from canonical tensor operators of su 3 and 3 to bi orthogonal coupling coefficients explicit expansion expansion of the matrix elements of su 3 and canonical tensor operators in terms of the bi orthogonal coupling coefficients and overlaps of the draayer akiyama construction are considered special bi orthogonal extremal isoscalar factors with subscripts as multiplicity labels and proportional to q racah q 6j coefficients or the generalized wilson rahman rational bi orthogonal functions in terms of balanced or hypergeometric series are used as a natural basis for extremal matrix elements of the highest weight component of the canonical tensor operators of su 3 in the generating function approach of biedenharn lohe and louck the expansion that is obtained triple sum together with previously derived asymmetric seed isofactors and elementary overlaps gives the explicit overlap coefficients and can be used to derive su 3 and canonical tensor operators as well as new explicit normalized seed isofactors with regge type symmetry specified for the minimal null space case

transformation formula for a double clausenian hypergeometric series its q analogue and its invariance group a transformation formula for a double basic hypergeometric series of type 1 2 20 2 2 is derived this transformation yields a double series analogue of sears transformation for a terminating 3 2 series in the limit q 1 the formula reduces to a transformation for a terminating double clausenian hypergeometric series of unit argument one of the proper kampe de feriet series f1 2 20 2 2 1 1 this formula is a double series analogue of whipple s terminating 3f2 transformation this transformation gives rise to a transformation group the invariance group acting on the parameters of the double series the invariance group is examined and shown to be a subgroup of a double copy of the symmetries of the square

picard fuchs equations for relative periods and abel jacobi map for calabi yau hypersurfaces we study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it we construct an inhomogeneous picard fuchs equation by applying a picard fuchs operator to the holomorphic top form on a calabi yau hypersurface in toric variety and deriving a general formula for the d exact form on one side of the equation we also derive a double residue formula giving a purely algebraic way to compute the inhomogeneous picard fuchs equations for the abel jacobi map which has played an important role in the recent study of d branes by morrison and walcher using the variation formalism we prove that the relative periods of toric b branes on calabi yau hypersurface satisfy the enhanced gkz hypergeometric system proposed in physics literature by alim hecht mayr and mertens and discuss the relations between several works in the recent study of open string mirror symmetry we also give the general solutions to the enhanced hypergeometric system

nonlinear oscillations in the rolling motion of euler s disk the fixed points of the dynamical system describing the rolling motion of a uniform disk or a uniform circular hoop on a rough horizontal plane without dissipation are analyzed all fixed points of the system are stated to be steady motion states which are either saddle points or centers the spectral analysis of the sound amplitude time series emitted by a toy of the rolling disk shows that the similar nonlinear oscillations are expected in the motion with dissipation if under adiabatic approximation a finite time singularity occurs the bifurcation in the disk motion can appear

effective superpotentials for compact d5 brane calabi yau geometries for compact calabi yau geometries with d5 branes we study n 1 effective superpotentials depending on both open and closed string fields we develop methods to derive the open closed picard fuchs differential equations which control d5 brane deformations as well as complex structure deformations of the compact calabi yau space their solutions encode the flat open closed coordinates and the effective superpotential for two explicit examples of compact d5 brane calabi yau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open closed coordinates by evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature finally we extract orbifold disk invariants from the superpotentials which up to overall numerical normalizations correspond to orbifold disk gromov witten invariants in the mirror geometry

towards the canonical tensor operators of uq 3 ii the denominator function problem the explicit denominator normalization function of the canonical tensor operators of the quantum algebra uq 3 corresponding to the maximal null space case is derived ab initio in terms of double basic hypergeometric series which cannot be obtained as any q extension of the su 3 denominator polynomial gb 1 x in terms of multiple double or triple balanced hypergeometric series introduced by biedenharn louck and their collaborators although their q 1 versions are shown being equivalent the corresponding orthonormal seed isoscalar factors of the coupling wigner clebsch gordan coefficients of uq 3 and su 3 with multiple irreducible representations are presented conjectured expression of the q polynomials which ratios appear in the uq 3 and new su 3 denominator functions for an arbitrary value of the canonical multiplicity label t of the repeating irreducible representations in terms of multiple partition dependent q series extension of the maximal and minimal null space versions i

on the denominator function for canonical su 3 tensor operators the definition of a canonical unit su 3 tensor operator is given in terms of its characteristic null space as determined by group theoretic properties of the intertwining number this definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique up to phases construction of all su 3 wcg coefficients wigner clebsch gordan using this construction an explicit su 3 invariant denominator function characterizing completely the canonically defined wcg coefficients is obtained it is shown that this denominator function squared is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials these polynomials denoted gtq are defined over three shift parameters and three barycentric coordinates the properties of these polynomials hence of the corresponding invariant denominator function are developed in detail these include a derivation of their degree symmetries and zeros the

transformation and summation formulas for double hypergeometric series abstract a number of new transformation formulas for double hypergeometric series are presented the series appearing here are the so called kampe de feriet functions of type f1 1 20 3 4 1 1 and f0 2 21 2 2 1 1 the transformation formulas relate such double series to a single hypergeometric series of 4f3 1 type by specializing certain parameters a list of new summation formulas for f0 2 21 2 2 1 1 series is obtained the origin of the results comes from studying symmetries of the 9 j coefficient appearing in quantum theory of angular momentum

biorthogonal coupling coefficients of uq n the coupling wigner clebsch gordan coefficients and their isofactors for the unitary quantum algebras uq n with the repeating irreducible representations in the coproduct decomposition are considered generalizing the u n case the biorthogonal systems of the uq n isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure a first construction correlated with the inverted littlewood richardson rules gives the bilinear combinations of isofactors after applying the proportionality of the q recoupling racah coefficients to the boundary q isofactors an alternative recursive construction gives the nonorthogonal q isofactors satisfying the most elementary boundary conditions and proportional to the uq n 1 recoupling coefficients for some less restricted values of parameters some multiplicity free and more general uq n recoupling coefficients are found the blocks bilinear combinations of which equal to the resubducing coefficients of the complementary chains of q algebras are proposed to use for the orthonormalization of some uq n biorthogonal isofactors including the general uq 3 case

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

hypergeometric series related to the 9 j coefficient of su 1 1 abstract we study various expressions for the 9 j coefficient of su 1 1 with an emphasis on multiple hypergeometric series an orthogonal polynomial in two discrete variables is associated to the 9 j coefficient some applications are considered dealing with formulas in which the 9 j coefficient appears as a connection or expansion coefficient and discussing some transformation formulas for double and triple hypergeometric series of unit argument

an algebraic algorithm for calculating clebsch gordan coefficients application to su 2 and su 3 a recent paper gave an explicit construction for inducing shift tensors of a compact reductive lie group from shift tensors of a suitably defined subgroup the shift tensors were defined on model spaces of holomorphic vector coherent state wave functions in this paper we use these shift tensors to obtain an algorithm for computing clebsch gordan coefficients the approach reproduces the known analytical results for su 2 and gives a simple algorithm for computing su 3 coefficients the algorithm is shown to yield analytical expressions for the multiplicity free su 3 couplings of type 20 10

multiplicity free uq n coupling coefficients the coupling wigner clebsch gordan coefficients of the unitary quantum algebras uq n are considered the tensorial properties of the generator powers and their ordered products used in the explicit projectors and weight lowering procedures are established different expressions for the multiplicity free isofactors of uq n coupling coefficients those coupling an arbitrary and symmetric representations are derived explicit expression of the arbitrary isofactors in terms of their boundary values are proposed proportionality of the semistretched isofactors to the stretched q recoupling coefficients of uq n 1 q analogues of 9j symbols is demonstrated the stretched isofactors of uq 3 are expressed in terms of clebsch gordan coefficients of uq 2

clebsch gordan coefficients in the asymptotic limit we investigate the graded structure of the tensor product space v 2 v 1 which arises in the coupling of two irreducible representations 1 and 2 of a semisimple lie algebra g in the limit in which one of the highest weights becomes asymptotically large the construction of asymptotic coupling coefficients is considered

the dual hahn q polynomials in the lattice and the q algebras and the dual q hahn polynomials in the non uniform lattice are obtained the main data for these polynomials are calculated the square of the norm of the coefficients of the three term recurrence relation etc as well as the lattice representation as a q hypergeometric series the connection with the clebsch gordan coefficients of the quantum algebras and is also given

tensor operators iii some fundamental tensor operator identities let s 2 and let a1 a2 as where s ai z and a1 a2 as 1 let 1 t s the method of undetermined coefficients was used to investigate natural transformations a1 a2 as and t a1 at as at necessary and sufficient conditions were found on the coefficients defining and sufficient conditions on the coefficients defining t explicit formulas for 1 and s are given herein

localization computation of one point disk invariants of projective calabi yau complete intersections we define one point disk invariants of a smooth projective calabi yau complete intersection in the presence of an anti holomorphic involution via localization we show that these invariants are rational numbers and obtain a formula for them which confirms in particular a conjecture by jinzenji shimizu int j geom method m 11 1 1456005 2014 conjecture 1

explicit canonical tensor operators and orthonormal coupling coefficients of su 3 the canonical unit su 3 tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators with the renormalization factors expressed in terms of the denominator functions of biedenharn gustafson lohe louck and milne the matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of asherova and smirnov the self conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique the corresponding extreme isoscalar factors of the clebsch gordan wigner coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of su 3 with its considerable simplication for the boundary values of parameters the general isofactors are also expanded in the different ways in terms of their boundary values the new classes of the generaliz

stabilization and positioning control of a rolling disk by using the bogdanov takens bifurcation abstract on the basis of bogdanov takens bifurcation we obtain two nonlinear control laws for stabilization and positioning of a rolling disk on a horizontal plane at a desired set point the first control law is applied to reach a set of predetermined points from previously calculated initial conditions which unavoidably implies a small position error to overcome this problem a second procedure based on a nonlinear inverse control law is examined by using adequate leaning tilting and rolling moments the system satisfies nonholonomic constraints it is underactuated and the analysis of the bogdanov takens bifurcation provides the admissible parameter values to reach the desired set point our analysis allows stabilization and positioning of the rolling disk through small control torques neglecting the internal servomechanism dynamics with adequate reaction forces satisfying the nonholonomic conditions the results of the analytical calculations are verified by full numerical simulations

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

vector coherent state theory as a theory of induced representations a general definition is given of vector coherent state vcs representation theory it is shown that the theory is more general than suggested by previous applications and that it incorporates the standard theories of induced representations as special cases the associated k matrix theory is also given a fuller treatment than hitherto and shown to provide a rather general algorithm both for projecting vcs representations from larger representations in which they are embedded and for determining the hermitian form with respect to which an isometric equivalent representation is in fact isometric

remodeling the b model we propose a complete new formalism to compute unambiguously b model open and closed amplitudes in local calabi yau geometries including the mirrors of toric manifolds the formalism is based on the recursive solution of matrix models recently proposed by eynard and orantin the resulting amplitudes are non perturbative in both the closed and the open moduli the formalism can then be used to study stringy phase transitions in the open closed moduli space at large radius this formalism may be seen as a mirror formalism to the topological vertex but it is also valid in other phases in the moduli space we develop the formalism in general and provide an extensive number of checks including a test at the orbifold point of ap fibrations where the amplitudes compute the t hooft expansion of vevs of wilson loops in chern simons theory on lens spaces we also use our formalism to predict the disk amplitude for the orbifold mathbb c 3 mathbb z 3

dynamics of an eccentric disk on a curved surface abstract this paper investigates the dynamic behavior of an eccentric disk rolling on a curve of arbitrary shape and then on a curve defined as a cubic function comparisons are made to a disk without eccentricity and the related point mass approximation the curve is subject to sinusoidal base excitation and the system is considered from the perspective of a potential well problem where escape is possible on one side the equations of motion are derived using a roll without slip constraint and the behavior is investigated by means of approximate analytical solutions numerically simulated frequency and amplitude sweeps and by considering the basins of attraction when initial conditions or forcing parameters are varied

su3 isoscalar factors a summary of the properties of the wigner clebsch gordan coefficients and iso scalar factors for the group su3 in the su2 u1 decomposition is presented the outer degeneracy problem is discussed in detail with a proof of a conjecture braunschweig s which has been the basis of previous work on the su3 coupling coefficients recursion relations obeyed by the su3 isoscalar factors are produced along with an algorithm which allows numerical determination of the factors from the recursion relations the algorithm produces isoscalar factors which share all the symmetry properties under permutation of states and conjugation which are familiar from the su2 case the full set of symmetry properties for the su3 wigner clebsch gordan coefficients and isoscalar factors are displayed

vector coherent state theory of the generic representations of so 5 in an so 3 basis for applications of group theory in quantum mechanics one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some hamiltonian of interest here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite dimensional irreps of the so 5 lie algebra in an so 3 basis the so 3 subgroup of so 5 is defined by regarding so 5 as linear transformations of the five dimensional space of an so 3 irrep of angular momentum two a need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations the algorithm has been implemented in maple and some tables of results are presented

on biorthogonal and orthonormal clebsch gordan coefficients of su 3 analytical and algebraic approaches minimal biorthogonal systems of the clebsch gordan wigner coefficients of su 3 u 2 are discussed as well as the dual coupled bases the closed system of analytical expressions for the dual isofactors reduced wigner coefficients and the overlaps of coupled states is obtained with the help of analytical inversion symmetry the regge type symmetry of the overlaps and the boundary orthonormal isofactors orthogonalization coefficients is discovered the polynomial structure of the alternative complete algebraic systems of the orthonormal su 3 isofactors characterized by the null spaces symmetries and additional selection rules and obtained by means of the hecht or gram schmidt process is considered the realizations of the external missing label operators of the third and the fourth orders in the minimal coupled bases which lead to preferable algorithms to evaluate the orthonormal su 3 coupling coefficients satisfying different symmetry properties are presented with the help of the 6j coef

on the denominator function for canonical su 3 tensor operators ii explicit polynomial form the canonical resolution of the multiplicity problem for tensor operators in su 3 is equivalent to the map the denominator mapping from the set of all su 3 unit tensor operators to su 3 invariant functions the denominator functions the denominator function vanishes precisely on that characteristic null space that specifies each operator uniquely since for su 3 the characteristic null spaces are known to be simply ordered each denominator function can be expressed up to explicitly known multiplicative factors as a ratio of two successive polynomials in the set gtq t 0 1 q 1 q 0 1 by obtaining explicitly the set of all polynomials gtq this paper completes the construction of all su 3 denominator functions

the racah wigner algebra and coherent tensors we present a set of tensors which are shift tensors wigner tensors in accordance with the definitions of biedenharn and louck and satisfy the coherence conditions of flath and towber our tensors are defined for all connected compact lie groups and for finite dimensional representations of connected reductive lie groups thus we have a realization of the coherent tensors in a rather general setting moreover this realization enables us to confirm most of the conjectures of flath and towber concerning the properties of coherent tensors

a calculus for su 3 leading to an algebraic formula for the clebsch gordan coefficients we develop a simple computational tool for su 3 analogous to bargmann s calculus for su 2 crucial new inputs are i explicit representation of the gelfand zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable ii an auxiliary gaussian measure with respect to which the gelfand zetlin states are orthogonal but not normalized iii simple generating functions for generating all basis states and also all invariants as an illustration of our techniques an algebraic formula for the clebsch gordan coefficients is obtained for the first time this involves only gaussian integrations thus su 3 is made as accessible for computations as su 2 is

bott s formula and enumerative geometry we outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of bott as an example gromov witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed the results are consistent with predictions made from mirror symmetry compu tations we also compute degrees of some loci in the linear system of plane curves of degrees less than 10 like those corresponding to sums of powers of linear forms and curves carrying inscribed polygons mathematical institute university of oslo p 0 box 1053 n 0316 oslo norway e mail address ellingsr0math uio no mathematical institute university of bergen alleg 55 n 5007 bergen norway e mail address strommeti i uib no this content downloaded from 157 55 39 29 on tue 12 apr 2016 10 23 14 utc all use subject to http about jstor org terms

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

the pattern calculus for tensor operators in quantum groups an explicit algebraic evaluation is given for all q tensor operators with unit norm belonging to the quantum group uq u n and having extremal operator shift patterns acting on arbitrary uq u n irreps these rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns a more conceptual derivation of these results is discussed using fundamental properties of the q 6j coefficients

towards the canonical tensor operators of uq 3 i the maximal null space case generalizing the su 3 canonical tensor operator concept biedenharn and louck to the quantum algebra uq 3 the wigner clebsch gordan coefficients of uq 3 with repeating irreducible representations are considered extremal projectors of the quantum algebra uq 3 in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq 3 canonical isoscalar factors explicit expressions of the uq 3 isofactors corresponding to the maximal null space case of the uq 3 unit canonical tensor operators and their normalization factors denominator functions are presented the transposition and conjugation phase factors for the su 3 and uq 3 canonical isofactors are correlated with phases and zeros of boundary isofactors invariance of the canonical isofactors or absence of such invariance under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance or numerical degeneracy of the usual splitting

on the triple sum formula for wigner 9j symbols we give a new proof of the triple sum formula for wigner 9j symbols due to alisauskas and jucys the proof uses explicit expressions for the coupling kernels recently introduced by the author parts of our results generalize to general recoupling coefficients

involutions obstructions and mirror symmetry abstract consider a maslov zero lagrangian submanifold diffeomorphic to a lie group on which an anti symplectic involution acts by the inverse map of the group we show that the fukaya a endomorphism algebra of such a lagrangian is quasi isomorphic to its de rham cohomology tensored with the novikov field in particular it is unobstructed formal and its floer and de rham cohomologies coincide our result implies that the smooth fibers of a large class of singular lagrangian fibrations are unobstructed and their floer and de rham cohomologies coincide this is a step in the syz and family floer cohomology approaches to mirror symmetry more generally our result continues to hold if the lagrangian has cohomology the free graded algebra on a graded vector space v concentrated in odd degree and the anti symplectic involution acts on the cohomology of the lagrangian by the induced map of negative the identity on v it suffices for the maslov class to vanish modulo 4

vector coherent state representations induced representations and geometric quantization i scalar coherent state representations coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model i classical realizations which are the starting point for geometric quantization ii induced unitary representations corresponding to prequantization and iii irreducible unitary representations obtained in geometric quantization by choice of a polarization these representations establish an intimate relation between coherent state theory and geometric quantization in the context of induced representations

explicit orthonormal clebsch gordan coefficients of su 3 the construction of the explicit algebraic polynomial expressions for the nonmultiplicity free orthonormal clebsch gordan wigner coefficients of su 3 u 2 is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the hecht or gram schmidt process the direct and inverse orthogonalization coefficients the first of them being equivalent to the boundary orthonormal isofactors are expressed up to explicitly given multiplicative factors in terms of the numerator and denominator polynomials related with the auxiliary a function of louck biedenharn and lohe that appears as a fragment of the denominator g functions of canonical su 3 tensor operators

the multiple sum formulas for 12j coefficients of su 2 and uq 2 the expressions for 12j coefficients of the both kinds without and with braiding of the su 2 group and the quantum algebra uq 2 are considered using dougall s summation formula of the very well poised hypergeometric 5f4 1 series and its q generalization several fourfold sum formulas with each sum related to the balanced 5f4 1 or 5 4 series for the q 12j coefficients of the second kind without braiding are derived applying q generalizations of rearrangement formulas of the very well poised hypergeometric 6f5 1 series which correspond to a new expression for the clebsch gordan coefficients of su 2 and uq 2 the new expressions with five sums of the 4f3 1 and 3f2 1 or 4 3 and 3 2 type are derived for the q 12j coefficients of the first kind with braiding instead of the usual expansions in terms of q 6j coefficients stretched and doubly stretched q 12j coefficients as triple double or single sums related to composed or separate hypergeometric 4f3 1 and 5f4 1 or 4 3 and 5 4 seri

transformation and summation formulas for kampe de feriet seriesf0 31 1 1 1 abstract the double hypergeometric kampe de feriet series f 0 3 1 1 1 1 depends upon 9 complex parameters we present three cases with 2 relations between those 9 parameters and show that under these circumstances f 0 3 1 1 1 1 can be written as a 4 f 3 1 series some limiting cases of these transformation formulas give rise to new summation results for special f 0 3 1 1 1 1 s the actual transformation results arose out of the study of 9 j coefficients

stability of landau ginzburg branes we evaluate the ideas of stability at the landau ginzburg lg point in moduli space of compact calabi yau manifolds using matrix factorizations to b model the topological d brane category the standard requirement of unitarity at the ir fixed point is argued to lead to a notion of r stability for matrix factorizations of quasihomogeneous lg potentials the d0 brane on the quintic at the landau ginzburg point is not obviously unstable aiming to relate r stability to a moduli space problem we then study the action of the gauge group of similarity transformations on matrix factorizations we define a naive moment maplike flow on the gauge orbits and use it to study boundary flows in several examples gauge transformations of nonzero degree play an interesting role for brane antibrane annihilation we also give a careful exposition of the grading of the landau ginzburg category of b branes and prove an index theorem for matrix factorizations

overlaps of the biorthogonal 3 coupling coefficients and related basic hypergeometric and other q factorial series the intermediate expansion technique is developed for the overlap coefficients of different biorthogonal coupled states and coupling coefficients of the quantum algebra and group su 3 with repeating irreducible representations and the role of separate basic and classical hypergeometric functions is demonstrated some expansion coefficients such as new prime overlap functions equivalent to the bilinear combinations of the different boundary isofactors are expressed in terms of the balanced saalschutzian basic or classical hypergeometric series these new overlap functions their analytical inversion some known triangular expansion matrices transition matrices related to definite racah coefficients the well poised or series and compositions of q factorial series resembling the well poised and series or equivalent to and series provide themselves as generators for mutual expansion of different non orthonormal systems of and su 3 isofactors which may be orthonormalized to the paracanonical or with a definite caution canonical version

calculation of clebsch gordan and racah coefficients using symbolic manipulatio n programs a procedure for calculating clebsch gordan and racah coefficients arising from the decomposition of n fold tensor products of the u n groups using symbolic manipulation programs is given basic states are realized as polynomials in a space equipped with a differentiation inner product the desired coefficients are then obtained by differentiating the relevant polynomials examples from su 2 and su 3 are given copyright 1989 academic press inc

irreducible tensor operators in the regular coaction formalisms of compact quantum group algebras the defining conditions for the irreducible tensor operators associated with the unitary irreducible corepresentations of compact quantum group algebras are deduced first in both the right and left regular coaction formalisms in each case it is shown that there are two types of irreducible tensor operator which may be called ordinary and twisted the consistency of the definitions is demonstrated and various consequences are deduced including generalizations of the wigner eckart theorem for both the ordinary and twisted operators also included are discussions within the regular coaction formalisms for compact quantum group algebras of inner products basis functions projection operators clebsch gordan coefficients and two types of tensor product of corepresentations the formulation of quantum homogeneous spaces for compact quantum group algebras is discussed and the defining conditions for the irreducible tensor operators associated with such quantum homogeneous spaces and with the uni

coherent state triplets and their inner products it is shown that if h is a hilbert space for a representation of a group g then there are triplets of spaces fh h fh in which fh is a space of coherent state or vector coherent state wave functions and fh is its dual relative to a conveniently defined measure it is shown also that there is a sequence of maps fh h fh which facilitates the construction of the corresponding inner products after completion if necessary the spaces fh h fh become isomorphic hilbert spaces it is shown that the inner product for h is often easier to evaluate in fh than in fh thus we obtain integral expressions for the inner products of coherent state and vector coherent state representations these expressions are equivalent to the algebraic expressions of k matrix theory but they are frequently more efficient to apply the construction is illustrated by many examples

transformation formulas for double hypergeometric series related to 9 j coefficients and their basic analogs in a recent paper alisauskas deduced different triple sum expressions for the 9 j coefficient of su 2 and suq 2 for a singly stretched 9 j coefficient these reduce to different double sum series using these distinct series we deduce a set of new transformation formulas for double hypergeometric series of kampe de feriet type and their basic analogs these transformation formulas are valid for rather general parameters of the series although a common feature is that all the series appearing here are terminating it is also shown that the transformation formulas deduced here generate a group of transformation formulas thus yielding an invariance group or symmetry group of particular double series

a numerical algorithm for the explicit calculation of su n and sl n c clebsch gordan coefficients we present an algorithm for the explicit numerical calculation of su n and sl n c clebsch gordan coefficients based on the gelfand tsetlin pattern calculus our algorithm is well suited for numerical implementation we include a computer code in an appendix our exposition presumes only familiarity with the representation theory of su 2

orthogonalization of the projected states and isofactors the explicit direct and inverse orthogonalization coefficients ocs are derived for the projected basis states of the two parametric covariant irreducible representations of including elliott s n 3 case and draayer with the alternative choices of the gram schmidt processes began respectively from the lowest vergados or highest absolute value of the intrinsic parameter k the different direct and inverse ocs the latter equivalent to the definite boundary isofactors or the boundary resubducing coefficients are found to be connected by the definite analytical continuation contraction procedures with mutually related ocs of the biorthogonal states and the boundary orthonormal pseudocanonical and paracanonical isofactors of the orthogonalization coefficients considered are presented as definite algebraic polynomial structures under the square root with the linear factors and numerator denominator polynomials in terms of the partition dependent functions of biedenharn and louck

clebsch gordan coefficients of su 3 in su 2 and so 3 bases new algorithms are developed for the purpose of optimizing the efficient calculation of su 3 clebsch gordan coefficients in both su 2 and so 3 coupled bases the new algorithms make use of the fact that highest weight states in a tensor product space are easily identified by vector coherent state methods the methods are developed for su 3 but apply to other compact semi simple lie groups

invariants of real symplectic 4 manifolds and lower bounds in real enumerative geometry we first build the moduli spaces of real rational pseudo holomorphic curves in a given real symplectic 4 manifold then following the approach of gromov and witten 3 19 11 we define invariants under deformation of real symplectic 4 manifolds these invariants provide lower bounds for the number of real rational j holomorphic curves which realize a given homology class and pass through a given real configuration of points

a note on computations of the d brane superpotential we develop some computational methods for the integrals over the 3 chains on the compact calabi yau 3 folds that play a prominent role in the analysis of the topological b model in the context of the open mirror symmetry we discuss such 3 chain integrals in two approaches in the first approach we provide an algorithm to obtain the inhomogeneous picard fuchs equations in the second approach we discuss the analytic continuation of the period integral to compute the 3 chain integral directly the latter direct integration method is applicable for both on shell and off shell formalisms

induced shift tensors in vector coherent state theory it is shown that the so called shift tensors of biedenharn and louck have a natural expression within the framework of vector coherent state vcs theory vcs theory was developed to give a systematic construction of the representations of lie groups and to derive the corresponding matrices representing the elements of their lie algebras in this paper there are explicit realizations of shift tensors within the framework of vcs theory for semisimple lie groups in terms of shift tensors for suitably defined subgroups

opening mirror symmetry on the quintic aided by mirror symmetry we determine the number of holomorphic disks ending on the real lagrangian in the quintic threefold we hypothesize that the tension of the domainwall between the two vacua on the brane which is the generating function for the open gromov witten invariants satisfies a certain extension of the picard fuchs differential equation governing periods of the mirror quintic we verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space we further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the a model and verifying ooguri vafa integrality this is the first exact result on open string mirror symmetry for a compact calabi yau manifold

a simple derivation of the quantum clebsch gordan coefficients for su 2 q to su 2 q the quantum deformation of su 2 the van der waerden method for calculating the clebsch gordan cg coefficients is genaralized the polynomial basis for irreducible representations of su 2 q the relevant polynomial invariants and the deduction of the q analog of the cg coefficients are given