on the haar measure of the quantumsu n group we prove that the haar state associated to the compact matrix quantum groupsu n is faithful for 1 1 0 and anyn 2

deformations of compact quantum groups via rieffel s quantization it is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under rieffel s quantization this is applied to showing that the two classes of compact quantum groupskqu andkq studied by levendorkii and soibelman are strict deformation quantization of each other and that the quantum groupsau m have many deformations

some quantum lie algebras of type dn positive a quantum lie algebra is constructed within the positive part of the drinfeld jimbo quantum group of type dn our quantum lie algebra structure includes a generalized antisymmetry property and a generalized jacobi identity closely related to the braid equation a generalized universal enveloping algebra of our quantum lie algebra of type dn positive is proved to be the drinfeld jimbo quantum group of the same type the existence of such a generalized lie algebra is reduced to an integer programming problem moreover when the integer programming problem is feasible we show by means of the generalized jacobi identity that the poincare birkhoff witt theorem basis is still true

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

classification of bicovariant differential calculi on quantum groups suppose thatq is not a root of unity we classify all bicovariant differential calculi of dimension greater than one on the quantum groupsgl q n o q n andsp q n for which the differentials du of the matrix entriesu generate the left module of first order forms our first classification theorem asserts that there are precisely two one parameter families of such calculi ongl q n forn 3 in the limitq 1 only two of these calculi give the ordinary differential calculus ongl n our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties ono q n andsp q n forn 4 this strengthens the corresponding result proved in our previous paper ss2 there are four such calculi ono q 3 we introduce two new 4 dimensional bicovariant differential calculi ono q 3

finite dimensional duality on the generalized lie algebra 2 q we show a local duality between the left and right finite dimensional representations on the enveloping algebra of 2 q the generalized lie algebra introduced by lyubashenko and sudbery this duality works in a general context of algebras satisfying certain algebraic properties our goal in this paper is to prove that the enveloping algebra of 2 q satisfies these abstract properties

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

examples of compact matrix pseudogroups arising from the twisting operation we construct multiparameter compact matrix pseudogroups of all types and show their representation theories are the same as their classical analogs

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

origins and uses of linear programming methods for treating l1 and l regressions corrections and comments on abstract this note comments on and corrects some of the referrals and comments of castillo et al castillo e minguez r castillo c confino a s 2008 dealing with the multiplicity of solutions of the l 1 and l regression models european journal of operational research 188 460 484 concerning the origins and uses of linear programming to estimate l 1 and l statistical regressions

schur weyl quasi duality and co triangular hopf quasigroups schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic lie theory in this paper we will introduce the notion of schur weyl quasi duality and study it for this generally we consider a braided vector space v c and its braided lie algebra endk v then we can construct its braided enveloping algebra u endk v which is a connected braided c cocommutative hopf algebra let h be a triangular hopf quasigroup with bijective antipode and b be a cotriangular hopf quasigroup with bijective antipode let v be any finite dimensional vector space in the category lq h r b of generalized long quasimodules we show that u endkv h b ksn v n is a schur weyl quasi duality under suitable conditions

on unitary representation theories of compact quantum groups categorical structure of unitary representation of compact quantum groups is studied with relation to a metrical structure encountered in the monoidal category of bimodules of finite jones index

quantum deformation of brst algebra we investigate theq deformation of the brst algebra the algebra of the ghost matter and gauge fields on one spacetime point using the result of the bicovariant differential caculus there are two nilpotent operations in the algebra the brst transformation b and the derivatived we show that one can define the covariant commutation relations among the fields and their derivatives consistently with these two operations as well as the operation the antimultiplicativ e inner involution

le langage des espaces et des groupes quantiques we study the foundations of the differential calculus in quantum geometry the notions of differential quantum space and cone are introduced generalizing a construction of manin to a quantum cone we associate the quantum group of its linear automorphisms preserving the differentials and deduce a de rham complex on this group we give examples of differential calculi on quantum hyperplanes and quantum linear groups

homological properties of braided matrices abstract let r be a multi parameter solution of the quantum yang baxter equation of type a n 1 in this note we prove that s majid s algebra of braided matrices b r is a noetherian auslander regular domain of dimension n 2 satisfying the cohen macaulay property for each n there is a formula dimensional family of such algebras

classification of bicovariant differential calculi on the jordanian quantum groups and and quantum lie algebras assuming that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators we classify all four dimensional first order bicovariant calculi on the jordanian quantum group and all three dimensional first order bicovariant calculi on the jordanian quantum group it is found that there are three one parameter families of four dimensional bicovariant first order calculi on and that there is a single unique three dimensional bicovariant calculus on this three dimensional calculus may be obtained through a classical like reduction from any one of the three families of four dimensional calculi on details of the higher order calculi and also the quantum lie algebras are presented for all calculi the quantum lie algebra obtained from the bicovariant calculus on is shown to be isomorphic to the quantum lie algebra we obtain as an ad submodule within the jordanian universal enveloping algebra and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module we also obtain the quantum killing form for this quantum lie algebra

braided affine geometry and q analogs of wave operators the main goal of this review is to compare different approaches to constructing the geometry associated with a hecke type braiding in particular with that related to the quantum group uq sl n we place emphasis on the affine braided geometry related to the so called reflection equation algebra rea all objects of such a type of geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators we begin by comparing the poisson counterparts of quantum varieties and describe different approaches to their quantization also we exhibit two approaches to introducing q analogs of vector bundles and defining the chern connes index for them on quantum spheres in accordance with the serre swan approach the q vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras besides we describe the basic properties of the rea used in this construction and compare different ways of defining q analogs of partial derivatives and differentials on the rea and algebras close to them in particular we present a way of introducing a q differential calculus via koszul type complexes the elements of the q calculus are applied to defining q analogs of some relativistic wave operators

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

moduli of quantum riemannian geometries on 4 points we classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously s majid commun math phys 225 131 2002 the full moduli space is found for 3 points and a restricted moduli space for 4 points generalized levi civita connections and their curvatures are found for a variety of models including models of a discrete torus the topological part of the moduli space is found for 9 points based on the known atlas of regular graphs we also remark on aspects of quantum gravity in this approach

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

the full classification of orthogonal easy quantum groups we study easy quantum groups a combinatorial class of orthogonal quantum groups introduced by banica speicher in 2009 we show that there is a countable descending chain of easy quantum groups interpolating between bichon s free wreath product with the permutation group s n and a semi direct product of a permutation action of s n on a free product this reveals a series of new commutation relations interpolating between a free product construction and the tensor product furthermore we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi direct product quantum groups recently studied by the authors this completes the classification of easy quantum groups we also study combinatorial and operator algebraic aspects of the new interpolating series

simple compact quantum groups i the notion of simple compact quantum group is introduced as non trivial noncommutative and noncocommutative examples the following families of compact quantum groups are shown to be simple a the universal quantum groups bu q for q gl n c satisfying qq in n 2 b the quantum automorphism groups aaut b of finite dimensional c algebras b endowed with the canonical trace when dim b 4 including the quantum permutation groups aaut xn on n points n 4 c the standard deformations kq of simple compact lie groups k and their twists kqu as well as rieffel s deformation kj

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

characterization of su q l 1 equivariant spectral triples for the odd dimensional quantum spheres the quantum group suq l 1 has a canonical action on the odd dimensional sphere s 2l 1 q all odd spectral triples acting on the l2 space of s 2l 1 q and equivariant under this action have been characterized this characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial k homology class these generalize the results of chakraborty pal for suq 2 ams subject classification no 58b34 46l87 19k33

bicovariant differential calculi on glp q 2 and quantum subgroups in a previous paper all bicovariant differential calculi on the two parameter quantum group glp q 2 were determined in this work the authors elaborate some of their properties and discuss relations with work by other authors furthermore they show that there are two different ways in which bicovariant differential calculi on glq 2 where p q induce corresponding calculi on the quantum subgroup slq 2 it is shown that on slq 2 there are only two different bicovariant differential calculi from these one obtains the 4d or calculi on suq 2 the classical limit of the two differential calculi on slq 2 is investigated revealing a relation with recent work on deformed differential calculi on commutative function algebras

the boundary of universal discrete quantum groups exactness and factoriality we study the c algebras and von neumann algebras associated with the universal discrete quantum groups they give rise to full prime factors and simple exact c algebras the main tool in our work is the study of an amenable boundary action yielding the akemann ostrand property finally this boundary can be identified with the martin or the poisson boundary of a quantum random walk

braided lie algebras and bicovariant differential calculi over co quasitriangular hopf algebras we show that if g is the quantum tangent space or quantum lie algebra in the sense of woronowicz of a bicovariant first order differential calculus over a co quasitriangular hopf algebra a r then a certain extension of it is a braided lie algebra in the category of a comodules this is used to show that the woronowicz quantum universal enveloping algebra u g is a bialgebra in the braided category of a comodules we show that this algebra is quadratic when the calculus is inner examples with this unexpected property include finite groups and quantum groups with their standard differential calculi we also find a quantum lie functor for co quasitriangular hopf algebras which has properties analogous to the classical one this functor gives trivial results on standard quantum groups oq g but reasonable ones on examples closer to the classical case such as the cotriangular jordanian deformations in addition we show that split braided lie algebras define generalized lie algebras in a different sense of deforming the adjoint representation we construct these and their enveloping algebras for oq sl n recovering the witten algebra for n 2

braided version of shirshov witt theorem the shirshov witt theorem claims that every subalgebra of a free lie algebra is free in character istic zero this theorem can be restated in terms of a free associative algebra every hopf subalgebra of a free algebra kxiwith the coproduct xi xi 1 1 xi is free and it is freely generated by primitive elements our aim is to extend this result to free algebras with a braided coproduct as far as possible by means of p m cohn theory we show that if a subalgebra is a right categorical right coideal then it is free we consider more thoroughly involutive braidings 2 id over a field of zero characteristic in this case every braided hopf subalgebra is generated by primitive elements moreover the space of all primitive elements forms a free lie algebra in the context of this re sult we discuss the situation that arises around the problem of embedding of a lie algebra in its associative universal enveloping algebra 2005 elsevier inc all rights reserved

classification of bicovariant differential calculi on quantum groups a representation theoretic approach the restricted dual of a quantized enveloping algebra can be viewed as the algebra of functions on a quantum group according to woronowicz there is a general notion of bicovariant differential calculus on such an algebra we give a classification theorem of these calculi the proof uses the notion due to reshetikhin and semenov tian shansky of a factorizable quasi triangular hopf algebra and relies on results of joseph and letzter on the way we also give a new formula for rosso s bilinear form

quantum lie algebras their existence uniqueness and q antisymmetry quantum lie algebras are generalizations of lie algebras which have the quantum parameter h built into their structure they have been defined concretely as certain submodules of the quantized enveloping algebras on them the quantum lie product is given by the quantum adjoint action

numerical solution of sturm liouville problems via fer streamers we address the numerical challenge of solving regular sturm liouville problems in liouville s normal form with a continuous and piecewise analytic potential and self adjoint separated boundary conditions the novelty of our approach which is based on a non standard truncation of fer expansions which we call fer streamers lies in the construction of a new numerical method which 1 does not impose any restriction on the step size for eigenvalues which are greater than or equal to the minimum of the potential 2 requires only a mild restriction on the step size for the remaining finite number of eigenvalues 3 can attain any convergence rate which grows exponentially with the number of terms and is uniform for every eigenvalue and 4 lends itself to a clear understanding of the manner in which the potential affects the local and global errors we provide our numerical method with its analytical underpinning but emphasize that it is at an early stage of development and that much remains to be done in particular we comment on our investigation of efficient discretization schemes for the integrals which arise in fer streamers

noncommutative geometry of angular momentum space u su 2 we study the standard angular momentum algebra xi xj i eijkxk as a noncommutative manifold r 3 we show that there is a natural 4d differential calculus and obtain its cohomology and hodge operator we solve the spin 0 wave equation and some aspects of the maxwell or electromagnetic theory including solutions for a uniform electric current density and we find a natural dirac operator we embed r 3 inside a 4d noncommutative space time which is the limit q 1 of q minkowski space and show that r 3 has a natural quantum isometry group given by the quantum double c su 2 u su 2 which is a singular limit of the q lorentz group we view r 3 as a collection of all fuzzy spheres taken together we also analyze the semiclassical limit via minimum uncertainty states j approximating classical positions in polar coordinates

connected components of compact matrix quantum groups and finiteness conditions abstract we introduce the notion of identity component of a compact quantum group and that of total disconnectedness as a drawback of the generalized burnside problem we note that totally disconnected compact matrix quantum groups may fail to be profinite we consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups these sequences have lengths 1 in the classical case but can be countably infinite for duals of discrete groups we give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1 we give necessary and sufficient conditions for normality of the identity component in the sense of wang and finiteness or profiniteness of the quantum component group among them we introduce an ascending chain condition on the representation ring called lie property which characterizes lie groups in the classical case and reduces to group noetherianity of the discrete group dual in the cocommutative case it is weaker than ring noetherianity but ensures existence of a generating representation the lie property and ring noetherianity are inherited by quotient quantum groups we show that a u f is not of lie type we discuss an example arising from the compact real form of u q sl 2 for q 0

classification of bicovariant differential calculi on quantum groups of type a b c and d under the assumptions thatq is not a root of unity and that the differentialsduji of the matrix entries span the left module of first order forms we classify bicovariant differential calculi on quantum groupsan 1 bn cn anddn we prove that apart one dimensional differential calculi and from finitely many values ofq there are precisely2n such calculi on the quantum groupan 1 slq n forn 3 all these calculi have the dimensionn2 for the quantum groupsbn cn anddn we show that except for finitely manyq there exist precisely twon2 dimensional bicovariant calculi forn 3 wheren 2n 1 forbn andn 2n forcn dn the structure of these calculi is explicitly described and the corresponding ad invariant right ideals of ker e are determined in the limitq 1 two of the 2n calculi foran 1 and one of the two calculi forbn cn anddn contain the ordinary classical differential calculus on the corresponding lie group as a quotient

quantum lie algebras associated to and quantum lie algebras are non associative algebras which are embedded into the quantized enveloping algebras of drinfeld and jimbo in the same way as ordinary lie algebras are embedded into their enveloping algebras the quantum lie product on is induced by the quantum adjoint action of we construct the quantum lie algebras associated to and we determine the structure constants and the quantum root systems which are now functions of the quantum parameter q they exhibit an interesting duality symmetry under

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

least squares ridge regression and goal programming constrained regression alternatives abstract ridge regression has been extensively reviewed in the literature of applied statistics as a method for dealing with problems of collinearity in least squares regressions in this paper ordinary least squares and least squares combined with ridge regressions are examined and compared with ordinary goal programming and goal programming combined with constraints on regressand values that admissible statistical estimates must satisfy the comparisons are effected by means of an example in chemical processing that has been extensively treated in the ridge regression literature

a note on the generalized lie algebra in a recent paper dobrev and sudbery classified the highest weight and lowest weight finite dimensional irreducible representations of the quantum lie algebra introduced by lyubashenko and sudbery the aim of this note is to add to this classification all the finite dimensional irreducible representations which have no highest weight and or no lowest weight in the case when q is a root of unity for this purpose we give a description of the enlarged centre

the double exponential sinc collocation method for singular sturm liouville problems sturm liouville problems are abundant in the numerical treatment of scientific and engineering problems in the present contribution we present an efficient and highly accurate method for computing eigenvalues of singular sturm liouville boundary value problems the proposed method uses the double exponential formula coupled with sinc collocation method this method produces a symmetric positive definite generalized eigenvalue system and has exponential convergence rate numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula

non commutative geometry of finite groups a finite set can be supplied with a group structure which can then be used to select classes of differential calculi on it via the notions of left right and bicovariance a corresponding framework has been developed by woronowicz more generally for hopf algebras including quantum groups a differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and moreover for the formulation of field theories and dynamics on finite sets associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures for a covariant calculus there are notions of invariance for linear connections and tensors all these concepts are explored for finite groups and illustrated with examples some results are formulated more generally for arbitrary associative hopf algebras in particular the problem of extension of a connection on a bimodule over an associative algebra to tensor products is investigated leading to the class of extensible connections it is shown that invariance properties of an extensible connection on a bimodule over a hopf algebra are carried over to the extension furthermore an invariance property of a connection is also shared by a dual connection which exists on the dual bimodule as defined in this work

woronowicz construction of compact quantum groups for functions on permutations classification result for n 3 abstract we provide a classification of compact quantum groups which can be obtained by the woronowicz construction when the arrays used in the twisted determinant condition are extensions of functions on permutations general properties of such quantum groups are revealed with the aid of operators intertwining tensor powers of the fundamental corepresentation two new families of quantum groups appear three versions of the quantum group su q 3 with complex q and a non commutative analogue of the semi direct product of two dimensional torus with the alternating group a 3

representations of the generalized lie algebra we construct finite dimensional irreducible representations of two quantum algebras related to the generalized lie algebra introduced by lyubashenko and sudbery we consider separately the cases of q generic and q at roots of unity some of the representations have no classical analogue even for generic q some of the representations have no analogue to the finite dimensional representations of the quantized enveloping while in those that do there are different matrix elements

a corrected spectral method for sturm liouville problems with unbounded potential at one endpoint abstract in this paper we shall derive a spectral matrix method for the approximation of the eigenvalues of weakly regular and singular sturm liouville problems in normal form with an unbounded potential at the left endpoint the method is obtained by using a galerkin approach with an approximation of the eigenfunctions given by suitable combinations of legendre polynomials we will study the errors in the eigenvalue estimates for problems with unsmooth eigenfunctions in proximity of the left endpoint the results of this analysis will be then used conveniently to determine low cost and effective procedures for the computation of corrected numerical eigenvalues finally we shall present and discuss the results of several numerical experiments which confirm the effectiveness of the approach

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

dealing with the multiplicity of solutions of the l1 and l regression models abstract it is well known that the least absolute value l and the least sum of absolute deviations l1 algorithms produce estimators that are not necessarily unique in this paper it is shown how the set of all solutions of the l1 and l regression problems for moderately large sample sizes can be obtained in addition if the multiplicity of solutions wants to be avoided two new methods giving the same optimal l1 and l values but supplying unique solutions are proposed the idea consists of using two steps in the first step the optimal values of the l1 and l errors are calculated and in the second step in case of non uniqueness of solutions one of the multiple solutions is selected according to a different criterion for the l the procedure is used sequentially but removing in each iteration the data points with maximum absolute residual and adding the corresponding constraints for keeping these residuals and this process is repeated until no change in the solution is obtained in this way not only the maximum absolute residual values are minimized in the modified method but also the maximum absolute residual values of the remaining points sequentially until no further improvement is possible in the l1 case a least squares criterion is used but restricted to the l1 residual condition thus in the modified l1 method not only the l1 residual is minimized but also the sum of squared residuals subject to the l1 residual the methods are illustrated by their application to some well known examples and their performances are tested by some simulations which show that the lack of uniqueness problem cannot be corrected for some experimental designs by increasing the sample size

generalized lie algebras of type an it is shown that the quantized enveloping algebra of sl n contains a generalized lie algebra defined by means of axioms similar to woronowicz s this gives rise to lie algebra like generators and relations for the locally finite part of the quantized enveloping algebra and suggests a canonical poincare birkhoff witt basis

quantum koszul complexes for majid s braided lie algebras a yang baxter operator was canonically associated to any cocommutative braided lie algebra as defined by majid applying previous results we construct quantum koszul complexes for braided lie algebras with respect to this operator in the usual case the complexes reduce to the chevalley eilenberg complexes for classical lie algebras

differential hopf algebras on quantum groups of type a abstract let a be a hopf algebra and be a bicovariant first order differential calculus over a it is known that there are three possibilities to construct a differential hopf algebra jthat contains as its first order part corresponding to the three choices of the idealj we distinguish the universal exterior algebra the second antisymmetrizer exterior algebra and woronowicz external algebra respectively let be one of then2 dimensional bicovariant first order differential calculi on the quantum groupglq n orslq n and letqbe a transcendental complex number for woronowicz external algebra we determine the dimension of the space of left invariant and of bi invariantk forms respectively bi invariant forms are closed and represent different de rham cohomology classes the algebra of bi invariant forms is graded anti commutative forn 3 the three differential hopf algebras coincide however in case of the 4d calculi onslq 2 the universal differential hopf algebra is strictly larger than woronowicz external algebra the bi invariant 1 form is not closed

the maxwell operator on q minkowski space and q hyperboloid algebras we introduce an analog of the maxwell operator on a q minkowski space algebra treated as a particular case of the so called reflection equation algebra and on certain of its quotients we treat the space of quantum differential forms as a projective module in the spirit of the serre swan approach also we use braided tangent vector fields which are q analogs of poisson vector fields associated with the lie bracket sl 2

galerkin schemes and the sinc galerkin method for singular sturm liouville problems abstract galerkin schemes for the computation of the eigenvalues of both regular and singular sturm liouville problems are compared to the sine galerkin method contrasted are numerical results as well as qualitative features the schemes discussed include finite element spectral and collocation the equivalence of the spectral method with sine basis functions to the sinc collocation scheme provides this technique with many distinctive features highlighted among these features through the course of the computational comparisons is its ease of implementation its exponential accuracy in the presence of singularities and its all around versatility

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

complex quantum group dual algebra and bicovariant differential calculus the method used to construct the bicovariant bimodule in ref csww is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group the complex quantum group fun q sl n c is defined by requiring that it contains fun q su n as a subalgebra analogously to the quantum lorentz group analyzing the properties of the fundamental bimodule we show that the dual algebra has the structure of the twisted product fun q su n fun q su n reg then the bicovariant differential calculi on the complex quantum group are constructed

bicovariant differential calculus on quantum groups su q n and so q n following woronowicz s proposal the bicovariant differential calculus on the quantum groupssu q n andso q n is constructed a systematic construction of bicovariant bimodules by using the hat r q matrix is presented the relation between the hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules and theq deformed universal enveloping algebras is given imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined as an application the maurer cartan equations and theq analogue of the structure constants are formulated

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

on quantum lie algebras and quantum root systems as a natural generalization of ordinary lie algebras we introduce the concept of quantum lie algebras we define these in terms of certain adjoint submodules of quantized enveloping algebras endowed with a quantum lie bracket given by the quantum adjoint action the structure constants of these algebras depend on the quantum deformation parameter q and they go over into the usual lie algebras when q 1 the notions of q conjugation and q linearity are introduced q linear analogues of the classical antipode and cartan involution are defined and a generalized killing form q linear in the first entry and linear in the second is obtained these structures allow the derivation of symmetries between the structure constants of quantum lie algebras the explicitly worked out examples of and illustrate the results

coherent states for quantum compact groups coherent states are introduced and their properties are discussed for simple quantum compact groupsal bl cl anddl the multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit the coherent state is interpreted as a holomorphic function on this orbit with values in the carrier hilbert space of an irreducible representation of the corresponding quantized enveloping algebra using gauss decomposition the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compactr matrix formulation generalizing this way theq deformed grassmann and flag manifolds the antiholomorphic realization of the irreducible representations of a compact quantum group the analogue of the borel weil construction is described using the concept of coherent state the relation between representation theory and non commutative differential geometry is suggested

a classification of unitary highest weight modules of the quantum analogue of the symmetric pair an an 1 abstract we study the irreducibility and unitarity of highest weight modules of the quantized enveloping algebra associated to s u n 1 we obtain a classification of these modules analogous to that of ehw in the classical context we also compute all commutation relations between root vectors in u q s u n 1

classifying spaces for quantum principal bundles a construction of a quantum analogue of principal bundles is discussed deformations of quantum groups in the sense of woronowicz allow to relax the condition of local triviality of a principal bundle the fibres need not be all identical any longer this leads to deformations of structure group and bundles there is still a classifying space in the sense that homotopy classes of bundles are classified by homotopy classes of maps from the base space into the classifying space

lie theory and coverings of finite groups we introduce the notion of an inverse property ip quandlec which we propose as the right notion of lie algebra in the category of sets to any ip quandle we construct an associated group gc for a class of ip quandles which we call locally skew and when gc is nite we show that the noncommutative de rham cohomology h 1 gc is trivial aside from a single generator that has no classical analogue if we start with a group g then any subsetc gnfeg which is ad stable and inversion stable naturally has the structure of an ip quandle ifc also generates g then we show that gc g with central kernel in analogy with the similar result for the simply connected covering group of a lie group we prove that this covering map gc g is an isomorphism for all nite crystallographic reection groups w withc the set of reections and that c is locally skew precisely in the simply laced case this implies that h 1 w k when w is simply laced proving in particular a conjecture for sn in 11 we obtain similar results for the dihedral groups d6m we also consider c zp 1 zp 1 as a locally skew ip quandle lie algebra of sl2 z and show that gc b3 the braid group on 3 strands the map b3 sl2 z which therefore arises naturally as a covering map in our theory coincides with the restriction of the usual universal covering map sl2 r sl2 r to the inverse image of sl2 z

a poincare birkhoff witt theorem for generalized lie color algebras a proof of the poincare birkhoff witt theorem is given for a class of generalized lie algebras closely related to the gurevich s lie algebras as concrete examples we construct the positive negative parts of the quantized universal enveloping algebras of type an and mp q e n k which is a nonstandard quantum deformation of gl n in particular we get for both algebras a unified proof of the poincare birkhoff witt theorem and we show that they are genuine universal enveloping algebras of certain generalized lie algebras

some aspects of braided geometry differential calculus tangent space gauge theory a new approach is suggested to quantum differential calculus on certain quantum varieties it consists in replacing quantum de rham complexes with differentials satisfying leibniz rule by those which are in a sense close to koszul complexes from gurevich 1991 leningrad math j 2 801 28 we also introduce the tangent space on a quantum hyperboloid equipped with an action on the quantum function space and define the notions of quantum pseudo metric and quantum connection partially defined on it all objects are considered from the viewpoint of flatness of quantum deformations the problem of constructing a flatly deformed quantum gauge theory is discussed as well

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

on statistical data analysis based on the l1 norm and related methods an introduction to statistical data analysis l 1 norm based y dodge the place of the l 1 norm in robust estimation p j huber i historical development the historical development of the l 1 and l estimation procedures r w farebrother bounded influence inference in regression a review e ronchetti ii computational algorithms and computer packages algorithms for unconstrained l 1 linear regression j e gentle s c narula v a sposito the reduced gradient algorithm m r osborne the l 1 estimate as limiting case of an l p or huber estimate h ekblom a review of computational methods for solving the nonlinear l 1 norm estimation problem r gonin a h money fitting data through homotopy methods j p schellhorn blinwdr a fortran program for robust and bounded influence regression r dutter solving bounded influence regression problems with robsys a marazzi xplore a computing environment for exploratory regression w hardle iii estimation characterization properties and selection of variables l 1 distances in statistical inference comparison of topological functional and statistical properties i vajda the role played by l 1 in data analysis b fichet l 1 embeddings of a data structure i d g le calve assessing the accuracy of the sample median estimated standard errors vs interpolated confidence intervals s j sheather the median of a finite measure on a banach space j h b kemperman variable selection in linear model based on trimmed least squares estimator j antoch numerical techniques for finding estimates which minimize the upper bound of the absolute deviation a gaivoronski asymptotic properties of restricted l 1 estimates of regression j dupacova adaptive combination of least squares and least absolute deviations estimators y dodge j jureckova iv statistical inference procedures a comparison of asymptotic testing methods for l 1 regression r koenker least absolute errors analysis of variance j w mckean r m schrader small sample properties of least absolute errors analysis of variance r m schrader j w mckean bootstrap and inference procedures for l 1 regression g stangenhaus invariant tests in bivariate models and the l 1 criterion b m brown t p hettmansperger using regression fractiles to identify outliers s portnoy various discrepancy measures in model testing two competing regression models v v fedorov v nonparametric analysis kernel estimates of the sparsity function a h welsh what does optimal bandwidth selection mean for nonparametric regression estimation j s marron density estimation from dependent sample l gyorfi vi cluster analysis clustering by means of medoids l kaufman p j rousseeuw l 1 in fuzzy clustering e trauwaert using the l 1 norm within cluster analysis h spath vii applications an application of l 1 to astronomy p j rousseeuw

quantization of lie group and algebra of g2 type in the faddeev reshetikhin takhtajan approach based on the quantized universal enveloping que algebras a quantization of the automorphism group of some nonassociative algebras is given in the formulation employing noncommuting matrix entries a quantum group of g2 type included in this scheme is studied in detail in the faddeev reshetikhin takhtajan frt approach also in the formulation employing noncommuting matrix entries the que algebra of g2 type is reconstructed through the pairing induced by the r matrix between the quantum group and the que algebra