gaussian quadrature for sums a rapidly convergent summation scheme gaussian quadrature is a well known technique for numerical integration recently gaussian quadrature with respect to discrete measures corresponding to finite sums has found some new interest in this paper we apply these ideas to infinite sums in general and give an explicit construction for the weights and abscissae of gaussian formulas the abscissae of the gaussian summation have a very interesting asymptotic distribution function with a kink singularity we apply the gaussian summation technique to two problems which have been discussed in the literature we find that the gaussian summation has a very rapid convergence rate for the hardy littlewood sum for a large range of parameters

liouville action and weil petersson metric on deformation spaces global kleinian reciprocity and holography we rigorously define the liouville action functional for the finitely generated purely loxodromic quasi fuchsian group using homology and cohomology double complexes naturally associated with the group action we prove that classical action the critical value of the liouville action functional considered as a function on the quasi fuchsian deformation space is an antiderivative of a 1 form given by the difference of fuchsian and quasi fuchsian projective connections this result can be considered as global quasi fuchsian reciprocity which implies mcmullen s quasi fuchsian reciprocity we prove that the classical action is a kahler potential of the weil petersson metric we also prove that the liouville action functional satisfies holography principle i e it is a regularized limit of the hyperbolic volume of a 3 manifold associated with a quasi fuchsian group we generalize these results to a large class of kleinian groups including finitely generated purely loxodromic schottky and quasi fuchsian groups and their free combinations

on the renormalized volume of hyperbolic 3 manifolds the renormalized volume of hyperbolic manifolds is a quantity motivated by the ads cft correspondence of string theory and computed via a certain regularization procedure the main aim of the present paper is to elucidate its geometrical meaning we use another regularization procedure based on surfaces equidistant to a given convex surface n the renormalized volume computed via this procedure is equal to what we call the w volume of the convex region n given by the usual volume of n minus the quarter of the integral of the mean curvature over n the w volume satisfies some remarkable properties first this quantity is self dual in the sense explained in the paper second it verifies some simple variational formulas analogous to the classical geometrical schlafli identities these variational formulas are invariant under a certain transformation that replaces the data at n by those at infinity of m we use the variational formulas in terms of the data at infinity to give a simple geometrical proof of results of takhtajan et al on the kahler potential on various moduli spaces

gaussian quadrature involving einstein and fermi functions with an application to summation of series polynomials 7rk rk d x k 0 1 2 are constructed which are orthogonal with respect to the weight distributions da t t et 1 dt and da t 1 et 1 dt r 1 2 on 0 oo moment related methods being inadequate a discretized stieltjes procedure is used to generate the coefficients ak pk in the recursion formula 7rk 1 t t ak k t fk7rk1 t k 0 1 2 so t 1 17j l t 0 the discretization is effected by the gauss laguerre and a composite fejer quadrature rule respectively numerical values of ak ipk as well as associated error constants are provided for 0 k 39 these allow the construction of gaussian quadrature formulae including error terms with up to 40 points examples of n point formulae n 5 5 40 are provided in the supplements section at the end of this issue such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series

potentials and chern forms for weil petersson and takhtajan zograf metrics on moduli spaces abstract for the tz metric on the moduli space m 0 n of n pointed rational curves we construct a kahler potential in terms of the fourier coefficients of the klein s hauptmodul we define the space s g n as holomorphic fibration s g n s g over the schottky space s g of compact riemann surfaces of genus g where the fibers are configuration spaces of n points for the tautological line bundles l i over s g n we define hermitian metrics h i in terms of fourier coefficients of a covering map j of the schottky domain we define the regularized classical liouville action s and show that exp s is a hermitian metric in the line bundle l i 1 n l i over s g n we explicitly compute the chern forms of these hermitian line bundles c 1 l i h i 4 3 tz i c 1 l exp s 1 2 wp we prove that a smooth real valued function s s i 1 n log h i on s g n a potential for this special difference of wp and tz metrics coincides with the renormalized hyperbolic volume of a corresponding schottky 3 manifold we extend these results to the quasi fuchsian groups of type g n

quantum liouville theory in the background field formalism i compact riemann surfaces using polyakov s functional integral approach and the liouville action functional defined in zt87c and tt03a we formulate quantum liouville theory on a compact riemann surface x of genus g 1 for the partition function x and correlation functions with the stress energy tensor components langle prod i 1 n t z i prod k 1 l bar t bar w k x rangle we describe feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution the hyperbolic metric on x extending analysis in tak93 tak94 tak96a tak96b we define the regularization scheme for any choice of the global coordinate on x for the schottky and quasi fuchsian global coordinates we rigorously prove that one and two point correlation functions satisfy conformal ward identities in all orders of the perturbation theory obtained results are interpreted in terms of complex geometry of the projective line bundle fancyscript e c lambda h c 2 over the moduli space mathfrak m g where c is the central charge and h is the hodge line bundle and provide the friedan shenker fs87 complex geometry approach to cft with the first non trivial example besides rational models

s orthogonality and construction of gauss turan type quadrature formulae using the theory of s orthogonality and reinterpreting it in terms of the standard orthogonal polynomials on the real line we develop a method for constructing gauss turan type quadrature formulae the determination of nodes and weights is very stable for finding all weights our method uses an upper triangular system of linear equations for the weights associated with each node numerical examples are included

on certain slowly convergent series occurring in plate contact problems a simple computational procedure is developed for accurately summing series of the form z 0 2k 1 pz2k 1 where z is complex with jzi 1 and p 2 or 3 as well as series of the type

schwarzian derivatives projective structures and the weil petersson gradient flow for renormalized volume to a complex projective structure on a surface thurston associates a locally convex pleated surface we derive bounds on the geometry of both in terms of the norms and 2 of the quadratic differential of given by the schwarzian derivative of the associated locally univalent map we show that these give a unifying approach that generalizes a number of important well known results for convex cocompact hyperbolic structures on 3 manifolds including bounds on the lipschitz constant for the nearest point retraction and the length of the bending lamination we then use these bounds to begin a study of the weil petersson gradient flow of renormalized volume on the space cc n of convex cocompact hyperbolic structures on a compact manifold n with incompressible boundary leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of dn the double of n

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

a class of slowly convergent series and their summation by gaussian quadrature series are considered whose general term is a rational function multiplied by a fractional power the summation of such series is reduced via laplace transformation techniques to a problem of quadrature which is then solved by gaussian quadrature relative to einstein and fermi weight functions a number of examples are worked out in detail

liouville action and holography on quasi fuchsian deformation spaces we study the liouville action for quasi fuchsian groups with parabolic and elliptic elements in particular when the group is fuchsian the contribution of elliptic elements to the classical liouville action is derived in terms of the bloch wigner functions we prove the first and second variation formulas for the classical liouville action on the quasi fuchsian deformation space we prove an equality expressing the holography principle which relates the liouville action and the renormalized volume for quasi fuchsian groups with parabolic and elliptic elements we also construct the potential functions of the kahler forms corresponding to the takhtajan zograf metrics associated to the elliptic elements in the quasi fuchsian groups

the hardy littlewood function an exercise in slowly convergent series the function in question is h x k 1 sin x k k in deference to the general theme of this conference a summation procedure is first described using orthogonal polynomials and polynomial rational gauss quadrature its effectiveness is limited to relatively small positive values of x direct summation with acceleration is shown to be more powerful for very large values of x such values are required to explore a in the meantime disproved conjecture of alzer and berg according to which h x is bounded from below by 2

quadrature formulas for functions with poles near the interval of integration on examine la construction des regles de quadrature de type interpolatoire utilisant uniquement des valeurs de fonctions reelles pour des fonctions ayant des paires complesex conjuguees de poles pres de l intervalle d integration

the spiral of theodorus numerical analysis and special functions theodorus of cyrene ca 460 399 b c teacher of plato und theaetetus is known for his proof of the irrationality of n n 2 3 5 17 he may have known also of a discrete spiral today named after him whose construction is based on the square roots of the numbers n 1 2 3 the subject of this lecture is the problem of interpolating this discrete angular spiral by a smooth if possible analytic spiral an interesting solution was proposed in 1993 by p j davis which is based on an infinite product the computation of this product gives rise to problems of numerical analysis in particular the summation of slowly convergent series and the identification of the product raises questions regarding special functions the former are solved by a method of integration in particular gaussian integration the latter by means of dawson s integral und the bose einstein distribution number theoretic questions also loom behind this work

a symplectic map between hyperbolic and complex teichmuller theory let s be a closed orientable surface of genus at least 2 the space th ml where th is the hyperbolic teichmuller space of s and ml is the space of measured geodesic laminations on s is naturally a real symplectic manifold the space cp of complex projective structures on s is a complex symplectic manifold a relation between these spaces is provided by thurston s grafting map gr we prove that this map although not smooth is symplectic the proof uses a variant of the renormalized volume defined for hyperbolic ends

on hermitian holomorphic classes related to uniformization the dilogarithm and the liouville action metrics of constant negative curvature on a compact riemann surface are critical points of the liouville action functional which in recent constructions is rigorously defined as a class in a cech de rham complex with respect to a suitable covering of the surface we show that this class is the square of the metrized holomorphic tangent bundle in hermitian holomorphic deligne cohomology we achieve this by introducing a different version of the hermitian holomorphic deligne complex which is nevertheless quasi isomorphic to the one introduced by brylinski in his construction of quillen line bundles we reprove the relation with the determinant of cohomology construction furthermore if we specialize the covering to the one provided by a kleinian uniformization thereby allowing possibly disconnected surfaces the same class can be reinterpreted as the transgression of the regulator class expressed by the bloch wigner dilogarithm

quasi fuchsian manifolds with particles we consider 3 dimensional hyperbolic cone manifolds which are convex cocompact in a natural sense with cone singularities along infinite lines such singularities are sometimes used by physicists as models for massive spinless point particles we prove an infinitesimal rigidity statement when the angles around the singular lines are less than any infinitesimal deformation changes either these angles or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines moreover any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone manifold structure these results hold also when the singularities are along a graph i e for interacting particles

tau function and chern simons invariant abstract we define a chern simons invariant for schottky hyperbolic 3 manifolds of infinite volume we then prove an expression relating the bergman tau function on a fiber space over the teichmuller space to the lifting of the function f defined by zograf on teichmuller space and a holomorphic function on this space which we introduce if the point in this space corresponds to a marked riemann surface x then this function is constructed from the renormalized volume and our chern simons invariant for the bounding 3 manifold of x given by schottky uniformization together with a regularized polyakov integral we also obtain a relation between the chern simons invariant and the eta invariant of the bounding 3 manifold with defect given by the phase of the bergman tau function of x

asymptotic summation of power series we give an asymptotic expansion in powers of n 1 of the remainder sum j n infty f jz j when the sequence f n has a similar expansion contrary to previous results explicit formulas for the computation of the coefficients are presented in the case of numerical series z 1 rigorous error estimates for the asymptotic approximations are also provided we apply our results to the evaluation of s z j 0 nu a b p sum j j 0 infty z j j b nu 1 j a p which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series

orthogonal polynomials constructive theory and applications we consider polynomials orthogonal with respect to some measure on the real line a basic problem in the constructive theory of such polynomials is the determination of their three term recurrence relation given the measure in question depending on what is known about this measure there are different ways to proceed if as is typical in applications in the physical sciences one knows the measure only through moment information the appropriate procedure is an algorithm that goes back to chebyshev the algorithm in effect implements the nonlinear map from the given moments or modified moments to the desired recursion coefficients the numerical effectiveness of this procedure is determined in an essential way by the numerical condition of this map it is known that the map is ill conditioned in the case of ordinary moments for modified moments it may or may not be well conditioned the matter depending on certain properties of the given measure and the additional measure defining modified moments a theorem to this effect will be given and illustrated on some typical examples if more is known about the given measure for example if it is absolutely continuous and can be evaluated pointwise then a procedure can be employed which is based on an observation of stieltjes stieltjes remarked that the desired recursion coefficients can be successively built up by alternating between known inner product formulae for these coefficients and the initial sections of the recurrence relation already obtained an effective implementation of this idea requires a suitable discretization of the inner product this requirement while possibly a weakness of the method also accounts for its beauty since it leaves room for imagination and ingenuity used skillfully the discretized stieltjes procedure is among the most widely applicable and effective methods for generating orthogonal polynomials some examples will be given to illustrate its use finally we show how our newly acquired capability of generating nonstandard orthogonal polynomials can be used to solve some special problems in approximation theory and in the summation of slowly convergent series a novel set of polynomials orthogonal on the semicircle will also be mentioned briefly in connection with cauchy principal value integrals

generating functional in cft and effective action for two dimensional quantum gravity on higher genus riemann surfaces we formulate and solve the analog of the universal conformal ward identity for the stress energy tensor on a compact riemann surface of genus g 1 and present a rigorous invariant formulation of the chiral sector in the induced two dimensional gravity on higher genus riemann surfaces our construction of the action functional uses various double complexes naturally associated with a riemann surface with computations that are quite similar to descent calculations in brst cohomology theory we also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the teichmuller space of compact riemann surfaces of genus g 1

quadratures with multiple nodes power orthogonality and moment preserving spline approximation quadrature formulas with multiple nodes power orthogonality and some applications of such quadratures to moment preserving approximation by defective splines are considered an account on power orthogonality s and orthogonal polynomials and generalized gaussian quadratures with multiple nodes including stable algorithms for numerical construction of the corresponding polynomials and cotes numbers are given in particular the important case of chebyshev weight is analyzed finally some applications in moment preserving approximation of functions by defective splines are discussed