segre product of artin schelter regular algebras of dimension 2 and embeddings in quantum p3 s the segre embedding of p1 p1as a smooth quadricqin p3corresponds to the surjection of the four dimensional polynomial ring onto the segre productsof two copies of the homogeneous coordinate ring of p1 we study segre products of noncommutative algebras if in particularaandbare two copies of a quantum p1thens i ai kbi is a twisted homogeneous coordinate ring of the quadricq the main result of this paper is the classification of all embeddings of the segre product of two quantum planes into so called quantum p3 s these are the proj of artin schelter regular algebrasrof global dimension four with the hilbert series of a commutative polynomial ring and which map ontos ifris not a twist of a polynomial ring then the point scheme ofreither is the union of the quadricqwith a line or is only the quadricq in the first case ris a central extension of a three dimensional artin schelter regular algebra and a twist of an algebra mapping onto the commutative homogeneous coordinate ring ofq in the second case such an algebraris the first known example of a four dimensional artin schelter regular algebra which is not determined by its point scheme

three dimensional free vibrations of thick skewed cantilevered plates abstract the natural frequencies of skewed cantilevered thick plates are determined by using the ritz method the present work is the first known three dimensional study of the problem assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly and which are mathematically complete accurate natural frequencies are calculated for skewed thick plates having arbitrary degrees of skewness detailed numerical studies reveal interesting trends concerning the variation of frequencies with increasing skew angle results obtained by using the present method are compared with those obtained by using three dimensional finite elements

higher symplectic reflection algebras and non homogeneous n koszul property abstract we study a class of associative algebras associated to finite groups acting on a vector space these algebras are non homogeneous n koszul algebra generalizations of symplectic reflection algebras p etingof v ginzburg symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism invent math 147 2002 243 348 the koszul property was generalized to homogeneous algebras of degree n 2 in r berger koszulity for nonquadratic algebras j algebra 239 2001 705 734 in the present paper the extension of the koszul property to non homogeneous algebras is realized through a pbw theorem this pbw theorem is the generalization to the n case of a quadratic result obtained by braverman and gaitsgory a braverman d gaitsgory poincare birkhoff witt theorem for quadratic algebras of koszul type j algebra 181 1996 315 328 see also a polishchuk l positselski quadratic algebras univ lecture ser vol 37 amer math soc providence ri 2005 we work in the general setting where the ground ring is an arbitrary von neumann regular ring

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

graded modules of gelfand kirillov dimension one over three dimensional artin schelter regular algebras abstract let a be a three dimensional artin schelter regular algebra we give a description of the category of finitely generated a modules of gelfand kirillov dimension one modulo those of finite dimension over the ground field the proof is based upon a result by gabriel which says that locally finite categories can be described by module categories over topological rings

a class of algebras similar to the enveloping algebra of sl 2 fix f e c x define r c a b h subject to the relations ha ah a hb bh b ab ba f h we study these algebras for different f and in particular show how they are similar to and different from u sl 2 the enveloping algebra of sl 2 c there is a notion of highest weight modules and a category a for such r for each n 0 if f x x 1 n l xn 1 then r has precisely n simple modules in each finite dimension and every finite dimensional r module is semisimple

noncommutative rational double points in this paper we study noncommutative analogues of rational double points the approach is to consider the action of a finite group g on certain noncommutative analogues of k x y which were studied by artin and stafford regular local rings of dimension 2 manuscript an explicit description in terms of generators and relations is given for a large class of such algebras when g is cyclic finally we show that these algebras are as gorenstein of dimension two have finite representation type and in many cases are regular in codimension one

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

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

poisson co homology of polynomial poisson algebras in dimension four sklyanin s case abstract in this paper we compute the poisson co homology of a polynomial poisson structure given by two casimir polynomial functions which define a complete intersection with an isolated singularity

regularity of algebras related to the sklyanin algebra this paper continues the research of ss by finding further examples of artin schelter regular rings of dimension four unlike the threedimensional case studied in atv1 2 these examples show that the fourdimensional regular rings are not uniquely determined by their associated geometric data indeed we find a one parameter family of regular algebras associated to this data

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

noetherian connected graded algebras of global dimension 3 abstract let a be a noetherian connected graded algebra of global dimension 3 we show that a is regular in the sense of artin and schelter if one of the following conditions holds 1 a is generated by two elements 2 the graded simple module has a standard resolution 3 the degrees of minimal relations are the same this includes the quadratic case some general properties of a are studied without assuming regularity

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

constructing clifford quantum p3s with finitely many points abstract we present an algebro geometric technique for constructing regular clifford algebras a of global dimension four with associated point scheme consisting of a prespecified finite number of points in particular if a has more than one point in its point scheme then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors these cubic divisors correspond to regular clifford subalgebras of a of global dimension three if a has exactly a finite number n of distinct points in its point scheme then n 1 3 4 5 13 14 16 18 20 and all these possibilities occur we also prove that if a regular clifford algebra r of global dimension d 2 has exactly a finite number n of distinct isomorphism classes of point modules then n is odd if and only if r is an ore extension of a regular clifford subalgebra of r of global dimension d 1

algebras associated to elliptic curves this paper completes the classification of artin schelter regular algebras of global dimension three for algebras generated by elements of degree one this has been achieved by artin schelter tate and van den bergh we are therefore concerned with algebras which are not generated in degree one we show that there exist some exceptional algebras each of which has geometric data consisting of an elliptic curve together with an automorphism just as in the case where the algebras are assumed to be generated in degree one in particular we study the elliptic algebras a a and a a where a p2 which were first defined in an earlier paper we omit a set s p2 consisting of 11 specified points where the algebras a a become too degenerate to be regular theorem let a represent a a or a a where a p2 s then a is an artin schelter regular algebra of global dimension three moreover a is a noetherian domain with the same hilbert series as the appropriately graded commutative polynomial ring in three variables this combined with our earlier results completes the classification

embedding a quantum nonsingular quadric in a quantum p3 a definition of regularity has been given for non commutative graded algebras and results of artin schelter tate and van den bergh classify the regular algebras of global dimension three that are generated by degree one elements our purpose is to classify a certain class of quadratic regular algebras of global dimension four letsbe a twisted homogeneous coordinate ring of a nonsingular quadricq p3 our interest is in algebrasrsuch that there is an embedding proj s proj r in this paper we classify all the quadratic regular algebrasrof global dimension four which have the same hilbert series as that of the polynomial ring on four variables and which map ontosvia a graded degree zero homomorphism our approach makes use of the point modules ofrand their associated geometric data we classify the algebrasraccording to their point scheme pand corresponding automorphism aut p those algebrasrwhich are determined by p belong to at most a five parameter family but those which are not determined by p belong to at most a four parameter family in the first case pis either p3or consists ofqtogether with a linel while in the second casep q it is also proved that under certain sufficient conditions the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism

basic properties of generalized down up algebras abstract we introduce a large class of infinite dimensional associative algebras which generalize down up algebras let k be a field and fix f k x and r s k define l l f r s to be the algebra generated by d u and h with defining relations d h r d 0 h u r u 0 d u s f h 0 included in this family are smith s class of algebras similar to u sl 2 le bruyn s conformal sl 2 enveloping algebras and the algebras studied by rueda the algebras l have gelfand kirillov dimension 3 and are noetherian domains if and only if rs 0 we calculate the global dimension of l and for rs 0 classify the simple weight modules for l including all finite dimensional simple modules simple weight modules need not be classical highest weight modules

homologically homogeneous rings in this paper we study the structure of a right noetherian ring r of finite right global dimesion integral over a central subring c and satisfying the following condition if v w are irreducible right r modules with rc v rc w then pr dim v

some finite quantum p3s that are infinite modules over their centers abstract a result of m artin j tate and m van den bergh asserts that a regular algebra of global dimension three is a finite module over its center if and only if the automorphism encoded in the point scheme has finite order we prove that the analogous result for a regular algebra of global dimension four is false by presenting families of quadratic noetherian regular algebras a of global dimension four such that i a is an infinite module over its center ii a has a finite point scheme which is the graph of an automorphism of finite order and iii a has a one parameter family of line modules such algebras are candidates for generic regular algebras of global dimension four the methods used to construct the algebras provide new techniques for creating other potential candidates

on koszul algebras and a new construction of artin schelter regular algebras abstract we prove the simple fact that the factor ring of a koszul algebra by a regular normal quadratic element is a koszul algebra this fact leads to a new construction of quadratic artin schelter regular algebras this construction generalizes the construction of artin schelter regular clifford algebras

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

quantum deformations of projective three space abstract we describe the possible noncommutative deformations of complex projective three space by exhibiting the calabi yau algebras that serve as their homogeneous coordinate rings we prove that the space parametrizing such deformations has exactly six irreducible components and we give explicit presentations for the generic members of each family in terms of generators and relations the proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic poisson structures in four dimensions which we extract from cerveau and lins neto s classification of degree two foliations on projective space corresponding to the exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical schwarzenberger bundles

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

on the three dimensional vibrations of the cantilevered rectangular parallelepiped a solution is presented for the three dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces this problem apparently is previously unsolved in the published literature the ritz method is used with displacements assumed in the form of algebraic polynomials convergence is studied numerical results are given for the first five frequencies of each of the four symmetry classes of vibration for five thick parallelepiped configurations including the cube contour plots are exhibited for the modal displacements of the cube the effects of varying poisson s ratio are also observed

transverse vibrations of thin elastic plates with concentrated masses and internal elastic supports abstract the fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined the case of an orthotropic rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the galerkin method then vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method finally the case of a circular plate elastically restrained against translation and rotation is considered

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

noetherianity and gelfand kirillov dimension of components abstract the category of all additive functors mod mod for a finite dimensional algebra were shown to be left noetherian if and only if is of finite representation type by m auslander here we consider the category of all additive graded functors from the category of associated graded category of mod to graded vector spaces this category decomposes into subcategories corresponding to the components of the auslander reiten quiver for a regular component we show that the corresponding graded functor category is left noetherian if and only if the section of the component is extended dynkin or infinite dynkin

some noncommutative minimal surfaces abstract in the ongoing programme to classify noncommutative projective surfaces connected graded noetherian domains of gelfand kirillov dimension three a natural question is to determine the minimal models within any birational class in this paper we show that the generic noncommutative projective plane corresponding to the three dimensional sklyanin algebra as well as noncommutative analogues of p 1 p 1 and the more general van den bergh quadrics satisfy very strong minimality conditions translated into an algebraic question where one is interested in a maximality condition we prove the following result theorem a let r be a sklyanin algebra or a van den bergh quadric that is infinite dimensional over its centre and let a r be any connected graded noetherian maximal order with the same graded quotient ring as r then up to taking veronese rings a is isomorphic to r let t be an elliptic algebra that is the coordinate ring of a noncommutative surface containing an elliptic curve then under an appropriate homological condition we prove that every connected graded noetherian overring of t is obtained by blowing down finitely many lines line modules of self intersection 1

3 d theory versus 2 d approximate theory of free orthotropic isotropic plate and shell vibrations part 1 derivation of governing equations in part 1 of the paper the basic assumptions of the shells with added masses considered are introduced they serve for a derivation of the governing equations which are analyzed in part 2 of the paper

down up algebras and their representation theory abstract a class of algebras called down up algebras was introduced by g benkart and t roby 1998 j algebra 209 305 344 we classify the finite dimensional simple modules over noetherian down up algebras and show that in some cases every finite dimensional module is semisimple we also study the question of when two down up algebras are isomorphic

coherent algebras and noncommutative projective lines abstract a well known conjecture says that every one relator group is coherent we state and partly prove a similar statement for graded associative algebras in particular we show that every gorenstein algebra a of global dimension 2 is graded coherent this allows us to define a noncommutative analogue of the projective line p 1 as a noncommutative scheme based on the coherent noncommutative spectrum qgra of such an algebra a that is the category of coherent a modules modulo the torsion ones this category is always abelian ext finite hereditary with serre duality like the category of coherent sheaves on p 1 in this way we obtain a sequence p n 1 n 2 of pairwise non isomorphic noncommutative schemes which generalize the scheme p 1 p 2 1

fat point modules over generalized laurent polynomial rings abstract recently a new construction of rings was introduced by cassidy goetz and shelton some of these rings called generalized laurent polynomial rings are quadratic artin schelter regular algebras of global dimension 4 we study a family of such algebras which have finite order point scheme automorphisms but which are not finitely generated over their centers our main result is the classification of all fat point modules for each algebra in the family we also consider the action of the shift functor and prove has infinite order on a fat point module f precisely when the center acts trivially on f the proofs of these facts use the noncommutative geometry of some cubic artin schelter regular algebras of global dimension 3

non commutative castelnuovo mumford regularity and as regular algebras abstract let a be a connected graded k algebra with a balanced dualizing complex we prove that a is a koszul as regular algebra if and only if that the castelnuovo mumford regularity and the ext regularity coincide for all finitely generated a modules this can be viewed as a non commutative version of t romer on the regularity over positively graded algebras j algebra 319 2008 1 15 theorem 1 3 by using castelnuovo mumford regularity we prove that any koszul standard as gorenstein algebra is as regular as a preparation to prove the main result we also prove that the following statements are equivalent 1 a is as gorenstein 2 a has finite left injective dimension 3 the dualizing complex has finite left projective dimension this generalizes i mori homological properties of balanced cohen macaulay algebras trans amer math soc 355 2002 1025 1042 corollary 5 9

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

vector valued modular forms on a three dimensional ball recently there came up growing interest in structure theorems for vector valued modular forms one reason may be harder s conjecture bg conjecture 3 in van der geer s talk different methods have been developed to get structure theorems for modules of vector valued modular forms ao cg do ib sa sat wi at the beginning the case of siegel modular forms has been studied but meanwhile also other groups have found interest for example in the paper cg clery and van der geer determined generators for some modules of vector valued picard modular forms on the two dimensional ball in this paper we consider the case of a three dimensional ball with the action of the picard modular group 3 3 see sect 3 the corresponding modular variety of dimension 3 is a copy of the segre cubic we obtain similar results as in cg but with completely different methods

some remarks on triple l functions we extend the garrett s result i 4 concerning triple l function in two aspects one concerns triple l functions of different weights and the other concerns special values of triple l functions of same weights at inner points of their critical strips let mk fn and sk fn be spaces of siegel modular forms and cuspforms of weight k and of degree n respectively for siegel modular forms f l fro of various degree and a field k we denote by k ft fro the field generated by all the fourier coefficients of f l fro over k we write fourier expansion of f sk f1

a comparison of optimality criteria methods for stress and displacement constraints abstract optimality criteria oc methods and the closely related dual method have become one of the most important class of optimization algorithms used in the practice of structural optimization since this class of algorithms search the optimum in the dual space of the lagrange multipliers corresponding to the active constraints the computational cost of the optimization is determined mostly by the number of active constraints the efficiency of oc methods has been greatly improved recently by an oc method termed dcoc which searches for the optimum in a usually much smaller dual subspace corresponding to only displacement multipliers however the relation of the dcoc with traditional oc methods is not clear and therefore it is still difficult even for people familiar with oc methods to understand dcoc algorithm this paper shows that the dcoc algorithm can be directly obtained from the traditional oc methods which provides insights into the method additionally a derivative based version of dcoc is developed which also helps in the understanding of the method

generalized laurent polynomial rings as quantum projective 3 spaces abstract given a ring r we introduce the notion of a generalized laurent polynomial ring over r this class includes the generalized weyl algebras we show that these rings inherit many properties from the ground ring r this construction is then used to create two new families of quadratic global dimension four artin schelter regular algebras we show that in most cases the second family has a finite point scheme and a defining automorphism of finite order nonetheless a generic algebra in this family is not finite over its center

a class of as regular algebras of dimension five abstract we classify 5 dimensional artin schelter regular algebras generated by two generators of degree 1 with three generating relations of degree 4 under a generic condition all the algebras obtained are proved to be strongly noetherian auslander regular and cohen macaulay with respect to the gelfand kirillov dimension

some problems of analysis and optimization of plates and shells the classical optimization problems of plates and shells to satisfy a priori given geometry and dynamical characteristics are considered orthotropic plates and shells with variable thickness and low transverse stiffness are analyzed first some useful theorems and their proofs are given then the finite approximation of the problem related to optimization of free vibrations of shells with transverse deformation and rotary inertia is discussed the varational iteration mvi and bubnov galerkin mb methods are applied and their convergence and suitability for application to plates and shells analysis are discussed and numerically evaluated

on certain vector valued siegel modular forms of degree two in this paper we explicitly construct vector valued siegel modular forms of degree two and the automorphic factor detk for even k where st denotes the standard representation of gl 2 c as an application we prove some congruences between eigenvalues of hecke operators for a positive integer n let f be the full siegel modular group of degree n and h the siegel upper half plane of degree n for m f and z h we put

zariski cancellation problem for non domain noncommutative algebras we study zariski cancellation problem for noncommutative algebras that are not necessarily domains

representation theory of two families of quantum projective 3 spaces abstract stephenson and vancliff recently introduced two families of quantum projective 3 spaces quadratic and artin schelter regular algebras of global dimension 4 which have the property that the associated automorphism of the scheme of point modules is finite order and yet the algebra is not finite over its center this is in stark contrast to theorems of artin tate and van den bergh in global dimension 3 we analyze the representation theory of these algebras we classify all of the finite dimensional simple modules and describe some zero dimensional elements of proj i e so called fat point modules in particular we observe that the shift functor on zero dimensional elements of proj which is closely related to the above automorphism actually has infinite order

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

an efficient dcoc algorithm based on high quality approximations for problems including eigenvalue constraints abstract an efficient dcoc algorithm for problems subject to the combination of static constraints and eigenvalue constraints is presented the high efficiency of the proposed method lies in the fact that the lagrange multipliers associated with stress constraints are evaluated explicitly and therefore the dual type problem involves only displacement and eigenvalue constraints which are highly reduced compared to traditional oc or dual methods fast convergence of the method is achieved by the high quality reciprocal linear approximation for displacement constraints and the rayleigh quotient approximation for eigenvalues

double extension regular algebras of type 14641 abstract we construct several families of artin schelter regular algebras of global dimension four using double ore extension and then prove that all these algebras are strongly noetherian auslander regular koszul and cohen macaulay domains many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an ore extension of an artin schelter regular algebra of global dimension three