hidden sl n symmetry in conformal field theories we prove that an irreducible representation of the virasoro algebra can be extracted from an irreducible representation space of thesl 2 current algebra by putting a constraint on the latter using the becchi rouet stora tyutin formalism thus there is asl 2 symmetry in the virasoro algebra but it is gauged and hidden this construction of the virasoro algebra is the quantum analogue of the hamiltonian reduction we then are naturally lead to consider a constrainedsl 2 wess zumino witten model this system is also related to quantum field theory of coadjoint orbit of the virasoro group based on this result we present a canonical derivation of thesl 2 current algebra in polyakov s theory of two dimensional gravity it is a manifestation of thesl 2 symmetry in conformal field theory hidden by the quantum hamiltonian reduction we also discuss the quantum hamiltonian reduction of thesl 2 current algebra and its relation to thew n algebra of zamolodchikov this makes it possible to define a natural generalization of the geometric action for thew n algebra despite its non lie algebraic nature

nonparametric inference of quantile curves for nonstationary time series the paper considers nonparametric specification tests of quantile curves for a general class of nonstationary processes using bahadur representation and gaussian approximation results for nonstationary time series simultaneous confidence bands and integrated squared difference tests are proposed to test various parametric forms of the quantile curves with asymptotically correct type i error rates a wild bootstrap procedure is implemented to alleviate the problem of slow convergence of the asymptotic results in particular our results can be used to test the trends of extremes of climate variables an important problem in understanding climate change our methodology is applied to the analysis of the maximum speed of tropical cyclone winds it was found that an inhomogeneous upward trend for cyclone wind speeds is pronounced at high quantile values however there is no trend in the mean lifetime maximum wind speed this example shows the effectiveness of the quantile regression technique

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

effect of mean on variance function estimation in nonparametric regression variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived we are particularly interested in the effect of the unknown mean on the estimation of the variance function our results indicate that contrary to the common practice it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth instead it is more desirable to use estimators of the mean with minimal bias on the other hand when the mean function is very smooth our numerical results show that the residual based method performs better but not substantial better than the first order difference based estimator in addition our asymptotic results also correct the optimal rate claimed in hall and carroll j roy statist soc ser b51 1989 3 14

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

space time dynamics from algebra representations we present a model for introducing dynamics into a space time geometry this space time structure is constructed from a c algebra defined in terms of the generators of an irreducible unitary representation of a finite dimensional lie algebra this algebra is included as a subalgebra in a bigger algebra the generators of which mix the representations of in a way that relates different space times and creates the dynamics this construction can be considered eventually as a model for two dimensional quantum gravity

functional data analysis for volatility abstract we introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data based recovery of the process from repeated observations a study of its asymptotic properties as the frequency of observed trades increases is complemented by simulations and an application to the analysis of intra day volatility patterns of the s p 500 index the proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility through the application of functional regression and prediction techniques

geometry of w algebras from the affine lie algebra point of view to classify the classical field theories with w symmetry one has to classify the symplectic leaves of the corresponding w algebra which are the intersection of the defining constraint and the coadjoint orbit of the affine lie algebra if the w algebra in question is obtained by reducing a wess zumino novikov witten wznw model the fields that survive the reduction will obey nonlinear poisson bracket or commutator relations in general for example the toda models are well known theories which possess such a nonlinear w symmetry and many features of these models can only be understood if one investigates the reduction procedure in this paper we analyse the sl n case from which the so called wn algebras can be obtained one advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the w algebra for the n 2 case corresponding to the coadjoint orbits of the virasoro algebra and for the n 3 case which gives rise to the zamolodchikov algebra our method in principle is capable of constructing explicit representatives on each leaf another attractive feature of this approach is the fact that the global nature of the w transformations can be explicitly described the reduction method also enables one to determine the classical highest weight hw states which are the stable minima of the energy on a w leaf these are important as only to those leaves can a hw representation space of the w algebra be associated which contains a classical hw state

sdiff s2 and the orbit method the group of area preserving diffeomorphisms of the two sphere sdiff s2 is one of the simplest examples of an infinite dimensional lie group it plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related newly defined symmetry groups we investigate its representation theory using the method of coadjoint orbits we describe the casimir functions and the cartan algebra then we evaluate the trace of a simple sdiff s2 operator using the atiyah bott fixed point formula the trace is divergent but we show that it has well defined truncations related to the structure of sdiff s2 finally we relate our results back to the recent appearances of sdiff s2 in black hole physics the group of area preserving diffeomorphisms of the two sphere sdiff s2 is one of the simplest examples of an infinite dimensional lie group it plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related newly defined symmetry groups we investigate its representation theory using the method of coadjoint orbits we describe the casimir functions and the cartan algebra then we evaluate the trace of a simple sdiff s2 operator using the atiyah bott fixed point formula the trace is divergent but we show that it has well defined truncations related to the structure of sdiff s2 finally we relate our results back to the recent appearances of sdiff s2 in black hole physics

on difference based variance estimation in nonparametric regression when the covariate is high dimensional summary we consider the problem of estimating the noise variance in homoscedastic nonparametric regression models for low dimensional covariates t d d 1 2 difference based estimators have been investigated in a series of papers for a given length of such an estimator difference schemes which minimize the asymptotic mean squared error can be computed for d 1 and d 2 however from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias we provide theoretical support for these findings in particular we show that with increasing dimension d this becomes more drastic if d4 these estimators even fail to be consistent a different class of estimators is discussed which allow better control of the bias and remain consistent when d4 these estimators are compared numerically with kernel type estimators which are asymptotically efficient and some guidance is given about when their use becomes necessary

non abelian harmonic oscillators and chiral theories we show that a large class of physical theories which has been under intensive investigation recently share the same geometric features in their hamiltonian formulation these dynamical systems range from harmonic oscillations to wzw like models and to the kdv dynamics on diff0s1 to the same class belong also the hamiltonian systems on groups of maps the common feature of these models are the chiral equations of motion for so called chiral decomposition of the phase space

universal teichmuller space and diff s 1 s 1 we point out that the coset space diffs1 s1 is a dense complex submanifold of the universal teichmuller spaces of compact riemann spaces of genus g 1 a holomorphic map ofs into the inifinite dimensional segal diskd1 is constructed this is the universal analogue of the map of teichmuller spaces into the siegel disk provided by the period matrix the kahler potential for the general homogenous metric on diffs1 s1 is computed explicitly using the map intod1 some applications to string theory are discussed

group quantization on configuration space new features of a previously introduced group approach to quantization are presented we show that the construction of the symmetry group associated with the system to be quantized the quantizing group does not require in general the explicit construction of the phase space of the system i e does not require the actual knowledge of the general solution of the classical equations of motion in many relevant cases an implicit construction of the group can be given directly on configuration space to show an application we construct the symmetry group for the conformally invariant massless scalar and electromagnetic fields and the scalar and dirac fields evolving in a symmetric curved space time or interacting with symmetric classical electromagnetic fields further generalizations of the present procedure are also discussed and in particular the conditions under which non abelian groups mainly affine groups and more general gauge groups can be included

quasi invariant lagrangians on lie groups and the method of coadjoint orbits starting with a two cocycle globally defined on a lie group g the lagrangian system on g is constructed whose lagrangian is quasi invariant under a right translation canonically lifted to the tangent bundle tg a direct consequence of the quasi invariance is the appearance of central extensions in the noether symmetry algebra then it is shown that the kinematical sector of such a model realizes through a symplectic reduction the kirillov kostant symplectic structure on the coadjoint orbit of the lie group with the central extensions the generalization to a higher dimensional theory of the wess zumino witten type is also discussed herein

estimation and tests of fit for the three parameter weibull distribution abstract estimation techniques are given for the three parameter weibull distribution with the location or origin parameter unknown and possibly also the shape and or scale parameters unknown tests of fit are described and tables are given for the edf statistics a 2 w 2 and u 2 to make the tests several examples are discussed

modular invariance on the torus and abelian chern simons theory the implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a group theoretical framework unlike the classical case at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance two cases must be considered depending on the cohomology class of the symplectic form on the torus if it is of integer cohomology class n then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if n is even or antiperiodic if n is odd if the symplectic form is of rational cohomology class n r a similar result holds the wave functions must be either periodic or antiperiodic on a torus r times larger in both directions depending on the parity of nr application of these results to the abelian chern simons theory is discussed

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

likelihood estimation and inference in a class of nonregular econometric models we study inference in structural models with a jump in the conditional density where location and size of the jump are described by regression curves two prominent examples are auction models where the bid density jumps from zero to a positive value at the lowest cost and equilibrium job search models where the wage density jumps from one positive level to another at the reservation wage general inference in such models remained a long standing unresolved problem primarily due to nonregularities and computational difficulties caused by discontinuous likelihood functions this paper develops likelihood based estimation and inference methods for these models focusing on optimal bayes and maximum likelihood procedures we derive convergence rates and distribution theory and develop bayes and wald inference we show that bayes estimators and confidence intervals are attractive both theoretically and computationally and that bayes confidence intervals based on posterior quantiles provide a valid large sample inference method

from geometric quantization to conformal field theory investigation of 2d conformal field theory in terms of geometric quantization is given we quantize the so called model space of the compact lie group virasoro group and kac moody group in particular we give a geometrical interpretation of the virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of cft minimal models 2d gravity wznw theory in the appendix we discuss the relation between classical constant r matrices and this geometrical approach

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

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

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

asymptotic efficiency in parametric structural models with parameter dependent support in certain auction search and related models the boundary of the support of the observed data depends on some of the parameters of interest for such nonregular models standard asymptotic distribution theory does not apply previous work has focused on characterizing the nonstandard limiting distributions of particular estimators in these models in contrast we study the problem of constructing efficient point estimators we show that the maximum likelihood estimator is generally inefficient but that the bayes estimator is efficient according to the local asymptotic minmax criterion for conventional loss functions we provide intuition for this result using le cam s limits of experiments framework

formal group laws for affine kac moody groups and group quantization we describe a method for obtaining formal group laws from the structure constants of affine kac moody groups and then apply a group manifold quantization procedure which permits construction of physical representations by using only canonical structures on the group as an intermediate step we get an explicit expression for two cocycles on loop groups the programme is applied to the affinesu 2 group

algebraic quantization on a group and nonabelian constraints a generalization of a previous group manifold quantization formalism is proposed in the new version the differential structure is circumvented so that discrete transformations in the group are allowed and a nonabelian group replaces the ordinary central u 1 subgroup of the heisenberg weyl like quantum group as an example of the former we obtain the wave functions associated with the system of two identical particles and the latter modification is used to account for the virasoro constraints in string theory

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

virasoro model space the representations of a compact lie groupg can be studied via the construction of an associated model space this space has the property that when geometrically quantized its hilbert space contains every irreducible representation ofg just once we construct an analogous space for the group diffs1 it is naturally a complex manifold with a holomorphic free action of diffs1 preserving a family of pseudo kahler structures all of the good coadjoint orbits are obtained from our space by hamiltonian constraint reduction we briefly discuss the connection to the work of alekseev and shatashvili

group approach to quantization of yang mills theories a cohomological origin of mass new clues for the best understanding of the nature of the symmetry breaking mechanism are revealed in this paper a revision of the standard gauge transformation properties of yang mills fields according to a group approach to the quantization scheme enables the gauge group coordinates to acquire dynamical content outside the null mass shell the corresponding extra internal field degrees of freedom are transferred to the vector potentials to conform massive vector bosons

extended conformal symmetries and u 1 currents the authors investigate the structure of extended conformal algebras generated by local chiral fields of spin 1 2 n using the gelfand dickey algebra of formal pseudodifferential operators for the gl n algebra the commutation relations of zamolodchikov s spin 3 operator algebra 1985 are derived in the presence of an additional u 1 conserved current

a quantum group structure in integrable conformal field theories we discuss a quantization prescription of the conformal algebras of a class ofd 2 conformal field theories which are integrable we first give a geometrical construction of certain extensions of the classical virasoro algebra known as classicalw algebras in which these algebras arise as the lie algebra of the second hamiltonian structure of a generalized korteweg de vries hierarchy this fact implies that thew algebras obtained as a reduction with respect to the nilpotent subalgebras of the kac moody algebra describe the integrability of a toda field theory subsequently we determine the coadjoint operators of thew algebras and relate these to classical yang baxter matrices the quantization of these algebras can be carried out using the concept of a so called quantum group we derive the condition under which the representations of these quantum groups admit a hilbert space completion by exploring the relation with the braid group then we consider a modification of the miura transformation which we use to define a quantumw algebra this leads to an alternative interpretation of the coset construction for kac moody algebras in terms of nonlinear integrable hierarchies subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of riemann surfaces to identify an action of thew algebras on the moduli space of stable curves and we give the invariants of this action this provides a generalization of the situation for the virasoro algebra where such as invariant is given by the so called mumford form which describes the partition function of the bosonic string

quantum mechanics of non linear systems a generalisation of weyl s correspondence rule and the associated moyal bracket for systems with non linear phase spaces that admit symplectic transitive group actions is presented this provides a simple account with explicit constructions of some of the general results of bayen et al 1978

quantum groups and wznw models an explanation of the appearance of quantum groups in chiral wznw models is given invariance of the theory under quantum group action is discussed

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

symplectic manifolds coherent states and semiclassical approximation the symplectic structure and hamiltonian dynamics for a class of grassmannian manifolds are described here using the two dimensional sphere s2 and disc d2 as illustrative cases we write their path integral representations using coherent state techniques these path integrals can be evaluated exactly by semiclassical methods thus providing examples of the localization formula along the way we also give a local coordinate description for a class of grassmannians

adaptive confidence intervals for regression functions under shape constraints adaptive confidence intervals for regression functions are constructed under shape constraints of monotonicity and convexity a natural benchmark is established for the minimum expected length of confidence intervals at a given function in terms of an analytic quantity the local modulus of continuity this bound depends not only on the function but also the assumed function class these benchmarks show that the constructed confidence intervals have near minimum expected length for each individual function while maintaining a given coverage probability for functions within the class such adaptivity is much stronger than adaptive minimaxity over a collection of large parameter spaces

field models from two cocycles on infinite dimensional lie groups and symplectic structures 2d gravity and chern simons theory in this paper the connection between field models and infinite dimensional lie groups is widely analysed on the bases of a new group quantization approach we also relate the poincare cartan form of variational calculus to the symplectic current structure of the covariant phase space formulation of higher derivative field theory the virasoro and kac moody groups are considered in the first case the action functional of the 2d induced gravity in the light cone formulation is derived the hidden sl 2 r simply appears as generated by the kernel of the lie algebra two cocycle and plays the role of a gauge type symmetry nevertheless it is shown that a proper space like formulation is out of reach of the virasoro group the corresponding symplectic structure of the nonlocal action functional is determined showing that it is related to the symplectic structure associated with the sl 2 r kac moody group this unravels the proper geometrical meaning of the hidden symmetry and differs from the analysis in related works based on the coadjoint orbit approach the relation between the kac moody groups and the chern simons gauge theory on a disc in the presence of a source is considered using the new approach

leibniz chern simons theory and phases of exceptional field theory we discuss a generalization of chern simons theory in three dimensions based on leibniz or loday algebras which are generalizations of lie algebras special cases of such theories appear in gauged supergravity where the leibniz algebra is defined in terms of the global lie symmetry algebra of the ungauged limit and an embedding tensor we show that the leibniz algebra of generalized diffeomorphisms in exceptional field theory can similarly be obtained from a lie algebra that describes the enhanced symmetry of an ungauged phase of the theory moreover we show that a topological phase of e8 8 exceptional field theory can be interpreted as a chern simons theory for an algebra unifying the three dimensional poincare algebra and the leibniz algebra of e8 8 generalized diffeomorphisms

testing composite hypotheses hermite polynomials and optimal estimation of a nonsmooth functional a general lower bound is developed for the minimax risk when estimating an arbitrary functional the bound is based on testing two composite hypotheses and is shown to be effective in estimating the nonsmooth functional 1 n i from an observation y n i n this problem exhibits some features that are significantly different from those that occur in estimating conventional smooth functionals this is a setting where standard techniques fail to yield sharp results a sharp minimax lower bound is established by applying the general lower bound technique based on testing two composite hypotheses a key step is the construction of two special priors and bounding the chi square distance between two normal mixtures an estimator is constructed using approximation theory and hermite polynomials and is shown to be asymptotically sharp minimax when the means are bounded by a given value m it is shown that the minimax risk equals 2 m 2 log log n log n 2 asymptotically where is the bernstein constant the general techniques and results developed in the present paper can also be used to solve other related problems

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

bayesian likelihood methods for estimating the end point of a distribution summary we consider maximum likelihood methods for estimating the end point of a distribution the likelihood function is modified by a prior distribution that is imposed on the location parameter the prior is explicit and meaningful and has a general form that adapts itself to different settings results on convergence rates and limiting distributions are given in particular it is shown that the limiting distribution is non normal in non regular cases parametric bootstrap techniques are suggested for quantifying the accuracy of the estimator we illustrate performance by applying the method to multiparameter weibull and gamma distributions

estimating conditional proportion curves by regression residuals researchers often derive a categorical outcome from an observed continuous measurement y for example human obesity status can be defined by the body mass index they proceed to estimate the conditional proportion curve p x p y c x x versus the covariate x by regular categorical data analysis in doing so they fail to fully use the rich information in the original continuous measurement researchers also face additional difficulties caused by the uncertainty of the cutoff c x which defines the categories for example some cutoff may cause artificial discontinuity or non smoothness in the conditional proportion curve p x we propose a fully nonparametric empirical estimator using the standardized regression residuals from the general location scale model y m x v x e for the original continuous measurement we prove the consistency of the proposed estimator in regular settings the proposed estimator has several attractive features it readily provides estimates for alternative cutoff values and easily accommodates the discontinuity in p x caused by special cutoff numerical examples demonstrate that our method has better performance in mean squared error than some alternative approaches we also discuss a design adjusted version of the proposed estimators to accommodate survey designs the proposed methods are applied to data on obesity measurements among female hispanic americans this new approach can be applied to other epidemiological and socio economic research that needs to define population categories from underlying continuous measurements copyright 2010 john wiley sons ltd

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

on the classification of the coadjoint orbits of the sobolev bott virasoro group the purpose of this paper is to give the classification of the bott virasoro coadjoint orbits with nonzero central charge in the functional analytic setting of smooth hilbert manifolds the central object of the paper is thus the completion of the bott virasoro group with respect to a sobolev topology giving rise to a smooth hilbert manifold and topological group called the sobolev bott virasoro group as a consequence of this approach analytic and geometric properties of the coadjoint orbits are studied c 2008 elsevier inc all rights reserved

variance function estimation in regression the effect of estimating the mean abstract the authors consider estimation of a variance function g in regression problems such estimation requires simultaneous estimation of the mean function f we obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g for example in nonparametric regression with two derivatives on g classical rates of convergence are possible if and only if the unknown f satisfies a lipschitz condition of order 1 3 or more if a parametric model is known for g then g may be estimated n 1 2 consistently if and only if f is lipschitz of order 1 2 or more optimal rates of convergence are attained by kernel estimators

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

variance estimation in nonparametric regression via the difference sequence method consider a gaussian nonparametric regression problem having both an unknown mean function and unknown variance function this article presents a class of difference based kernel estimators for the variance function optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized and asymptotic normality is also established we also show that for suitable asymptotic formulations our estimators achieve the minimax rate

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

short distance expansion from the dual representation of infinite dimensional lie algebras we develop a method for computing the short distance expansion of fields or operators that live in the coadjoint representation of an infinite dimensional lie algebra by using only properties of the adjoint representation and its dual we explicitly implement this method by computing the short distance expansion for the duals of the virasoro algebra affine lie algebras and the geometrically realized n extended supersymmetric virasoro algebra this method can also be used to compute short distance expansions between fields that transform in the adjoint and those that transform in the coadjoint representations

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

quantum dynamics on super groups constraints and the brst supergroup the smallest supergroup b k containing among its generators those of a constraint group k the brst charge and the corresponding ghost and antighost operators is constructed the supergroup b k can be enlarged to include the ordinary dynamical variables of the unconstrained physical system in a group approach to quantization b k generalizes the ordinary u 1 phase invariance of wave functions in particular this mechanism reproduces the essential features of the brst quantization when k is diff s1 the critical values of string theory are those for which the space of polarized functions on the enlarged brst supergroup is not irreducible

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

aspects of infinite dimensional l super galilean conformal algebra in this work we construct an infinite dimensional l super galilean conformal algebra which is a generalization of the l 1 algebra found in the literature we give a classification of central extensions the vector field representation the coadjoint representation and the operator product expansion of the infinite dimensional l super galilean conformal algebra keeping possible applications in physics and mathematics in mind

algebraic quantization good operators and fractional quantum numbers the problems arising when quantizing systems with periodic boundary conditions are analysed in an algebraic group quantization scheme and the failure of the ehrenfest theorem is clarified in terms of the already defined notion ofgood andbad operators the analysis of constrained heisenberg weyl groups according to this quantization scheme reveals the possibility for quantum operators without classical analogue and for new quantum fractional numbers extending those allowed for chern classes in traditional geometric quantization this study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus both examples featuring anomalous operators non equivalent quantization and the latter fractional quantum numbers these provide the rationale behind flux quantization in superconducting rings and fractional quantum hall effect respectively

non parametric estimation of residual moments and covariance the aim of non parametric regression is to model the behaviour of a response vector y in terms of an explanatory vector x based only on a finite set of empirical observations this is usually perf

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

a new attempt towards the unification of space time and internal gauge symmetries the neat formulation that describes the gauge interactions associated with internal symmetries is extended to the case of a simple yet non trivial symmetry group structure which mixes gravity and electromagnetism by associating a gauge symmetry with a central extension of the poincare group

geodesic flow on super bott virasoro group and harry dym family we show various harry dym type equations and super harry dym equations introduced by brunelli das and popowicz 2003 supersymmetric extensions of the harry dym hierarchy j math phys 44 4756 4767 2003 follow from the geodesic flows on the bott virasoro group and its supersymmetric generalization superconformal group in fact their bi hamiltonian structures can be derived from the lie poisson structures on the super bott virasoro orbit we also show that that m2kdv or calogero degasperis equation is also connected to the bott virasoro group

nonparametric estimation in a regression model with additive and multiplicative noise abstract in this paper we consider an unknown functional estimation problem in a general nonparametric regression model with the feature of having both multiplicative and additive noise we propose two new wavelet estimators in this general context we prove that they achieve fast convergence rates under the mean integrated square error over besov spaces the obtained rates have the particularity of being established under weak conditions on the model a numerical study in a context comparable to stochastic frontier estimation with the difference that the boundary is not necessarily a production function supports the theory

construction of local one parameter subgroups generated by the virasoro operators via feynman path integrals abstract it is well known that the virasoro algebra acts on the representations of an infinite dimensional heisenberg algebra called the fock models we calculate feynman path integrals on an coadjoint orbit of the infinite dimensional heisenberg group to give explicit formulae of actions of local one parameters subgroups generated by the virasoro operators

dynamics on sl 2 r tilde x u 1 a complete analysis of the quantum dynamical system defined by the quantum group sl 2 r tilde x u 1 is made this quantum group which is primarily related to the relativistic harmonic oscillator is also shown to provide the quantum dynamics of a free particle moving in 1 1 dimensional anti de sitter spacetime the latter interpretation illustrates the capabilities of the present group approach to quantisation in dealing with dynamical systems in more general spacetimes

symmetry and quantization higher order polarization and anomalies the concept of full polarization subalgebra in a group approach to quantization on a lie group g as a generalization of the analogous concept in geometric or standard quantization is discussed the lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition a generalization of the subalgebra of full polarization is then provided made out of higher order differential operators in the enveloping algebra of g higher order polarizations can also be used to quantize nonanomalous theories in different representations numerous examples are analyzed including the finite dimensional dynamics associated with the schrodinger group which presents an anomaly and an infinite dimensional anomalous system associated with the virasoro group in the last example the operators in the higher order polarization are in one to one correspondence with the null vectors in the verma module approach

invariant measures on polarized submanifolds in group quantization we provide an explicit construction of quasi invariant measures on polarized coadjoint orbits of a lie group g the use of specific trivial central extensions of g by the multiplicative group r allows us to restore the strict invariance of the measures and accordingly the unitarity of the quantization of coadjoint orbits as an example the representations of sl 2 r are recovered

higher order polarizations on the virasoro group and anomalies in a previous paper the authors showed that the space of first order polarized functions on the virasoro group is not in general irreducible the full reduction was explicitly achieved by taking the orbit of the enveloping algebra through the vacuum this additional step provided the proper quantization in the strong coupling domain 0 c 1 in this paper we introduce the concept of higher order polarization as a generalization of that of polarization we prove that the imposing of the additional higher order polarization conditions is equivalent to the taking of the above mentioned orbit this demonstrates that the generalized higher order polarization conditions suffice to obtain the irreducible hilbert spaces we also discuss the need for higher order polarizations in terms of anomalies

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

maximum likelihood estimation of the lower tail of a probability distribution summary we consider the estimation of the three parameters of the lower tail of a distribution function based on the k smallest out of n observations the likelihood function has a singularity but it is argued that a local maximum when it exists should be taken as the m l e asymptotic results as k oo n oo k n 0 show that such a local maximum does exist and provides consistent estimators whenever the shape parameter is greater than one otherwise there is no local maximum and likelihood inference fails we also discuss interval estimation and propose a test to distinguish between the type i and weibull limit laws

estimating tails of probability distributions we study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold a key idea is the use of pickands generalised pareto distribution and its fitting in most cases by the method of maximum likelihood the results cover all three limiting types of extreme value theory we propose a new estimator for an index of regular variation and show that it often performs better than hill s estimator we give new results for estimating the endpoint of a distribution extending earlier work by hall and by smith and weissman finally we give detailed results for the domain of attraction of exp e x and show that in most cases our proposed estimator is more efficient than two others one based on the exponential distribution and the other due to davis and resnick we also touch briefly on the problem of large deviations from a statistical point of view the results make extensive use of existing work on rates of convergence

quantum computation as gravity we formulate nielsen s geometric approach to circuit complexity in the context of two dimensional conformal field theories where series of conformal transformations are interpreted as unitary circuits built from energy momentum tensor gates we show that the complexity functional in this setup can be written as the polyakov action of two dimensional gravity or equivalently as the geometric action on the coadjoint orbits of the virasoro group this way we argue that gravity sets the rules for optimal quantum computation in conformal field theories

estimating parameters in continuous univariate distributions with a shifted origin summary a general method of estimating parameters in continuous univariate distributions is proposed it is especially suited to cases where one of the parameters is an unknown shifted origin this occurs for example in the three parameter lognormal gamma and weibull models for such distributions it is known that maximum likelihood ml estimation can break down because the likelihood is unbounded and this can lead to inconsistent estimators properties of the proposed method are described in particular it is shown to give consistent estimators with asymptotic efficiency equal to ml estimators when these exist moreover it gives consistent asymptotically efficient estimators in situations where ml fails examples are given including numerical ones showing the advantages of the method

classical exchange algebra of the superstring on s 5 with ads time a classical exchange algebra of the superstring on s5 with ads time is shown on the light like plane to this end we use the geometrical method of which consistency is guaranteed by the classical yang baxter equation the dirac method does not work since there are constraints in which first class and second class constraints are mixed and one can hardly disentangle with each other keeping the isometry

group quantization on configuration space gauge symmetries and linear fields a new configuration space picture of a formalism of group quantization the gaq formalism is presented in the context of a previous algebraic generalization this presentation serves to make a comprehensive discussion in which other extensions of the formalism principally to incorporate gauge symmetries are developed as well both images are combined in order to analyze in a systematic manner and with complete generality the case of linear fields abelian current groups to illustrate these developments we particularize them for several fields and in particular we carry out the quantization of the abelian chern simons models over an arbitrary closed surface in detail

quantization on the virasoro group the quantization of the virasoro group is carried out by means of a previously established group approach to quantization we explicitly work out the two cocycles on the virasoro group as a preliminary step in our scheme the carrier space for all the virasoro representations is made out of polarized functions on the group manifold it is proved that this space does not contain null vector states even forc 1 although it is not irreducible the full reduction is achieved in a striaghtforward way by just taking a well defined invariant subspace c h the orbit of the enveloping algebra through the vacuum which is irreducible for any value ofc andh c h is a proper subspace of the space of polarized functions for those values ofc andh for which the kac determinant is zero we give the local version of these group representations as well as the associated classical phase space structures i e symplectic form and noether invariants

quantization as a consequence of the symmetry group an approach to geometric quantization a method is proposed to obtain the dynamics of a system which only makes use of the group law it incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics the classical or quantum character of the theory is related to the choice of the group avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems i e the classical limit the method is applied to the free particle dynamics and the harmonic oscillator

geometric dynamics of a harmonic oscillator arbitrary minimal uncertainty states and the smallest step 3 nilpotent lie group the paper presents a new method of geometric solution of a schrodinger equation by constructing an equivalent first order partial differential equation with a bigger number of variables the equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform the method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three lie group the group also known under many names e g quartic group we obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector in contrast it is shown that the well known fock segal bargmann transform and the heisenberg group require a specific fiducial vector to produce a geometric solution a technical aspect considered in this paper is that a certain modification of a coherent state transform is required although the irreducible representation of the group is square integrable modulo a subgroup the obtained dynamic is transverse to the homogeneous space

corrected maximum likelihood in non regular problems on suggere que l utilisation du produit des densites pour la fonction de vraisemblance est une mauvaise application de la methode du maximum de vraisemblance dans des cas non reguliers