a graph theoretic method for determining generating sets of prime ideals in quantum matrices abstract we take a graph theoretic approach to the problem of finding generators for those prime ideals of o q m m n k which are invariant under the torus action k m n launois 2004 15 has shown that the generators consist of certain quantum minors of the matrix of canonical generators of o q m m n k and in launois 2004 14 gives an algorithm to find them in this paper we modify a classic result of lindstrom 1973 17 and gessel and viennot 1985 6 to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex disjoint directed paths in an associated directed graph

cauchon diagrams for quantized enveloping algebras abstract let g be a finite dimensional complex simple lie algebra k a commutative field and q a nonzero element of k which is not a root of unity to each reduced decomposition of the longest element w 0 of the weyl group w corresponds a pbw basis of the quantised enveloping algebra u q g and one can apply the theory of deleting derivation to this iterated ore extension in particular for each decomposition of w 0 this theory constructs a bijection between the set of prime ideals in u q g that are invariant under a natural torus action and certain combinatorial objects called cauchon diagrams in this paper we give an algorithmic description of these cauchon diagrams when the chosen reduced decomposition of w 0 corresponds to a good ordering in the sense of lusztig 1990 lus90 of the set of positive roots this algorithmic description is based on the constraints that are coming from lusztig s admissible planes lusztig 1990 lus90 each admissible plane leads to a set of constraints that a diagram has to satisfy to be cauchon moreover we explicitly describe the set of cauchon diagrams for explicit reduced decomposition of w 0 in each possible type in any case we check that the number of cauchon diagrams is always equal to the cardinal of w in cauchon and meriaux 2008 cm08 we use these results to prove that cauchon diagrams correspond canonically to the positive subexpressions of w 0 so the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of w 0 corresponding to a good ordering

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

double adjacency graphs based discriminant neighborhood embedding discriminant neighborhood embedding dne is a typical graph based dimensionality reduction method and has been successfully applied to face recognition by constructing an adjacency graph aiming to keep the local structure for original data in the subspace it is able to find the optimal discriminant direction effectively not for every sample does dne set up a link between it and its heterogeneous samples when constructing the adjacency graph which would result in a small between class scatter motivated by this fact we develop an extension of dne called double adjacency graphs based discriminant neighborhood embedding dag dne by introducing two adjacency graphs or homogeneous and heterogeneous neighbor adjacency graphs in dag dne neighbors belonging to the same class are compact while neighbors belonging to different classes become separable in the subspace thus dag dne could keep the local structure of a given data and find a good projection matrix for them to investigate the performance of dag dne we compare it with the state of the art dimensionality reduction techniques such as dne and mfa on several publicly available datasets experimental results show the feasibility and effectiveness of the proposed method highlightsa supervised dimensionality method dag dne is proposed for classification tasks dag dne constructs two neighbor adjacency graphs to balance links dag dne keeps the local structure of data and finds a good projection for them experimental results show the feasibility and effectiveness of dag dne

discriminative sparsity preserving projections for image recognition previous works have demonstrated that image classification performance can be significantly improved by manifold learning however performance of manifold learning heavily depends on the manual selection of parameters resulting in bad adaptability in real world applications in this paper we propose a new dimensionality reduction method called discriminative sparsity preserving projections dspp different from the existing sparse subspace algorithms which manually construct a penalty adjacency graph dspp employs sparse representation model to adaptively build both intrinsic adjacency graph and penalty graph with weight matrix and then integrates global within class structure into the discriminant manifold learning objective function for dimensionality reduction extensive experimental results on four image databases demonstrate the effectiveness of the proposed approach we analyzed the vertex points of penalty graph using sparse representation we adaptively constructed both penalty adjacency graph and intrinsic graph our method simultaneously considered the global and local geometric structure of data

the dixmier map for nilpotent super lie algebras in this article we prove that there exists a dixmier map for nilpotent super lie algebras in other words if we denote by mathrm prim mathcal u mathfrak g the set of graded primitive ideals of the enveloping algebra mathcal u mathfrak g of a nilpotent lie superalgebra mathfrak g and mathcal a d 0 the adjoint group of mathfrak g 0 we prove that the usual dixmier map for nilpotent lie algebras can be naturally extended to the context of nilpotent super lie algebras i e there exists a bijective map i mathfrak g 0 mathcal a d 0 rightarrow mathrm prim mathcal u mathfrak g defined by sending the equivalence class of a functional to a primitive ideal i of mathcal u mathfrak g and which coincides with the dixmier map in the case of nilpotent lie algebras moreover the construction of the previous map is explicit and more or less parallel to the one for lie algebras a major difference with a previous approach cf 18 one key fact in the construction is the existence of polarizations for super lie algebras generalizing the concept defined for lie algebras as a corollary of the previous description we obtain the isomorphism mathcal u mathfrak g i lambda simeq rm cliff q k otimes a p k where p q rm dim mathfrak g 0 mathfrak g 0 lambda 2 rm dim mathfrak g 1 mathfrak g 1 lambda we get a direct construction of the maximal ideals of the underlying algebra of mathcal u mathfrak g and also some properties of the stabilizers of the primitive ideals of mathcal u mathfrak g

global and krull dimensions of quantum weyl algebras abstract given a hecke symmetry r a giaquinto and j zhang defined a natural quantization an r of the nth weyl algebra an based on r and studied many ring theoretic properties of rings a2 ja b arising from the jordan hecke symmetry and an q pi j arising from the standard multiparameter hecke symmetry here we compute the global and krull dimensions in the cases that were left open namely we show that over any field k of characteristic zero gldim a2 ja b kdim a2 ja b 3 for any a b k with a b and gldim an 1 pi j kdim an 1 pi j n

two dimensional supervised local similarity and diversity projection this paper presents a novel manifold learning method namely two dimensional supervised local similarity and diversity projection 2dslsdp for feature extraction the proposed method defines two weighted adjacency graphs namely similarity graph and diversity graph the affinity matrix of similarity graph is determined by the spatial relationship between vertices of this graph while affinity matrix of diversity graph is determined by the diversity information of vertices of its graph using these two graphs the proposed method constructs local similarity scatter and diversity scatter respectively a concise feature extraction criterion is then raised via minimizing the ratio of the local similarity scatter to local diversity scatter thus 2dslsdp can well preserve not only the adjacency similarity structure but also the diversity of data points which is important for the classification experiments on the ar and umist databases show the effectiveness of the proposed method

down up algebras and ambiskew polynomial rings abstract we show that the down up algebras of g benkart 1998 in recent progress in algebra contemporary mathematics vol 224 am math soc providence and g benkart and t roby 1998 j algebra209 305 344 lie in a certain class of iterated skew polynomial rings called ambiskew polynomial rings in two indeterminates x and y over a commutative ring b in such rings commutation of the indeterminates with elements of b involve the same endomorphism of b but from different sides that is yb b y and bx x b and for some scalar p yx pxy b in previous studies of ambiskew polynomial rings was required to be an automorphism but in order to cover all down up algebras this requirement must be dropped the noetherian down up algebras are those where is an automorphism and in this case we apply existing results on ambiskew polynomial rings to determine the finite dimensional simple modules and the prime ideals we adapt the methods underlying these results so as to apply to the non noetherian down up algebras for which they reveal a surprisingly rich structure

centred bimodules over prime rings closed submodules and applications to ring extensions let m be a centred bimodule over a prime ring r in this paper we define and study a very useful class of sub bimodules of m the class of closed sub bimodules there is a canonical torsion free extension of m to a q bimodule m which is always free over q where q is the complete ring of right quotients of r we prove that closed sub bimodules of m are in one to one correspondence with closed sub bimodules of m the results are applied to study the torsion free rank of a sub bimodule of m and to study non singular and strongly closed sub bimodules also the results are applied to study prime ideals in centred extensions and intermediate extensions in particular we complete and extend the results obtained in m ferrero j algebra 148 1992 1 16

algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

duo rings and ore extensions abstract we show that in an ore polynomial ring every left ideal is two sided only in the trivial commutative case it is unknown whether the analogous statement about right ideals is true we establish a variety of restrictions on any putative counterexample

localization and ideal theory in noetherian strongly group graded rings abstract we study primality hypercentrality simplicity and localization and the second layer condition in skew groups rings and group graded rings we give necessary and sufficient conditions for the skew group ring of a torsion free nilpotent group to be a simple ring and if the coefficient ring is commutative we give necessary and sufficient conditions for the skew group ring of an abelian group to be simple our method involves showing certain group graded rings are hypercentral our main results show that if g is a polycyclic by finite group and r is an artinian ring or a commutative noetherian ring then a strongly g graded ring with base ring r satisfies the second layer condition we discuss consequences of this for localization in such rings

irreducible modules for the quantum affine algebra uq slˆ2 and its borel subalgebra abstract let u q b denote the standard borel subalgebra of the quantum affine algebra u q sl 2 we show that the following hold for any choice of scalars 0 1 from the set 1 1 i let v be a finite dimensional irreducible u q b module of type 0 1 then the action of u q b on v extends uniquely to an action of u q sl 2 on v the resulting u q sl 2 module structure on v is irreducible and of type 0 1 ii let v be a finite dimensional irreducible u q sl 2 module of type 0 1 when the u q sl 2 action is restricted to u q b the resulting u q b module structure on v is irreducible and of type 0 1

spectre premier de oq mn k image canonique et separation normale abstract given any commutative field k denote r oq mn k the coordinate ring of quantum n n matrices over k and assume q is a nonzero element in k which is not a root of unity recall that r is generated by n2 variables xi i 1 n 2 subject only to the following relations if x y z t is any 2 2 sub matrix of x x i then a yx q 1xy zx q 1xz tz q 1zt ty q 1yt zy yz b tx xt q q 1 yz denote r the k algebra generated by the same variables xi subject to the same relations except relations b which are replaced by c tx xt so that r is just the algebra of regular functions on some quantum affine space of dimension n2 over k the theory of derivative elimination defines a natural embedding spec r spec r and asserts that the canonical image spec r is a union of strata spec w r in the sense of goodearl letzter in cms conf proc vol 22 1998 39 58 where w describes some subset w of p 1 n 2 the sets spec w r 1 spec w r w w define the goodearl letzter h stratification of spec r in the sense of goodearl letzter trans amer math soc 352 2000 1381 1403 in this paper we give the precise description of the set w and we compute its cardinality using that description and the derivative elimination algorithm we can verify theorems 6 3 1 6 3 2 that h spec r has an h normal separation in the sense of goodearl in lecture notes in pure and appl math 210 2000 205 237 so that spec r has normal separation in the sense of brown goodearl trans amer math soc 348 1996 2465 2502 this property was conjectured by k brown and k goodearl since r is auslander regular and cohen macaulay this implies by goodearl lenagan j pure appl algebra 111 1996 123 142 that r is catenary and satisfies the tauvel s height formula

prime spectra of quantum semisimple groups we study the prime ideal spaces of the quantized function algebras rq g for g a semisimple lie group and q an indeterminate our method is to examine the structure of algebras satisfying a set of seven hypotheses and then to demonstrate using work of joseph hodges and levasseur that the algebras rq g satisfy this list of assumptions rings satisfying the assumptions are shown to satisfy normal separation and therefore jategaonkar s strong second layer condition for such rings much representation theoretic information is carried by the graph of links of the prime spectrum and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms bounds on the ranks of these groups are obtained in the case of rq g in the final section the results are specialized to the case g sln c where detailed calculations can be used to illustrate the general results as a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n space satisfy our axiom scheme when the group generated by the parameters is torsionfree 0 introduction and background 0 1 let p and q be prime ideals of a noetherian ring r we say that p is linked to q and write p q if there is an ideal a of r with pq c a c p n q such that p n q a is a nonzero torsionfree left r p module and a torsionfree right r q module this condition can be simplified in case r satisfies the second layer condition see below in that case p q if and only if p n q pq is faithful as a left r p module and as a right r q module 39 10 1 4 the graph of links of r is the directed graph whose vertices are the points of spec r the set of prime ideals of r with a directed edge from p to q if and only if p q the primes in the connected component of the graph of links containing p constitute the clique of p denoted clique p there is a close connection between the graph of links of r and the representation theory of r cf 20 4 this connection is particularly strong when r satisfies the strong second layer condition which requires that whenever u is a cyclic uniform r module with the annihilator received by the editors november 4 1994 and in revised form september 5 1995 1991 mathematics subject classification primary 16d30 16d60 16p40 17b37 the research of the second author was partially supported by a grant from the national science foundation usa part of the work was carried out while he visited the university of glasgow mathematics department during october 1993 supported by the edinburgh and london mathematical societies work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the department of mathematics of the university of washington whom both thank for its hospitality the travel costs of the first author were in part covered by a grant from the carnegie trust for the universities of scotland 1996 american mathematical society

localization and ideal theory in iterated differential operator rings abstract in this paper we study primality hypercentrality simplicity and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings we give sufficient conditions for the skew enveloping algebra of a nilpotent lie algebra with coefficient ring containing the rational numbers to be a simple ring and we give necessary and sufficient conditions in the case that the lie algebra is abelian our main results show that if l is a finite dimensional solvable lie algebra over a field k of characteristic zero and r is an artinian or a commutative noetherian algebra over k then the skew enveloping algebra r u l satisfies the second layer condition we discuss consequences of this for localization and use the localization theory to state a classical krull dimension versus global dimension inequality when k is uncountable

evolution semigroups and sums of commuting operators a new approach to the admissibility theory of function spaces this paper is concerned with conditions for the admissibility of a translation invariant function space m with respect to a well posed linear evolution equation du dt au f t t r we propose a new approach to this problem by considering the sum of two commuting operators d dt dm and the operator of multiplication by a on m on the one hand the closure of this operator is the infinitesimal generator of the so called evolution semigroup associated with on the other hand the generator g of this semigroup relates a mild solution u of to the forcing term f by the rule gu f consequently various spectral criteria of the type dm a for the admissibility of the function space m with respect to can be proved in an elegant manner moreover they can be naturally extended to general classes of differential equations including higher order and abstract functional differential equations applications and examples are provided to illustrate the obtained results

r commutative geometry and quantization of poisson algebras abstract an r commutative algebra is an algebra a equipped with a yang baxter operator r a a a a satisfying m mr where m a a a is the multiplication map together with the compatibility conditions r a 1 1 a r 1 a a 1 r id m m id r 2 r 1 and r m id id m r 1 r 2 the basic notions of differential geometry extend from commutative or supercommutative algebras to r commutative algebras examples of r commutative algebras obtained by quantization of poisson algebras include the weyl algebra noncommutative tori quantum groups and certain quantum vector spaces in many of these cases the r commutative de rham cohomology is stable under quantization

the launois lenagan conjecture abstract in this note we prove the launois lenagan conjecture on the classification of the automorphism groups of the algebras of quantum matrices r q m n of square shape for all positive integers n base fields k and deformation parameters q k which are not roots of unity

generalized vector valued almost periodic and ergodic distributions abstract schwartz s almost periodic distributions are generalized to the case of banach space valued distributions d ap r x and furthermore for a given arbitrary class a to d a r x t d r x t a for test functions d r c it is shown that this extension process a d a is iteration complete i e d d a d a moreover the t from d a are characterized in various ways also tempered distributions s r x d p r x with p x valued functions of polynomial growth are shown under suitable assumptions d a r x n 0 m n a d a r x l loc 1 r x n 0 m n a where ma f l loc 1 m h f 1 h 0 h f s d s a for all h 0 m a is defined with the corresponding extension of m h with an extension of the indefinite integral from l loc 1 to d r x a distribution analogue to the bohl bohr amerio kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained also for almost automorphic levitan almost periodic and recurrent functions similar for a result of levitan concerning ergodic indefinite integrals for many of the above results a new condition is needed we show that it holds for most of the a needed in applications also an application to the study of asymptotic behavior of distribution solutions of neutral integro differential difference systems is given

field of fractions of certain quantum algebras abstract in this paper we show that the field of fractions of certain algebras appearing in the theory of quantum groups are isomorphic to the field of fractions of quantum n space when the parameter q is not a root of 1 more precisely the quantum serre relations of type a n and the relations defining the quantum weyl algebra induce q communtation relations in the field of fractions also we study the problem of separation of these skew fields using the standard embeddings in non commutative series

invariant prime ideals in quantizations of nilpotent lie algebras de concini kac and procesi defined a family of subalgebras u of a quantized universal enveloping algebra uq g associated to the elements of the corresponding weyl group w they are deformations of the universal enveloping algebras u n adw n where n are the nilradicals of a pair of dual borel subalgebras based on results of gorelik and joseph and an interpretation of u as quantized algebras of functions on schubert cells we construct explicitly the h invariant prime ideals of each u and show that the corresponding poset is isomorphic to w where h is the group of group like elements of uq g moreover for each h prime of u w we construct a generating set in terms of demazure modules related to fundamental representations using results of ramanathan and kempf we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related poisson structures on schubert cells in flag varieties

quantum and poisson structures of multi parameter symplectic and euclidean spaces abstract a class of poisson algebras a n p q considered as a poisson version of the multiparameter quantized coordinate rings k n p q of symplectic and euclidean 2n spaces is constructed and poisson structures of a n p q are described in particular it is proved that the prime poisson and poisson primitive spectra of a n p q are homeomorphic to the prime and primitive spectra of k n p q in the case when the multiplicative subgroup of k generated by all parameters in k n p q is torsion free and as a corollary that the prime and primitive spectra of k n p q are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring

les ideaux premiers invariants de oq mm p c abstract for q generic we give a positive answer to a conjecture of goodearl and lenagan the h invariant prime ideals of o q m m p c are generated by quantum minors

the influence of ideal structure on representation theory the purpose of this paper is to explore the representation theoretic consequences of the two sided ideal structure of a noetherian ring with special emphasis on an additional combinatorial structure on the prime spectrum of the ring the graph of links we obtain existence theorems for extensions and for rings satisfying the strong second layer condition we obtain extensive information about the indecomposable injective modules particularly detailed results are obtained when the ring is polynormal

quantized rank r matrices abstract first some old as well as new results about p i algebras ore extensions and degrees are presented then quantized n r matrices as well as certain quantized factor algebras m r 1 q n of m q n are analyzed for r 1 n 1 m r 1 q n is the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all r 1 r 1 quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one in almost all cases the quantum parameter is a primitive m th root of unity the degrees and centers of the algebras are determined when m is a prime and the general structure is obtained for arbitrary m

real spectra of quantum groups abstract the only noncommutative ring for which the real spectrum has been described so far is the quantum affine ring r q x y j algebra 212 1999 190 the aim of this paper is to describe the real spectra of quantum affine rings k q x 1 x n where k is a formally real affine r algebra and q m n r as a by product we describe the real spectra of quantized enveloping algebra u q s l 2 r and quantum special linear group o q sl 2 r formal reality and semireality is characterized for the following classes of quantum groups quantum affine rings quantized enveloping algebras quantized function algebras quantized weyl algebras

feature extraction using constrained maximum variance mapping in this paper an efficient feature extraction method named as constrained maximum variance mapping cmvm is developed the proposed algorithm can be viewed as a linear approximation of multi manifolds learning based approach which takes the local geometry and manifold labels into account the cmvm and the original manifold learning based approaches have a point in common that the locality is preserved moreover the cmvm is globally maximizing the distances between different manifolds after the local scatters have been characterized the proposed method focuses on developing a linear transformation that can maximize the dissimilarities between all the manifolds under the constraint of locality preserving compared to most of the up to date manifold learning based methods this trick makes contribution to pattern classification from two aspects on the one hand the local structure in each manifold is still kept on the other hand the discriminant information between manifolds can be explored finally feret face database cmu pie face database and usps handwriting data are all taken to examine the effectiveness and efficiency of the proposed method experimental results validate that the proposed approach is superior to other feature extraction methods such as linear discriminant analysis lda locality preserving projection lpp unsupervised discriminant projection udp and maximum variance projection mvp

quasi hom lie algebras central extensions and 2 cocycle like identities this paper introduces the notion of a quasi hom lie algebra or simply a qhl algebra which is a natural generalization of hom lie algebras introduced in a previous paper j t hartwig d larsson s d silvestrov deformations of lie algebras using sigma derivations math qa 0408064 quasi hom lie algebras include also as special cases color lie algebras and superalgebras and can be seen as deformations of these by maps twisting the jacobi identity and skew symmetry the natural realm for these quasi hom lie algebras is generalizations deformations of the witt algebra delta of derivations on the laurent polynomials c t t 1 we also develop a theory of central extensions for qhl algebras which can be used to deform and generalize the virasoro algebra by centrally extending the deformed witt type algebras constructed here in addition we give a number of other interesting examples of quasi hom lie algebras among them a deformation of the loop algebra less

effacement des derivations et spectres premiers des algebres quantiques abstract given any commutative field k and any iterated ore extension r k x1 x2 2 2 xn n n satisfying some suitable assumptions we construct the so called derivative elimination algorithm it consists of a sequence of changes of variables inside the division ring f fract r starting with the indeterminates x1 xn and terminating with new variables t1 tn these new variables generate some quantum affine space r such that f fract r this algorithm induces a natural embedding spec r spec r which satisfies the following property if p is in spec r then fract r p fract r p we study both the derivative elimination algorithm and natural embedding and use them to produce for the general case a common proof of the quantum gelfand kirillov property for the prime homomorphic images of the following quantum algebras u q w g b q w g w w r q g where g denotes any complex semi simple connected simply connected lie group with associated lie algebra g and weyl group w quantum matrices algebras quantum weyl algebras and quantum euclidean respectively symplectic spaces another application will be given in g cauchon j algebra to appear in the general case the prime spectrum of any quantum matrices algebra satisfies the normal separation property

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

existence of bounded uniformly continuous mild solutions on r of evolution equations and their asymptotic behaviour abstract we prove that u a u has on r a mild solution u b u c r x that is bounded and uniformly continuous where a is the generator of a c 0 semigroup on the banach space x with resolvent satisfying r i t a o t t for some 1 2 l r x and i s p a 0 sp beurlingspectrum as a consequence it is shown that if f is the space of almost periodic almost automorphic bounded levitan almost periodic or one of certain classes of recurrent functions and as above is such that only m h 1 h 0 h s d s f for each h 0 then u f b u c r x these results seem new and strengthen several recent theorems

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

asymptotic behavior of solutions of systems of neutral and convolution equations suppose j for some r or j r and let x be a banach space we study asymptotic behavior of solutions on j of neutral system of equations with values in x this reduces to questions concerning the behavior of solutions of convolution equations h v 0 b where h hj k i s anr r matrix hj k d 1 b bj and bj d r x for 1 j kr we prove that if 0 is a bounded uniformly continuous solution of with b from some translation invariant suitably closed class a then 0 belongs to a provided for example that det h has countably many zeros on r and c0 3 x in particular if b is asymptotically almost periodic almost automorphic or recurrent 0 is too our results extend theorems of bohr neugebauer bochner doss basit and zhikov and also certain theorems of fink madych staffans and others also we investigate bounded solutions of this leads to an extension of the known classes of almost periodicity to larger classes called mean classes we explore mean classes and prove that bounded solutions of belong to mean classes provided certain conditions hold these results seem new even for the simplest difference equation 0 t 1 0 t b t with j x r and b stepanoff almost periodic 1998 academic press

on the composition of q skew derivations in banach algebras abstract let a be a banach algebra let be an automorphism of a and let d be q skew derivations of a we show that if d a is quasi nilpotent for any a a then d a 3 lies in the radical of a for all a a this result simultaneously generalizes the previous results obtained by bresar and semrl 5 chebotar et al 7 and abdelali 1

solution of a q difference noether problem and the quantum gelfand kirillov conjecture for mathfrak gl _n it is shown that the q difference noether problem for all classical weyl groups has a positive solution simultaneously generalizing well known results on multisymmetric functions of mattuck proc am math soc 19 764 765 1968 and miyata nagoya math j 41 69 73 1971 in the case q 1 and q deforming the noncommutative noether problem for the symmetric group futorny et al in adv math 223 773 796 2010 it is also shown that the quantum gelfand kirillov conjecture for mathfrak gl n for a generic q follows from the positive solution of the q difference noether problem for the weyl group of type d n the proof is based on the theory of galois rings futorny and ovsienko in j algebra 324 598 630 2010 from here we obtain a proof of the quantum gelfand kirillov conjecture for mathfrak gl n and for a certain extension of mathfrak sl n previously the case of mathfrak sl n was shown by fauquant millet j algebra 218 93 116 1999 and by alev and dumas j algebra 170 229 265 1994 for n 2 3 moreover we give an explicit description of the skew fields of fractions for u q mathfrak gl n and u q mathrm ext mathfrak sl n which generalizes the results of alev and dumas j algebra 170 229 265 1994

torus invariant prime ideals in quantum matrices totally nonnegative cells and symplectic leaves the algebra of quantum matrices of a given size supports a rational torus action by automorphisms it follows from work of letzter and the first named author that to understand the prime and primitive spectra of this algebra the first step is to understand the prime ideals that are invariant under the torus action in this paper we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size as a corollary we obtain explicit generating sets of quantum minors for the torus invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over mathbb q

a taxonomy of 2 primal rings abstract various conditions on a noncommutative ring imply that it is 2 primal i e the ring s prime radical coincides with the set of nilpotent elements of the ring we will examine several such conditions and show that their known interdependencies are their only ones of particular interest will be the ps i condition on a ring i e every factor ring modulo the right annihilator of a principal right ideal is 2 primal we will see that even within a fairly narrow class of rings ps i is a strictly stronger condition than 2 primal we will show that the ps i condition is left right asymmetric we will also study the interplay between various types of semilocal rings and various types of 2 primal rings the kothe conjecture will make a cameo appearance in section 6 we will examine subideals of prime ideals of commutative rings that are invariant under derivations

affiliated series and extensions of modules the main purpose of this paper is to examine the influence on the representation theory of a noetherian ring of the prime ideal structure of the ring the results concern the affiliated primes of a module defined below we show how the affiliated primes are connected with the annihilator primes and we prove a uniqueness theorem for affiliated primes under certain conditions we then proceed to analyse what affiliated primes can appear in terms of the ideal structure of the ring with special reference to the fundamental primes of injective modules our starting point is a theorem of jategaonkar which analyses the structure of modules with exactly two affiliated primes and which already has consequences for uniserial modules of length two we give a generalization of this result for modules with longer affiliated series and in particular for uniserial modules of length greater than two in particular we show that some of the possibilities one might have envisioned in the light of jategaonkar s theorem cannot in fact occur and we demonstate a somewhat unexpected connection between affiliated primes and annihilator primes to put all this material in context we begin section one with an analysis of what it all means for uniserial modules of length three we then turn to the uniqueness question for affiliated primes here and in the rest of the paper we find it desirable to have some kind of dimension 164 0021 8693 91 3 00

a strong dixmier moeglin equivalence for quantum schubert cells abstract dixmier and moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite dimensional complex lie algebra they showed that the primitive rational and locally closed ideals coincide in modern terminology they showed that the universal enveloping algebra of a finite dimensional complex lie algebra satisfies the dixmier moeglin equivalence we define quantities which measure how close an arbitrary prime ideal of a noetherian algebra is to being primitive rational and locally closed if every prime ideal is equally close to satisfying each of these three properties then we say that the algebra satisfies the strong dixmier moeglin equivalence using the example of the universal enveloping algebra of sl 2 c we show that the strong dixmier moeglin equivalence is strictly stronger than the dixmier moeglin equivalence for a simple complex lie algebra g a non root of unity q 0 in an infinite field k and an element w of the weyl group of g de concini kac and procesi have constructed a subalgebra u q w of the quantised enveloping k algebra u q g these quantum schubert cells are known to satisfy the dixmier moeglin equivalence and we show that they in fact satisfy the strong dixmier moeglin equivalence along the way we show that commutative affine domains uniparameter quantum tori and uniparameter quantum affine spaces satisfy the strong dixmier moeglin equivalence

examples of ore extensions which are maximal orders whose based rings are not maximal orders abstract let r be a prime goldie ring and be a skew derivation on r it is well known that if r is a maximal order then the ore extension r x is a maximal order it was a long standing open question that the converse is true or not in case 1 and 0 we give an example of non maximal order r with a skew derivation on r 1 0 such that r x is a maximal order

derivations of skew polynomial rings abstract let f be a field of characteristic 0 and let i j f for 1 i j n define r f x 1 x 2 x n to be the skew polynomial ring with x i x j i j x j x i and let s f x 1 x 2 x n x 11 x 12 x 1n be the corresponding laurent polynomial ring in a recent paper kirkman procesi and small considered these two rings under the assumption that s is simple and showed for example that the lie ring of inner derivations of s is simple furthermore when n 2 they determined the automorphisms of s related its ring of inner derivations to a certain block algebra and proved that every derivation of r is the sum of an inner derivation and a derivation which sends each xi to a scalar multiple of itself in this paper we extended these results to a more general situation specifically we study twisted group algebras ft g where g is a commutative group and f is a field of any characteristic furthermore we consider certain subalgebras ft h where h is a subsemigroup of g which generates g as a group finally if e g g f is a skew symmetric bilinear form then we study the lie algebra fe g associated with e and we consider its relationship to the lie structure defined on various twisted group algebras ft g

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

links between prime ideals in differential operator rings in recent years much work has been done on a localization theory for noetherian rings links between prime ideals play a central role in this development when a ring r is not commutative it is rarely possible to localize r at a prime ideal p having a link between p and another prime ideal is an obstruction to localization in this paper we will investigate the links between prime ideals in differential operator rings showing that it is possible to localize these rings in a certain sense recall that a multiplicatively closed subset s of a ring r is a right ore set if for all r in r and all s in s we have rs n sr qr the set s is an ore set if it is both a right and a left ore set it is always possible to construct a ring rf where the elements of s become units however if we insist on writing the elements of rf as right fractions rs with r in r and s in s then the set s must be a right ore set 9 sect 12 11 let p be a prime ideal of r and let 7 p r e r 1 r p is regular in r p we say that p is a localizable prime ideal if the set u p is an ore set we will shortly give an example of a prime ideal which is not localizable but first we introduce the differential operator ring let r be a ring and let 6 be an additive map from r to itself we call 6 a derivation if it satisfies the product rule 6 ab a b ad b the differential operatqr ring over r denoted r 0 s is a free left r module with basis 1 8 13 thus every element of r 0 s is a polynomial in 0 with coefficients from r the addition in r l s is defined as usual for polynomials but multiplication is extended from r by the rule br rti 6 r the differential operator ring is an associative ring and is determined up to isomorphism by an obvious universal property 9 sect 12 21

from quantum ore extensions to quantum tori via noncommutative ufds abstract all iterated skew polynomial extensions arising from quantized universal enveloping algebras of kac moody algebras are special examples of a very large axiomatically defined class of algebras called cgl extensions for the purposes of constructing initial clusters for quantum cluster algebra structures on an algebra r and classification of the automorphisms of r one needs embeddings of r into quantum tori t which have the property that r contains the corresponding quantum affine space algebra a we explicitly construct such an embedding a r t for each cgl extension r using the methods of noncommutative noetherian unique factorization domains and running a gelfand tsetlin type procedure with normal instead of central elements along the way we classify the homogeneous prime elements of all cgl extensions and we prove that each cgl extension r has an associated maximal torus which covers the automorphisms of r corresponding to all normal elements for symmetric cgl extensions we describe the relationship between our quantum affine space algebra a and cauchon s quantum affine space algebra generated by elements obtained via deleting derivations

on theq commutations inuq n at roots of one abstract let e be a root of one and g a semisimple lie algebra with triangular decomposition g n h n letu e resp ures e be the nonrestricted resp restricted quantum enveloping algebra of n we prove that fract u eis a quantum weyl field we then give a description of the e center ofu e letufin ebe the finite part ofures e via the drinfeld correspondence theufin e covariant space of a weyl module is e central in case g s l n this enables us to describe this space in terms of semistandard young tableaux

supervised projection approach for boosting classifiers in this paper we present a new approach for boosting methods for the construction of ensembles of classifiers the approach is based on using the distribution given by the weighting scheme of boosting to construct a non linear supervised projection of the original variables instead of using the weights of the instances to train the next classifier with this method we construct ensembles that are able to achieve a better generalization error and are more robust to noise presence it has been proved that adaboost method is able to improve the margin of the instances achieved by the ensemble moreover its practical success has been partially explained by this margin maximization property however in noisy problems likely to occur in real world applications the maximization of the margin of wrong instances or outliers can lead to poor generalization we propose an alternative approach where the distribution of the weights given by the boosting algorithm is used to get a supervised projection then the supervised projection is used to train the next classifier using a uniform distribution of the training instances the proposed approach is compared with three boosting techniques namely adaboost gentleboost and madaboost showing an improved performance on a large set of 55 problems from the uci machine learning repository and less sensitiveness to noise in the class labels the behavior of the proposed algorithm in terms of margin distribution and bias variance decomposition is also studied

classification of factorial generalized down up algebras we determine when a generalized down up algebra is a noetherian unique factorisation domain or a noetherian unique factorisation ring

combinatorics of h primes in quantum matrices abstract for q c transcendental over q we give an algorithmic construction of an order isomorphism between the set of h primes of o q m n c and the sub poset s of the reverse bruhat order of the symmetric group s 2 n consisting of those permutations that move any integer by no more than n positions further we describe the permutations that correspond via this bijection to rank t h primes more precisely we establish the following result imagine that there is a barrier between positions n and n 1 then a 2n permutation s corresponds to a rank t h invariant prime ideal of o q m n c if and only if the number of integers that are moved by from the right to the left of this barrier is exactly n t the existence of such an order isomorphism was conjectured by goodearl and lenagan

the frt construction via quantum affine algebras and smash products abstract for every element w in the weyl group of a simple lie algebra g de concini kac and procesi defined a subalgebra u q w of the quantized universal enveloping algebra u q g the algebra u q w is a deformation of the universal enveloping algebra u n w n we construct smash products of certain finite type de concini kac procesi algebras to obtain ones of affine type we have analogous constructions in types a n and d n we show that the multiplication in the affine type de concini kac procesi algebras arising from this smash product construction can be twisted by a cocycle to produce certain subalgebras related to the corresponding faddeev reshetikhin takhtajan bialgebras

the global dimension of a q skew polynomial ring abstract a q skew polynomial ring is a mixed case skew polynomial ring in which the defining automorphism and derivation commute up to a scalar multiple this paper establishes criteria on the maximal ideals of a commutative noetherian ring with finite global dimension which determine the global dimension of a q skew polynomial ring extension

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

simple ambiskew polynomial rings abstract we determine simplicity criteria in characteristics 0 and p for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring one obstruction to simplicity is the possible existence of a canonical normal element z in the case where this element exists we give simplicity criteria for the rings obtained by inverting z and the rings obtained by factoring out the ideal generated by z the results are illustrated by numerous examples including higher quantized weyl algebras and generalizations of some low dimensional symplectic reflection algebras

pi degree parity in q skew polynomial rings abstract for k a field of arbitrary characteristic and r a k algebra we show that the pi degree of an iterated skew polynomial ring r x 1 1 1 x n n n agrees with the pi degree of r x 1 1 x n n when each i i satisfies a q i skew relation for q i k and extends to a higher q i skew i derivation we confirm the quantum gel fand kirillov conjecture for various quantized coordinate rings and calculate their pi degrees we extend these results to completely prime factor algebras

quotients premiers deoq mn k resume consider a fieldk some nonzero elementqofkwhich is not a root of unity and some nonnegative integern k r goodearl and e s letzter have proved 1991 prime factor algebras of the coordinate ring of quantum matrices preprint that any prime factor algebra of the coordinate ringoq m n k of quantumn nmatrices overkis a domain in that paper it is proved that the division ring of fractions of such a domain is always isomorphic to the division ring of fractions of the coordinate ring of some quantum space of dimension at mostn2over some fieldkwhich is an extension ofk a similar result is also proved for quantum weyl algebras

prime and primitive ideals of a class of iterated skew polynomial rings hodges and levasseur described the primitive spectra of quantized coordinate rings of sl3 14 and then of sln 15 these results established a close connection between primitive ideals torus action and poisson geometry the proofs relied on explicit computations involving generators and relations subsequently joseph generalized the hodges levasseur program to semisimple algebraic quantum groups 17 hodges et al then expanded joseph s work to include certain multiparameter deformations 16 these papers rely less on concrete calculations and more on deeper more conceptual techniques it is a natural and important question then as to how the preceding theory might apply to other algebras particularly other algebras arising in the study of quantum groups goodearl and letzter established parallel results for certain iterated skew polynomial rings 13 with application to quantum weyl algebras and to some quantum coordinate rings

ore extensions of jacobson rings abstract it is shown that under suitable conditions an ore extension r x of a jacobson ring r i e a ring all of whose prime ideals are semiprimitive with a monomorphism will be jacobson these conditions are satisfied by the class of rings studied by pearson stephenson and watters and by left noetherian rings giving a theorem of goldie and michler

kernel self optimization learning for kernel based feature extraction and recognition kernel learning is becoming an important research topic in the area of machine learning and it has wide applications in pattern recognition computer vision image and signal processing kernel learning provides a promising solution to nonlinear problems including nonlinear feature extraction classification and clustering however in kernel based systems the problem of the kernel function and its parameters remains to be solved methods of choosing parameters from a discrete set of values have been presented in previous studies but these methods do not change the data distribution structure in the kernel based mapping space accordingly performance is not improved because the current kernel optimization does not change the data distribution based on this problem this paper presents a uniform framework for kernel self optimization with the ability to adjust the data structure the data dependent kernel is extended and applied to kernel learning and optimization equations with two criteria for measuring data discrimination are used to solve the optimal parameter values some experiments are performed to evaluate the performance in popular kernel learning methods including kernel principal components analysis kpca kernel discriminant analysis kda and kernel locality preserving projection klpp these evaluations show that the framework of kernel self optimization is feasible for enhancing kernel based learning methods

cyclic homology of differential operators the virasoro algebra and a q analogue we show how methods from cyclic homology give easily an explicit 2 cocycle on the lie algebra of differential operators of the circle such that restricts to the cocycle defining the virasoro algebra the same methods yield also aq analogue of as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity we use an algebra ofq difference operators andq analogues of koszul and the rham complexes to construct these quantum cocycles

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

fields of fractions of quantum solvable algebras abstract we introduce the notion of pure q solvable algebra the quantum matrices quantum weyl algebra u q n are the examples it is proved that the skew field of fractions of a pure q solvable algebra r is isomorphic to the skew field of twisted rational functions this is a quantum version of the gelfand kirillov conjecture for solvable algebraic lie algebras

algebraic q skew derivations abstract let r be a prime ring c its extended centroid and r f resp q its left resp symmetric martindale quotient ring let be a derivation of r where is an automorphism of r we show the equivalence of k polynomials resp k identities of and cv polynomials resp semi invariant polynomials in the ore extension q x we prove the existence of k polynomials of in certain rather general family of maps as applications the following are proved among other things consider the expression x i 0 n a i i x where a i r f and a n 0 1 if dim c r c then either r is a gpi ring or x 0 for all x r f 2 if r c then either r is commutative or x 0 for all x r f

quasi lie structure of σ derivations of c t 1 hartwig larsson and silvestrov in j t hartwig d larsson s d silvestrov deformations of lie algebras using derivations j algebra 295 2 2006 314 361 defined a bracket on derivations of a commutative algebra we show that this bracket preserves inner derivations and based on this obtain structural results providing new insights into derivations on laurent polynomials in one variable

rapid and brief communication local structure based supervised feature extraction in the past few years the computer vision and pattern recognition community has witnessed the rapid growth of a new kind of feature extraction method the manifold learning methods which attempt to project the original data into a lower dimensional feature space by preserving the local neighborhood structure among them locality preserving projection lpp is one of the most promising feature extraction techniques however when lpp is applied to the classification tasks it shows some limitations such as the ignorance of the label information in this paper we propose a novel feature extraction method called locally discriminating projection ldp ldp utilizes class information to guide the procedure of feature extraction in ldp the local structure of the original data is constructed according to a certain kind of similarity between data points which takes special consideration of both the local information and the class information the similarity has several good properties which help to discover the true intrinsic structure of the data and make ldp a robust technique for the classification tasks we compare the proposed ldp approach with lpp as well as other feature extraction methods such as pca and lda on the public available data sets feret and ar experimental results suggest that ldp provides a better representation of the class information and achieves much higher recognition accuracies

a quadratic poisson gel fand kirillov problem in prime characteristic the quadratic poisson gel fand kirillov problem asks whether the field of fractions of a poisson algebra is poisson birationally equivalent to a poisson affine space i e to a polynomial algebra k x1 xn with poisson bracket defined by xi xj ijxixj for some skew symmetric matrix ij mn k this problem was studied in 9 over a field of characteristic 0 by using a poisson version of the deleting derivation homomorphism of cauchon in this paper we study the quadratic poisson gel fand kirillov problem over a field of arbitrary characteristic in particular we prove that the quadratic poisson gel fand kirillov problem is satisfied for a large class of poisson algebras arising as semiclassical limits of quantised coordinate rings for we introduce the concept of higher poisson derivation which allows us to extend the poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic when a torus is acting rationally by poisson automorphisms on a poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring we prove under some technical assumptions that quotients by poisson prime torus invariant ideals also satisfy the quadratic poisson gel fand kirillov problem in particular we show that coordinate rings of determinantal varieties satisfy the quadratic poisson gel fand kirillov problem