information theoretic lengths of jacobi polynomials the information theoretic lengths of the jacobi polynomials p n x which are information theoretic measures renyi shannon and fisher of their associated rakhmanov probability density are investigated they quantify the spreading of the polynomials along the orthogonality interval 1 1 in a complementary but different way as the root mean square or standard deviation because contrary to this measure they do not refer to any specific point of the interval the explicit expressions of the fisher length are given the renyi lengths are found by the use of the combinatorial multivariable bell polynomials in terms of the polynomial degree n and the parameters the shannon length which cannot be exactly calculated because of its logarithmic functional form is bounded from below by using sharp upper bounds to general densities on 1 1 given in terms of various expectation values moreover its asymptotics is also pointed out finally several computational issues relative to these three quantities are carefully analyzed

representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

class groups of number fields numerical heuristics extending previous work of h w lenstra jr and the first author we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four given the signature and the galois group of the galois closure the theoretical justifications for these conjectures will appear elsewhere but the agreement with the existing tables is quite good 1 introduction and notations in 3 h w lenstra jr and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree in a forthcoming paper 4 we shall show that this technique can be extended to a much wider class of number fields and also to relative extensions the aim of the present paper is to rapidly make available the numerical conjec tures obtained for people not really interested in our heuristic reasoning or not wanting to wait for 4 to appear hence apart from a total lack of justifications for the conjectures that we present this paper is essentially self contained the plan is as follows in the rest of this section we present the notations used in the sequel some of them being nonstandard and in general differing from the notations of 3 we urge the reader to read the notations carefully before applying the conjectures in the next section we present templates for the subsequent conjectures and then the conjectures themselves illustrated by numerical examples first for their own sake and second as a double check for the reader to understand the templates these conjectures are given for all types of fields of degree less than or equal to four in the final section we comment on the consistency of the conjectures with existing tables which is quite good combinatorial notations if x is a set xi denotes its cardinality

multiparametric quantum gl 2 lie bialgebras quantum r matrices and non relativistic limits multiparametric quantum deformations of gl 2 are studied through a complete classification of gl 2 lie bialgebra structures from these the non relativistic limit leading to harmonic oscillator lie bialgebras is implemented by means of a contraction procedure new quantum deformations of gl 2 together with their associated quantum r matrices are obtained and other known quantizations are recovered and classified several connections with integrable models are outlined

some properties of convolution operators in the class pα β abstract making use of several families of convolution operators we introduce and study a certain general class p 0 u we also investigate the relationships between the class p and the hardy space h of bounded analytic functions in u finally we consider some interesting applications of the results presented here to a class of generalized hypergeometric functions

symplecton for uh sl 2 and representations of slh 2 polynomials of boson creation and annihilation operators which form irreducible tensor operators for jordanian quantum algebra uh sl 2 called h symplecton are introduced and their properties are investigated it is shown that many properties of symplecton for lie algebra sl 2 are extended to h symplecton the h symplecton is also a basis of irreducible representation of slh 2 dual to uh sl 2 as an application of the procedure used to construct h symplecton we construct the representation bases of slh 2 on the quantum h plane

a certain one parameter additive family of operators defined on analytic functions abstract y komatu proved several interesting results for a certain one parameter additive family of operators defined on analytic functions regular in the unit disk and presented two conjectures for univalent functions in the unit disk it is the purpose of this paper to prove a number of results relevant to komatu s conjectures for certain classes of univalent functions with negative coefficients in the unit disk

some properties of α convex and α quasiconvex functions with respect to n symmetric points the main object of the present paper is to investigate convex and quasiconvex functions with respect to n symmetric points some interesting properties such as inclusion relationships integral representations convolution conditions are obtained

some properties of a certain subclass of multivalent analytic functions involving the liu owa operator by making use of the principle of subordination connecting analytic functions and the liu owa operator we introduce a certain subclass of multivalent analytic functions such results as subordination and superordination properties convolution properties distortion theorems inequality properties and sufficient conditions for multivalent starlikeness are proved

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

universal t matrix representations of ospq 1 2 and little q jacobi polynomials we obtain a closed form expression of the universal t matrix encapsulating the duality between the quantum superalgebra uq osp 1 2 and the corresponding supergroup ospq 1 2 the classical q 1 limit of this universal t matrix yields the group element of the undeformed ospq 1 2 supergroup the finite dimensional representations of the quantum supergroup ospq 1 2 are readily constructed employing the above mentioned universal t matrix and the known finite dimensional representations of the dually related deformed uq osp 1 2 superalgebra proceeding further we derive the product law the recurrence relations and the orthogonality of the representations of the quantum supergroup ospq 1 2 it is shown that the entries of these representation matrices are expressed in terms of the little q jacobi polynomials with q q two mutually complementary singular maps of the universal t matrix on the universal r matrix are also presented

the algebra of weyl symmetrised polynomials and its quantum extension the algebra of weyl symmetrised polynomials in powers of hamiltonian operatorsp andq which satisfy canonical commutation relations is constructed this algebra is shown to encompass all recent infinite dimensional algebras acting on two dimensional phase space in particular the moyal bracket algebra and the poisson bracket algebra of which the moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra we propose the introduction of a second deformation by the replacement of the heisenberg algebra forp q with aq deformed commutator and construct algebras ofq symmetrised polynomials

duality for exotic bialgebras in the classification of hietarinta three triangular 4 4 r matrices lead via the frt formalism to matrix bialgebras which are not deformations of the trivial one in this paper we find the bialgebras which are in duality with these three exotic matrix bialgebras we note that the l t duality of frt is not sufficient for the construction of the bialgebras in duality we find also the quantum planes corresponding to these bialgebras both by the wess zumino r matrix method and by manin s method

on real cyclic sextic fields a table of units and class numbers of real cyclic sextic fields with conductor 16 1 leopoldt s unit index and the signature rank of the unit group

some functional inequalities and inclusion relationships associated with certain families of integral operators by making use of certain familiar integral operators the authors introduce and investigate several new subclasses of starlike convex close to convex and quasi convex functions among other results presented here the authors establish a number of inclusion relationships associated with some of these integral operators some of the results established in this paper would provide extensions of those given in earlier works

duality for the matrix quantum group glp q 2 c it is shown that the algebra up q dual to glp q 2 c is isomorphic to u pq 1 2 sl 2 c z as a commutation algebra where z is a subalgebra central in up q the subalgebra z is a hopf subalgebra of up q while the commutation subalgebra u pq 1 2 sl 2 c is not a hopf subalgebra

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

some inclusion relationships and integral preserving properties of certain subclasses of meromorphic functions associated with a family of integral operators the object of the present paper is to investigate a family of integral operators defined on the space of normalized meromorphic functions by making use of these novel integral operators we introduce and investigate several new subclasses of starlike convex close to convex and quasi convex meromorphic functions in particular we establish some inclusion relationships associated with the aforementioned integral operators several interesting integral preserving properties are also considered

the hardy space of analytic functions associated with certain one parameter families of integral operators abstract recently h m srivastava and s owa j math anal appl 118 1986 80 87 proved a number of results relevant to komatu s conjectures cf y komatu bull fac sci engrg chuo univ ser i math 22 1979 1 22 for a certain one parameter additive family of operators defined on analytic functions regular in the open unit disk the object of the present paper is to investigate several one parameter families of integral operators and to prove certain inclusion theorems involving the hardy space h p of analytic functions and the class of functions whose derivative has a positive real part an application involving a generalized hypergeometric function is also considered

braided chains of q deformed heisenberg algebras given m copies of a q deformed weyl or clifford algebra in the defining representation of a quantum group we determine a prescription to embed them into a unique inclusive covariant algebra the different copies are coupled to each other and are naturally ordered into a chain in the case a modified prescription yields an inclusive algebra which is even explicitly covariant where is a symmetry relating the different copies by the introduction of these inclusive algebras we significantly enlarge the class of covariant deformed weyl clifford algebras available for physical applications

on clebsch gordan coefficients and matrix elements of representations of the quantum algebra uq su2 clebsch gordan coefficients and matrix elements of irreducible representations of the quantum algebra uq su2 were considered in several papers in particular a few expressions for them were derived an approach to clebsch gordan coefficients and to matrix elements of representations of uq su2 on the base of the theory of basic hypergeometric functions is given this approach allows one to obtain q analogs of all well known classical expressions for clebsch gordan coefficients most of them were absent new symmetry relations generating functions and recurrence formulas for clebsch gordan coefficients of uq su2 are obtained unlike other papers clebsch gordan coefficients and matrix elements are considered on the base of minimal theoretical constructions in fact without using the notion of a c algebra and of a hopf algebra

1 1 schrodinger lie bialgebras and their poisson lie groups all lie bialgebra structures for the 1 1 dimensional centrally extended schrodinger algebra are explicitly derived and proved to be of coboundary type therefore since all of them come from a classical r matrix the complete family of schrodinger poisson lie groups can be deduced by means of the sklyanin bracket all possible embeddings of the harmonic oscillator extended galilei and gl 2 lie bialgebras within the schrodinger classification are studied as an application new quantum hopf algebra deformations of the schrodinger algebra including their corresponding quantum universal r matrices are constructed

computing all power integral bases in orders of totally real cyclic sextic number fields an algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields the orders considered are in most cases the maximal orders of the fields the corresponding index form equation is reduced to a relative thue equation of degree 3 over the quadratic subfield and to some inhomogeneous thue equations of degree 3 over the rationals at the end of the paper numerical examples are given

the quantum group glh 2 among all quantum group structures on the space mat 2 of 2 by 2 matrices how many have a central quantum determinant so that one can define quantum sl 2 out of quantum gl 2 up to isomorphism there are two such structures glq 2 and glh 2 the former is well known the latter is described in this letter

deforming maps for lie group covariant creation and annihilation operators any deformation of a weyl or clifford algebra a can be realized through a deforming map i e a formal change of generators in a this is true in particular if a is covariant under a lie algebra g and its deformation is induced by some triangular deformation uhg of the hopf algebra ug we propose a systematic method to construct all the corresponding deforming maps together with the corresponding realizations of the action of uhg the method is then generalized and explicitly applied to the case that uhg is the quantum group uhsl 2 a preliminary study of the status of deforming maps at the representation level shows in particular that deformed fock representations induced by a compact uhg can be interpreted as standard undeformed fock representations describing particles with ordinary bose or fermi statistics

on cyclic cubic fields a table of class numbers and units in cyclic cubic fields with conductor o for3 f 2 a 6 mod9 b 3 or 6 mod9 b o for 3 if introduced by hasse 8 p 12 let 6 6 6 denote the gaussian periods for a generating cubic character of k taken with a suitable sign then k q o and see

twist deformation of the rank one lie superalgebra the drinfeld twist is applied to deform the rank one orthosymplectic lie superalgebra the twist element is the same as for the sl 2 lie algebra due to the embedding of the sl 2 into the superalgebra the r matrix has the direct sum structure in the irreducible representations of the dual quantum group is defined using the frt formalism it includes the jordanian quantum group as subalgebra and grassmann generators as well

non standard quantum so 2 2 and beyond a new non standard quantization of the universal enveloping algebra of the split natural real form so 2 2 of d2 is presented some classical graded contractions of so 2 2 associated to a z2 z2 grading are studied and the automorphisms defining this grading are generalized to the quantum case thus providing quantum contractions of this algebra this produces a new family of non standard quantum algebras some of these algebras can be realized as 2 1 kinematical algebras we explicitly introduce a new deformation of poincare algebra which is naturally linked to the null plane basis another realization of these quantum algebras as deformations of the conformal algebras for the two dimensional euclidean galilei and minkowski spaces is given and its new properties are emphasized

on the q symplecton realization of the quantum group suq 2 a q symplecton realization of the quantum group suq 2 is constructed giving explicitly the product law for the weyl ordered characteristic polynomials a new form of the q racah coefficients is obtained which is especially adapted to proving a new symmetry for these functions a surprising result is that unlike the su 2 case there are two distinct invariant q triangle functions in the q symplecton extension

on combined standard nonstandard or hybrid q h deformations combined q h deformations proposed by kupershmidt and ballesteros herranz parashar are studied in each case a transformation is shown to lead to an equivalent standard q deformation we briefly indicate that appropriate singular limits of the same type of transformations can however lead from standard biparametric p q deformations to nonhybrid but biparametric nonstandard g h ones finally a case of hybrid q h deformation is recalled related to the superalgebra gl 1 1

representations and clebsch gordan coefficients for the jordanian quantum algebra representation theory for the jordanian quantum algebra is developed closed form expressions are given for the action of the generators of on the basis vectors of finite dimensional irreducible representations it is shown how representation theory of has a close connection to the combinatorial identities involving summation formulae a general formula is obtained for the clebsch gordan coefficients of these clebsch gordan coefficients are shown to coincide with those of su 2 for but for they are in general a non zero monomial in

some preserving subordination and superordination of analytic functions involving the liu owa integral operator in this paper we obtain some subordination and superordination preserving results of analytic functions involving the liu owa integral operator sandwich type result is also obtained

representation functions for jordanian quantum group slh 2 and jacobi polynomials the explicit expressions of the representation functions d functions for jordanian quantum group slh 2 are obtained by combination of tensor operator technique and drinfeld twist it is shown that the d functions can be expressed in terms of jacobi polynomials as the undeformed d functions can some of the important properties of the d functions for slh 2 such as wigner s product law recurrence relations and rtt type relations are also presented

notes on jung kim srivastava integral operator abstract the object of the present paper is to investigate some properties of certain integral operator q introduced and studied recently by jung kim and srivastava j math anal appl 176 1993 138 147

finite dimensional representations of the quantum group glp q 2 using the exponential map from up q gl 2 using the fronsdal galindo formula for the exponential mapping from the quantum algebra up q gl 2 to the quantum group glp q 2 we show how the 2j 1 dimensional representations of glp q 2 can be obtained by exponentiating the well known 2j 1 dimensional representations of up q gl 2 for j 1 3 2 j 1 2 corresponds to the defining two dimensional t matrix the earlier results on the finite dimensional representations of glq 2 and slq 2 or suq 2 are obtained when p q representations of uq q 2 q in c mod r and uq 2 q in r mod 0 are also considered the structure of the clebsch gordan matrix for up q gl 2 is studied the same clebsch gordan coefficients are applicable in the reduction of the direct product representations of the quantum group glp q 2

exponential mapping for non semisimple quantum groups the concept of a universal t matrix introduced by fronsdal and galindo 1993 in the framework of quantum groups is discussed here as a generalization of the exponential mapping new examples related to inhomogeneous quantum groups of physical interest are developed the duality calculations are explicitly presented and it is found that in some cases the universal t matrix as for lie groups is expressed in terms of usual exponential series

duality for multiparametric quantum gl n we show that the hopf algebra uuq dual to the multiparameter matrix quantum group gluq n may be realized a la sudbery 1990 i e tangent vectors at the identity furthermore we give the cartan weyl basis of uuq and show that this is consistent with the duality we show that as a commutation algebra uuq equivalent to uu sl n c x uu z where z is one dimensional and uu z is a central algebra in uuq however as a co algebra uuq cannot be split in this way and depends on all parameters

irreducible decomposition for tensor product representations of jordanian quantum algebras tensor products of irreducible representations of the jordanian quantum algebras and are considered for both the highest weight finite dimensional representations of and the lowest weight infinite dimensional ones of it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding lie algebras

duality for the jordanian matrix quantum group we find the hopf algebra dual to the jordanian matrix quantum group as an algebra it depends only on the sum of the two parameters and is split into two subalgebras with three generators and with one generator the subalgebra is a central hopf subalgebra of the subalgebra is not a hopf subalgebra and its co algebra structure depends on both parameters we discuss also two one parameter special cases g h and g h the subalgebra is a hopf algebra and coincides with the algebra introduced by ohn as the dual of the subalgebra is isomorphic to u sl 2 as an algebra but has a nontrivial co algebra structure and again is not a hopf subalgebra of

tensor operators for quantum groups and applications tensor operators are defined for hopf algebras if the hopf algebra is quasitriangular the properly defined tensor product of tensor operators is a tensor operator in one application the problem of defining invariant correlation functions for the uq sl 2 symmetric spin 1 2 heisenberg chain is considered in another application it is shown that suitably modified bosonic and fermionic q creation and annihilation operators are tensor operators of uq sl n

universal r matrix for non standard quantum a universal r matrix for the non standard jordanian quantum deformation of is presented a family of solutions of the quantum yang baxter equation is obtained from some finite dimensional representations of this lie bialgebra quantization of