a fast algorithm for prandtl s integro differential equation collocation and quadrature methods for singular integro differential equations of prandtl s type are studied in weighted sobolev spaces a fast algorithm basing on the quadrature method is proposed convergence results and error estimates are given

the geometry of the universal teichmuller space and the euler weil petersson equation on the identity component of the universal teichmuller space endowed with the takhtajan teo topology the geodesics of the weil petersson metric are shown to exist for all time this component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle viewed this way the regularity of its elements is shown to be h3 2 epsilon for all epsilon 0 the evolutionary pde associated to the spatial representation of the geodesics of the weil petersson metric is derived using multiplication and composition below the critical sobolev index 3 2 geodesic completeness is used to introduce special classes of solutions of this pde analogous to peakons our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the weil petersson metric in imaging c 2015 elsevier inc all rights reserved

an integral operator induced by a zygmund function abstract we introduce a kernel function induced by a continuous function f on the unit circle and show that the corresponding integral operator on the standard bergman space is a bounded compact or hilbert schmidt operator precisely when f belongs to the big zygmund class the little zygmund class or the sobolev class h 3 2 respectively this may be considered as the infinitesimal version of the main result obtained in hu and shen 2012 9 by two of the authors

on the stability of the classical collocation method for cauchy type singular integral equations abstract the classical collocation method for cauchy type singular integral equations of the second kind on an interval using gaussian quadrature rules with respect to jacobi nodes is investigated in general the nodes and the weights of the quadrature rules can only be evaluated approximately in the present paper the question is answered how precise our knowledge about them must be in order to ensure that the corresponding algebraic system of equations is solvable yet and its solution is close to the solution of the exact system

sharp characters of finite groups abstract if is a virtual generalized character of a finite group g with n and l g g g g 1 g g g g 1 then g dividesfl n where fl x is the monic polynomial of least degree having l as its set of roots this generalises a result of the second author for permutation characters we say that the pair g is l sharp if g f l n we characterise the l sharp pairs for various sets l sometimes under additional hypotheses and give a number of examples

sharp characters of finite groups of type 1 1 1 1 satisfying a weaker condition than sharpness is studied especially for a pair g x in that class the group structure of g o g is determinted where o g is the largest normal subgroup of g of odd order the theorem is proved by picking the sharp pairs among pairs in the class

zygmund vector fields hilbert transform and fourier coefficients in shear coordinates we parametrize the space cal z of zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the farey tesselation then we express the hilbert transform and the fourier coefficients of the zygmund vector fields in terms of the above parametrization by infinitesimal shear functions finally we compute the weil petersson metric on the teichm uller space of a punctured surface in terms of shears

geometric quantization of n 2 superstring a nonperturbative geometric formulation of the n 2 neveu schwarz superstring theory which has been recently interpreted by h ooguri and c vafa mod phys lett a5 1389 1990 as a consistent quantum theory of self dual gravity in four dimensions is constructed it is shown that the natural complex structure over the loop superspace md d associated to the n 2 neveu schwarz fermionic string is invariant under symmetry group osp 2 2 superdiffs1 2 moreover it is proved that there is a unique lorentz and osp 2 2 invariant complex structure on md d this result implies that the superspace of all admissible complex structures over md d is isomorphic to the homogeneous kahler supermanifold superdiffs1 2 osp 2 2 the ricci curvature of superdiffs1 2 osp 2 2 is calculated applying the method of geometric quantization to the n 2 neveu schwarz superstring theory along the lines suggested by m j bowick and s g rajeev nucl phys b361 469 1991 a representation is constructed of nonperturbativ

weil petersson teichmuller space revisited abstract in our previous paper 23 it was proved that a sense preserving homeomorphism g on the unit circle s 1 belongs to the weil petersson class wp s 1 namely g can be extended to a quasiconformal mapping to the unit disk whose beltrami coefficient is square integrable in the poincare metric if and only if g is absolutely continuous such that log g belongs to the sobolev class h 1 2 in this sequel to 23 we show that the smooth hilbert manifold structure on wp s 1 inherited from h 1 2 by the pullback g log g is compatible with the standard hilbert manifold structure introduced by takhtajan teo 27 this enables us to give a fast approach to some results in our previous papers 23 and 24

characters of the bms group in three dimensions using the frobenius formula we evaluate characters associated with certain induced representations of the centrally extended bms3 group this computation involves a functional integral over a coadjoint orbit of the virasoro group a delta function localizes the integral to a single point allowing us to obtain an exact result the latter is independent of the specific form of the functional measure and holds for all values of the bms3 central charges and all values of the chosen mass and spin it can also be recovered as a flat limit of virasoro characters

a period mapping in universal teichmuller space in previous work it had been shown that the remarkable homogeneous space m diff s 1 psl 2 r sits as a complex analytic and kahler submanifold of the universal teichmuller space there is a natural immersion of m into the infinite dimensional version due to segal of the siegel space of period matrices that map is proved to be injective equivariant holomorphic and kahler isometric with respect to the canonical metrics regarding a period mapping as a map describing the variation of complex structure we explain why is an infinite dimensional period mapping

calculus on manifolds of conformal maps and cft in conformal field theory cft on simply connected domains of the riemann sphere the natural conformal symmetries under self maps are extended in a certain way to local symmetries under general conformal maps and this is at the basis of the powerful techniques of cft conformal maps of simply connected domains naturally have the structure of an infinite dimensional groupoid which generalizes the finite dimensional group of self maps we put a topological structure on the space of conformal maps on simply connected domains which makes it into a topological groupoid further we almost extend this to a local manifold structure based on the infinite dimensional frechet topological vector space of holomorphic functions on a given domain a from this we develop the notion of conformal a differentiability at the identity our main conclusion is that quadratic differentials characterizing cotangent elements on the local manifold enjoy properties similar to those of the holomorphic stress energy tensor of cft these properties underpin the local symmetries of cft applying the general formalism to cft correlation functions we show that the stress energy tensor is exactly such a quadratic differential this is at the basis of constructing the stress energy tensor in conformal loop ensembles it also clarifies the relation between cardy s boundary conditions for cft on simply connected domains and the expression of the stress energy tensor in terms of metric variations

a fast and well conditioned spectral method for singular integral equations abstract we develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing chebyshev and ultraspherical polynomials to reformulate the equations as almost banded infinite dimensional systems this is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels the resulting system can be solved in o m 2 n operations using an adaptive qr factorization where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution the complexity is reduced to o m n operations by pre caching the qr factorization when the same operator is used for multiple right hand sides stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity applications considered include the faraday cage and acoustic scattering for the helmholtz and gravity helmholtz equations including spectrally accurate numerical evaluation of the far and near field solution the julia software package singularintegralequations jl implements our method with a convenient user friendly interface

the ricci curvature of diff s1 sl 2 r previous calculations of the ricci curvature for the manifold diff diff s1 s1 are extended to diff s1 sl 2r these manifolds are distinguished by being coadjoint orbits of diff s1 which admit compatible symphectic and complex structures making them kahler manifolds

cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution cauchy singular integral equations on an interval are studied in weighted spaces of continuous functions convergence rates for a general method for their numerical solution are proved in weighted uniform norms applications of these results to the collocation method and the quadrature method are given

on the tangent space to the bmo teichmuller space abstract in a previous paper 31 we established the complex analytic manifold theory of the bmo teichmuller space in this paper we identify the function space which is the tangent space to the bmo teichmuller space

new error bounds for the quadrature method for the solution of cauchy singular integral equations the quadrature method for singular integral equations with cauchy kernels on the arc 1 1 is investigated and a comprehensive theory of this method s convergence in the continuous maximum norm is presented this approach is based on new results in the theory of orthogonal polynomials on a regularity property for the solution of some singular integral equations and on the concept of regular convergent operator sequences new and improved estimates for the rates of convergence are obtained

remarks on the gauss quadrature rule for a particular class of finite part integrals the method of finite part hadamard type or hypersingular integrals is already a standard approach in applied numerical methods in engineering and mainly in acoustics fluid dynamics elasticity and fracture mechanics here we study the convergence of the non classical possessing a negative node outside the integration interval as well as a negative weight corresponding to this node gauss legendre quadrature rule for finite part integrals on the interval 0 1 with the weight function 1 y and we prove its validity for integrands possessing a continuous first derivative the case of integrands possessing a higher order derivative is also considered and the rate of convergence is established these results are based on the derivation of a bound for the negative node in the above quadrature rule and further on the theory of convergence of the classical gauss quadrature rule numerical results verifying the results of this paper are also displayed the present results can be generalized to the gauss laguerre similarly non classical quadrature rule for finite part integrals on the interval 0 as well as to the two special but important in engineering applications non classical gaussian quadrature rules for cauchy type principal value integrals and mangler type finite part integrals have been proposed by tsamasphyros and dimou

infinitesimal bending and twisting in one dimensional dynamics an infinitesimal theory for bending and earthquaking in one dimensional dynamics is developed it is shown that any tangent vector to teichmuller space is the initial data for a bending and for an earthquaking or dinary differential equation the discussion involves an analysis of infinitesimal bendings and earthquakes the hilbert transform natural bounded linear oper ators from a banach space of measures on the mobius strip to tangent vectors to teichmuller space and the construction of a nonlinear right inverse for these operators the inverse is constructed by establishing an infinitesimal version of thurston s earthquake theorem

diff s 1 and the teichmuller spaces precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations widehat diff s 1 1 carry in a natural way the structure of infinite dimensional holomorphically homogeneous complex analytic kahler manifolds these aren diff s1 rot s1 andm diff s1 mob s1 note thatn is a holomorphic disc fiber space overm now m can be naturally considered as embedded in the classical universal teichmuller spacet 1 simply by noting that a diffeomorphism ofs1 is a quasisymmetric homeomorphism t 1 is itself a homomorphically homogeneous complex banach manifold we prove in the first part of the paper that the inclusion ofm int 1 iscomplex analytic

weil petersson teichmuller space ii smoothness of flow curves of h32 vector fields abstract given a continuous vector field t of sobolev class h 3 2 on the unit circle s 1 the flow maps g t of the differential equation d d t t 0 are known to be quasisymmetric homeomorphisms very recently gay balmaz ratiu 15 conjectured that the flow curve g t is in the weil petersson class wp s 1 and is continuously differentiable with respect to the hilbert manifold structure of wp s 1 introduced by takhtajan teo 40 the first assertion had already been demonstrated in our previous paper 36 in this sequel to 36 we will continue to deal with the weil petersson class wp s 1 and completely solve this conjecture in the affirmative

coadjoint orbits of the virasoro group the coadjoint orbits of the virasoro group which have been investigated by lazutkin and pankratova and by segal should according to the kirillov kostant theory be related to the unitary representations of the virasoro group in this paper the classification of orbits is reconsidered with an explicit description of the possible centralizers of coadjoint orbits the possible orbits are diff s1 itself diff s1 s1 and diff s1 sl n 2 r withsl n 2 r a certain discrete series of embeddings ofsl 2 r in diff s1 and diffs1 t wheret may be any of certain rather special one parameter subgroups of diffs1 an attempt is made to clarify the relation between orbits and representations it appears that quantization of diffs1 s1 is related to unitary representations with nondegenerate kac determinant unitary verma modules while quantization of diffs1 sl n 2 r is seemingly related to unitary representations with null vectors in leveln a better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of virasoro representation theory in the process of investigating virasoro coadjoint orbits we observe the existence of left invariant symplectic structures on the virasoro group manifold as is described in an appendix these give rise to lie bialgebra structures with the virasoro algebra as the underlying lie algebra

geometric investigations of a vorticity model equation abstract this article consists of a detailed geometric study of the one dimensional vorticity model equation t u x 2 u x 0 h u x t r x s 1 which is a particular case of the generalized constantin lax majda equation wunsch showed that this equation is the euler arnold equation on diff s 1 when the latter is endowed with the right invariant homogeneous h 1 2 metric in this article we prove that the exponential map of this riemannian metric is not fredholm and that the sectional curvature is locally unbounded furthermore we prove a beale kato majda type blow up criterion which we then use to demonstrate a link to our non fredholmness result finally we extend a blow up result of castro cordoba to the periodic case and to a much wider class of initial conditions using a new generalization of an inequality for hilbert transforms due to cordoba cordoba

on sharp characters of rank 2 with rational values abstract for a finite group g and its character let l be the set of distinct values of on non identity elements of g and f l 1 denote the monic polynomial of least degree having l as its set of roots blichfeldt a theorem concerning the invariants of linear homogeneous groups with some applications to substitution groups trans amer math soc 5 1904 461 466 2 showed that f l 1 g is a rational integer cameron and kiyota sharp character of finite groups j algebra 115 1988 125 143 called the pair g l sharp if g f l 1 and posed the problem of determining all the l sharp pairs g for a given set l for several cases those problems have already been studied by many authors our purpose is to determine the sharp pairs of types 2 1 and 1 2 having non trivial center

some notes on quasisymmetric flows of zygmund vector fields abstract in an important paper 24 reimann showed that the flow mappings of a continuous vector field of zygmund class are quasisymmetric homeomorphisms in this paper we will discuss the flow mappings when the vector field belongs to the smooth zygmund class or the sobolev class h 3 2

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

anomalies and curvature of w manifolds we study the holomorphic structure of certain complex manifolds associated withw algebras namely the flag manifoldsw t andw1 t1 and the spacesw sl r andw1 gl r wheret andt1 are the maximal tori inw andw1 we compute their ricci curvature and show how the results are related to the anomaly freedom conditions forw andw1 we discuss the relation of these manifolds with extensions of universal teichmuller space