub problems and then using the fusion algorithm of valuation networks to solve the sub problems sequential valuation networks inherit many of the strengths of sequential decision diagrams and valuation networks while overcoming many of their shortcomings we illustrate our technique by representing and solving a modified version of covaliu and oliver s manage sci 41 12 1995 1860 reactor problem in complete detail

bispectral operators of prime order the aim of this paper is to solve the bispectral problem for bispectral operators whose order is a prime number more precisely we give a complete list of such bispectral operators we use systematically the operator approach and in particular dixmier ideas on the first weyl algebra when the order is 2 the main theorem is exactly the result of duistermaat grunbaum on the other hand our proofs seem to be simpler

necessary and sufficient conditions for rearrangeable log sub d n m p we extend previous results on sufficient conditions for rearrangeable log sub 2 n m p networks to log sub d n m p networks we show that the original argument using path intersecting graphs cannot be extended to d 2 and we give a new argument further we show our sufficient conditions are also necessary

influence diagrams for bayesian decision analysis abstract influence diagrams for representing bayesian decision problems are redefined in a formal way using conditional independence this makes the graphs somewhat more helpful for exploring the consequences of a clients state beliefs some important results about the manipulation of influence diagrams are extended and reviewed as is an algorithm for computing an optimal policy two new results about the manipulation of influence diagrams are derived a novel influence diagram representing a practical decision problem is used to illustrate the methodologies presented in this paper

differential operators on an affine curve let x denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero denote by 3 x the sheaf of differential operators on x and 2 x f x 3 x the ring of global differential operators on x the following is established theorem 3 x is a finitely generated k algebra and a noetherian ring furthermore 2 x has a unique minimal non zero ideal j and 3 x j is a finite dimensional k algebra let x denoted the normalisation of x and n x x the projection map the main technique is to compare 2 x with 3 x

mad subalgebras of rings of differential operators on curves we study the maximal abelian ad nilpotent mad subalgebras of the domains morita equivalent to the first weyl algebra we give a complete description both of the individual mad subalgebras and of the space of all such a surprising consequence is that this last space is independent of our results generalize some classic theorems of dixmier about the weyl algebra

choosing the best log sub k n m p strictly nonblocking networks we extend the log sub 2 n m p network proposed by shyy and lea 1991 to base k we give a unifying proof instead of three separate cases as done by shyy and lea for the condition of being strictly nonblocking and a simpler expression of the result we compare the number of crosspoints for log sub k n m p over various k

graded self injective algebras are trivial extensions abstract for a positively graded artin algebra a n 0 a n we introduce its beilinson algebra b a we prove that if a is well graded self injective then the category of graded a modules is equivalent to the category of graded modules over the trivial extension algebra t b a consequently there is a full exact embedding from the bounded derived category of b a into the stable category of graded modules over a it is an equivalence if and only if the 0 th component algebra a 0 has finite global dimension

noncommutative bispectral darboux transformations we prove a general theorem establishing the bispectrality of noncommutative darboux transformations it has a wide range of applications that establish bispectrality of such transformations for differential difference and q difference operators with values in all noncommutative algebras all known bispectral darboux transformations are special cases of the theorem using the methods of quasideterminants and the spectral theory of matrix polynomials we explicitly classify the set of bispectral darboux transformations from rank one differential operators and airy operators with values in matrix algebras these sets generalize the classical calogero moser spaces and wilson s adelic grassmannian

automorphisms and ideals of the weyl algebra let a 1 be the first weyl algebra and let g be its automorphism group we study the natural action of g on the space of isomorphism classes of right ideals of a 1 equivalently of finitely generated rank 1 torsion free right a 1 modules we show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety our results are strikingly similar to those for the commutative algebra of polynomials in two variables however we do not know of any general principle that would allow us to predict this in advance as a key step in the proof we obtain a new description of the bispectral involution of w1 we also make some comments on the group g from the viewpoint of shafarevich s theory of infinite dimensional algebraic groups

automorphisms of the heisenberg weyl algebra and d orthogonal polynomials we show that the d orthogonal charlier and hermite polynomials appear naturally as matrix elements of nonunitary transformations corresponding to automorphisms of the heisenberg weyl oscillator algebra basic properties duality recurrence and difference equations of the d orthogonal charlier polynomials can be derived directly from representations of the heisenberg weyl algebra

the noncommutative schemes of generalized weyl algebras abstract the first weyl algebra over k a 1 k x y x y y x 1 admits a natural z grading by letting deg x 1 and deg y 1 smith showed that gr a 1 is equivalent to the category of quasicoherent sheaves on a certain quotient stack using autoequivalences of gr a 1 smith constructed a commutative ring c graded by finite subsets of the integers he then showed gr a 1 gr c z fin in this paper we prove analogues of smith s results by using autoequivalences of a graded module category to construct rings with equivalent graded module categories for certain generalized weyl algebras we use autoequivalences defined in a companion paper so that these constructions yield commutative rings

ideals of cubic algebras and an invariant ring of the weyl algebra hasselt univ dept wni b 3590 diepenbeek belgium univ antwerp dept wiskunde informat b 2020 antwerp belgium de naeghel k hasselt univ dept wni agoralaan gebouw d b 3590 diepenbeek belgium koendenaeghel hotmail com nicolas marconnet ua ac be

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

log sub 2 n m p strictly nonblocking networks a class of self routing strictly nonblocking switching networks called the log sub 2 n m p network is introduced the proposed networks are self routing and have o log sub 2 n stages between each inlet outlet pair the self routing capability makes it attractive for high speed switching the small number of crosspoints between each inlet outlet pair also makes the network well suited for many nonlogic photonic switching technologies the complexity and the fault tolerant capability of the strictly nonblocking log sub 2 n m p network are analyzed to show the merits of the proposed network the authors compare it to the cantor network which is also strictly nonblocking and has similar characteristics

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

qualitative possibilistic influence diagrams based on qualitative possibilistic utilities this paper proposes a new approach for decision making under uncertainty based on influence diagrams and possibility theory the so called qualitative possibilistic influence diagrams extend standard influence diagrams in order to avoid difficulties attached to the specification of both probability distributions relative to chance nodes and utilities relative to value nodes in fact generally it is easier for experts to quantify dependencies between chance nodes qualitatively via possibility distributions and to provide a preferential relation between different consequences in such a case the possibility theory offers a suitable modeling framework different combinations of the quantification between chance and utility nodes offer several kinds of possibilistic influence diagrams this paper focuses on qualitative ones and proposes an indirect evaluation method based on their transformation into possibilistic networks the proposed approach is implemented via a possibilistic influence diagram toolbox pidt

a comparison of graphical techniques for decision analysis recently we proposed a new method for representing and solving decision problems based on the framework of valuation based systems the new representation is called a valuation network and the new solution method is called a fusion algorithm in this paper we compare valuation networks to decision trees and influence diagrams we also compare the fusion algorithm to the backward recursion method of decision trees and to the arc reversal method of influence diagrams recently we proposed a new method for representing and solving decision problems based on the axiomatic framework of valuation based systems shenoy 1993 1992a the new representation is called a valuation network and the new solution method is called a fusion algorithm in this paper we compare valuation networks to decision trees and influence diagrams we also compare the fusion algorithm to the backward recursion method of decision trees and the arc reversal method of influence diagrams a popular method for representing and solving decision problems is decision trees decision trees have their genesis in the pioneering work of von neumann and morgenstern 1944 on extensive form games decision trees graphically depict all possible scenarios the decision tree representation allows computation of an optimal strategy by the backward recursion method of dynamic programming raiffa and schlaifer 1961 raiffa and schlaifer 1961 call the dynamic

a note on kirkwood s algebraic method for decision problems the main objectives of this note are to point out some strengths and some limitations of kirkwood s algebraic method for decision problems and to propose a modification to address the limitations the modification is based on the valuation network technique and it enables us to better relate kirkwood s method to the graphical techniques of valuation networks and influence diagrams

rings graded equivalent to the weyl algebra abstract we consider the first weyl algebra a in the euler gradation and completely classify graded rings b that are graded equivalent to a that is the categories gr a and gr b are equivalent this includes some surprising examples in particular we show that a is graded equivalent to an idealizer in a localization of a we obtain this classification as an application of a general morita type characterization of equivalences of graded module categories given a z graded ring r an autoequivalence f of gr r and a finitely generated graded projective right r module p we show how to construct a twisted endomorphism ring end r f p and prove theorem the z graded rings r and s are graded equivalent if and only if there are an autoequivalence f of gr r and a finitely generated graded projective right r module p such that the modules f n p generate gr r and s end r f p

valuation network representation and solution of asymmetric decision problems abstract this paper deals with asymmetric decision problems we describe a generalization of the valuation network representation and solution technique to enable efficient representation and solution of asymmetric decision problems the generalization includes the concepts of indicator valuations and effective frames we illustrate our technique by solving howard s used car buyer s problem in complete detail we highlight the contribution of this paper over the symmetric valuation network technique

the picard group of the graded module category of a generalized weyl algebra abstract the first weyl algebra a 1 k x y x y y x 1 is naturally z graded by letting deg x 1 and deg y 1 sue sierra studied gr a 1 category of graded right a 1 modules computing its picard group and classifying all rings graded equivalent to a 1 in this paper we generalize these results by studying the graded module category of certain generalized weyl algebras we show that for a generalized weyl algebra a f with base ring k z defined by a quadratic polynomial f the picard group of gr a f is isomorphic to the picard group of gr a 1

a quotient stack related to the weyl algebra abstract let a denote the ring of differential operators on the affine line with its two usual generators t and d d t given degrees 1 and 1 respectively let x be the stack having coarse moduli space the affine line spec k z and isotropy groups z 2 at each integer point then the category of graded a modules is equivalent to the category of quasi coherent sheaves on x

structuring conditional relationships in influence diagrams an influence diagram is a graphical representation of a decision problem that is at once a formal description of a decision problem that can be treated by computers and a representation that is easily understood by decision makers who may be unskilled in the art of complex probabilistic modeling the power of an influence diagram both as an analysis tool and a communication tool lies in its ability to concisely summarize the structure of a decision problem however when confronted with highly asymmetric problems in which particular acts or events lead to very different possibilities many analysts prefer decision trees to influence diagrams in this paper we extend the definition of an influence diagram by introducing a new representation for its conditional probability distributions this extended influence diagram representation combining elements of the decision tree and influence diagram representations allows one to clearly and efficiently represent asymmetric decision problems and provides an attractive alternative to both the decision tree and conventional influence diagram representations

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

endomorphisms of right ideals of the weyl algebra let a a k be the first weyl algebra over an infinite field k let p be any noncyclic projective right ideal of a and set s end p we prove that as k algebras s f a in contrast there exists a noncyclic projective right ideal q of s such that s end q thus despite the fact that they are morita equivalent s and a have surprisingly different properties for example under the canonical maps autk a pick a pick s in contrast autk s has infinite index in pick s introduction given a comnmutative domain r the first weyl algebra a r is defined to be the associative r algebra thus r is central subring generated by elements x and y subject to the relation xy yx 1 when no ambiguity is possible we write a for a r let f be a field of characteristic zero then a a f is a simple ring and indeed may be thought of as one of the nicest and most important examples of simple noetherian rings the initial motivation for this paper was the following result of smith if p is the noncyclic right ideal p x2a xy 1 a of a then end p t a as f algebras 11 note that as char f 0 a is a simple hereditary ring and so end p is automatically morita equivalent to a even worse as pedp ada 17 the full matrix rings m2 a and m2 end p are isomorphic thus any proof of smith s result must be fairly subtle and may therefore provide useful invariants for a the first main aim of this paper is to generalize smith s result to an arbitrary projective right ideal of a while the proof of this is harder than that of smith s result it does provide a more informative proof in the sense that unlike smith we do not and cannot require an explicit description of end p for the rest of this introduction k will denote a field of arbitrary characteristic and all isomorphisms of rings will be k algebra isomorphisms theorem a let p be a projective right ideal of a a k then end p a if and only if p is a cyclic right ideal of a see theorem 3 1 this has some easy corollaries corollary b let p and q be projective right ideals of a a k then end p end q if and only if p to q for some t e d k the division ring of fractions of a and a e autk a the group of k automorphisms of a received by the editors february 13 1986 1980 mathematics subject classification 1985 revision primary 16a65 16a72 16a19 supported in part by an nsf grant 1987 american mathematical society 0002 9947 87 1 00 25 per page

cusps and d modules 1 1 v modules on singular varieties let y denote a variety over a field and let vy denote the full sheaf of differential operators on y in characteristic zero vy is the familiar sheaf of differential operators and in characteristic p the sheaf vy will contain all divided powers of operators d it is well known that on a general singular variety y the ring of differential operators is badly behaved in particular it need not be noetherian nor if y is affine a simple ring see bgg following grothendieck there is an alternative notion of a sheaf with infinitesi mal parallel transport called a co stratification see be2 be3 bdi or section 2 which agrees with the notion of left respectively right py module when y is smooth but will not agree in general when y is singular our first result presents a simple vanishing condition that guarantees that the notions of x module and co stratification on a singular variety coincide

on the symplectic structure of instanton moduli spaces abstract we study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of su 2 instantons on both commutative and non commutative r 4 we identify global darboux coordinates and quadratic hamiltonians on classical phase spaces for which these quiver varieties are natural completions we also show that the group of non commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non commutative instantons this paper should be viewed as a step towards extending known results for calogero moser spaces to the instanton moduli spaces

dixmier groups and borel subgroups abstract in this paper we study a family g n n 0 of infinite dimensional ind algebraic groups associated with algebras morita equivalent to the weyl algebra a 1 c we give a geometric presentation of these groups in terms of amalgamated products generalizing classical theorems of dixmier and makar limanov our main result is a classification of borel subgroups of g n for all n we show that the conjugacy classes of non abelian borel subgroups of g n are in bijection with the partitions of n furthermore we prove an infinite dimensional analogue of the classical theorem of steinberg 52 that characterizes borel subgroups in purely group theoretic terms combined together the last two results imply that the g n are pairwise non isomorphic as abstract groups this settles an old question of stafford 51

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

on invertible bimodules and automorphisms of noncommutative rings in this article we attempt to generalize the result that for a commu tative ring r the outer automorphism group of i automorphisms of m r is abelian of exponent n it is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings we also show that the automorphism group of an affine commutative domain of positive di mension acts faithfully on the spectrum of the domain we investigate other questions involving bimodules and automorphisms and extend a result of smith on the first weyl algebra as a fixed ring let r be a ring with 1 in this article we are interested in comparing the automorphism group of mn r the ring of x matrices over r and the automorphism group of r in particular if r is commutative it is a classical result that the outer automorphism group of r automorphisms of mn r is an abelian group of exponent dividing this can be restated in terms of invertible bimodules the result is equivalent to the easy fact that any projective rank one module x with nx free over a commutative ring has order dividing in the picent group of r see rz and ko in the noncommutative case there is still a well known relationship between automorphisms and the picard

graphical and recursive models for contingency tables summary we discuss two classes of models for contingency tables graphical and recursive models both of which arise from restrictions that are expressible as conditional independencies of variable pairs the first of these is a subclass of hierarchical log linear models each of its models can be represented by an undirected graph in the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables we derive decomposable or multiplicative models as the intersecting class this result has useful consequences for exploratory types of analysis as well as for the model interpretation we can give an aid for detecting well fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph

decision analysis by augmented probability simulation we provide a generic monte carlo method to find the alternative of maximum expected utility in a decision analysis we define an artificial distribution on the product space of alternatives and states and show that the optimal alternative is the mode of the implied marginal distribution on the alternatives after drawing a sample from the artificial distribution we may use exploratory data analysis tools to approximately identify the optimal alternative we illustrate our method for some important types of influence diagrams

kp hierarchy for the cyclic quiver we introduce a generalisation of the kp hierarchy closely related to the cyclic quiver and the cherednik algebra hk zm this hierarchy depends on m parameters one of which can be eliminated with the usual kp hierarchy corresponding to the m 1 case generalising the result of wilson invent math 133 1 1 41 1998 we show that our hierarchy admits solutions parameterised by suitable quiver varieties the pole dynamics for these solutions is shown to be governed by the classical calogero moser system for the wreath product zm sn and its new spin version these results are further extended to the case of the multi component hierarchy