gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

hodge integrals and degenerate contributions hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the gromov witten theory of 3 folds in the calabi yau case the contributions are compared to related m theoretic calculations in the fano case the contributions suggest new integrality conditions

real doubles of hurwitz frobenius manifolds new frobenius structures on hurwitz spaces are found a hurwitz space is considered as a real manifold therefore the number of coordinates is twice as large as the number of coordinates on hurwitz frobenius manifolds of dubrovin simple branch points of a ramified covering and their complex conjugates play the role of canonical coordinates on the constructed frobenius manifolds corresponding solutions to wdvv equations and g functions are obtained

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

the chen ruan cohomology of moduli of curves of genus 2 with marked points in this work we describe the chen ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points in the first half of the paper we compute the additive structure of the

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

on the benney hierarchy free energy string equation and quantization the bi hamiltonian structure of the benney hierarchy is revisited we show that the compatibility condition of the poisson brackets provides the genus zero free energy of a topological field theory coupled to two dimensional gravity we calculate the correlation functions via the landau ginzburg formulation and derive the string equation based on the twistor construction moreover by using the approach of dubrovin and zhang we compute the genus one correction of the poisson brackets and compare them with the oevel strampp brackets of the kaup broer hierarchy

intersection numbers and rank one cohomological field theories in genus one we obtain a simple recursive presentation of the tautological and classes on the moduli space of curves in genus 0 and 1 in terms of boundary strata graphs we derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials as an application we describe the moduli space of normalized restricted rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products our results simplify and generalize those of kaufmann manin and zagier

on wielandt s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix a relatively obscure eigenvalue inequality due to wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix the asymptotic distributions are obtained under a fairly general setting an application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given 1 introduction and summary the derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis although the main idea behind most of the derivations is quite basic i e the expansion of the sample roots about the population roots the derivations themselves are often quite involved these complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem one of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices the method is based upon a relatively obscure eigenvalue inequality

enumeration of real curves in mathbb c mathbb p 2n 1 and a witten dijkgraaf verlinde verlinde relation for real gromov witten invariants

descendant invariants and characteristic numbers on a stack of stable maps the cotangent line classes are modified by subtracting certain boundary divisors these modified cotangent line classes are compatible with forgetful morphisms and are well suited to enumerative geometry tangency conditions allow simple expressions in terms of modified cotangent line classes topological recursion relations are established among their top products in genus 0 yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety in higher genus the obtained numbers are only virtual due to contributions from spurious components of the space of maps for the projective plane the necessary corrections are determined in genus 1 and 2 to give the characteristic numbers in these cases

a change of coordinates on the large phase space of quantum cohomology the gromov witten invariants of a smooth projective variety v when twisted by the tautological classes on the moduli space of stable maps give rise to a family of cohomological field theories and endow the base of the family with coordinates we prove that the potential functions associated to the tautological classes the large phase space and the classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions our result is a generalization of the work of manin zograf who studied the case where v is a point we utilize this change of variables to derive the topological recursion relations associated to the classes from those associated to the classes

a genus 3 topological recursion relation in this paper we give a new genus 3 topological recursion relation for gromov witten invariants of compact symplectic manifolds this formula also applies to intersection numbers on moduli spaces of spin curves a by product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1 pointed genus 3 stable curves

diagnostics for assessing regression models abstract we concern ourselves with diagnostics for checking the overall and local goodness of fit of a model s x used in the regression of y on x u 0 1 d the model for s x is a functional form that depends on a finite number of unknown parameters two statistics and w are proposed that measure the level of agreement between the model fit to the data and the nonparametric kernel estimator on m preselected points in u conditions are given under which and w are asymptotically equivalent both of these statistics measure overall lack of fit and are related to the deviance their asymptotic distribution under the null model and under local alternatives is derived this work is motivated by the local mean deviance plot of landwehr pregibon and shoemaker for assessing overall lack of fit in logistic regression their plot is summarized by our test statistics and is extended to other likelihood based regressions of y on x

the structure of the tautological ring in genus one we prove getzler s claims about the cohomology of the moduli space of stable curves of genus one that is that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the witten dijkgraaf verlinde verlinde wdvv relation and getzler s relation in particular the even cohomology ring is isomorphic to the tautological ring

semiparametric estimates of the relation between weather and electricity sales abstract a nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data this accommodates introduction of covariates timing adjustments due to the actual billing schedules and serial correlation the procedure is an extension of smoothing splines with the smoothness parameter estimated from minimization of the generalized cross validation criterion introduced by craven and wahba 1979 estimates are presented for residential sales for four electric utilities and are compared with models that represent the weather using only heating and cooling degree days or with piecewise linear splines

the algebraic curve of classical superstrings on ads5 s5 we investigate the monodromy of the lax connection for classical iib superstrings on ads5 s5 for any solution of the equations of motion we derive a spectral curve of degree 4 4 the curve consists purely of conserved quantities all gauge degrees of freedom have been eliminated in this form the most relevant quantities of the solution such as its energy can be expressed through certain holomorphic integrals on the curve this allows for a classification of finite gap solutions analogous to the general solution of strings in flat space the role of fermions in the context of the algebraic curve is clarified finally we derive a set of integral equations which reformulates the algebraic curve as a riemann hilbert problem they agree with the planar one loop open image in new window supersymmetric gauge theory proving the complete agreement of spectra in this approximation

spline smoothing in a partly linear model on donne des estimations qui minimisent la somme des sommes residuelles des carres et une penalite de difficulte

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

bandwidth choice for average derivative estimation abstract the average derivative is the expected value of the derivative of a regression function kernel methods have been proposed as a means of estimating this quantity the problem of bandwidth selection for these kernel estimators is addressed here asymptotic representations are found for the variance and squared bias these are compared with each other to find an insightful representation for a bandwidth optimizing terms of lower order than n 1 it is interesting that for dimensions greater than 1 negative kernels have to be used to prevent domination of bias terms in the asymptotic expression of the mean squared error the extent to which the theoretical conclusions apply in practice is investigated in an economical example related to the so called law of demand

the classical exchange algebra of a green schwarz sigma model on supercoset target space with z4m grading we investigate the classical exchange algebra of the monodromy matrix for a green schwarz sigma model on supercoset target space with z4m grading by using a first order hamiltonian formulation and by adding to the lax connection terms proportional to constraints this enables us to show that the conserved charges of the theory are in involution in the poisson bracket sense our calculation is based on a general world sheet metric taking a particular case of m 1 and a particular choice of supergroup our results coincide with those of the green schwarz superstring theory in ads5 s5 background obtained by magro j high energy phys 0901 021 2009 10 1088 1126 6708 2009 01 021

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

from zwiebach invariants to getzler relation we introduce the notion of zwiebach invariants that generalize gromov witten invariants and homotopical algebra structures we outline the induction procedure that induces the structure of zwiebach invariants on the sub bicomplex that gives the structure of gromov witten invariants on sub bicomplex with zero differentials we propose to treat hodge dgbv with 1 12 axiom as the simplest set of zwiebach invariants and explicitly prove that it induces wdvv and getzler equations in genera 0 and 1 respectively

relative virtual localization and vanishing of tautological classes on moduli spaces of curves we prove a localization formula for the moduli space of stable relative maps as an application we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i g 1 genus 0 components as consequences we prove and generalize various conjectures and theorems about various moduli spaces of curves due to getzler ionel faber looijenga pandharipande diaz and others this theorem appears to be the geometric content behind these results the rest is straightforward graph combinatorics the theorem also suggests the importance of the stratification of the moduli space by number of rational components

the semi classical approximation for modular operads we study the contribution of one loop graphs the semi classical expansion problem in the setting of modular operads as an application we calculate the betti numbers of the deligne mumford knudsen moduli spaces of stable curves of genus 1 with n marked smooth points

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

the gorenstein conjecture fails for the tautological ring of mathcal overline m _ 2 n we prove that for n equal to at least one of the integers 8 12 16 20 the tautological ring r bullet overline mathcal m 2 n is not gorenstein in fact our n equals the smallest integer such that there is a non tautological cohomology class of even degree on overline mathcal m 2 n by work of graber and pandharipande such a class exists on overline mathcal m 2 20 and we present some evidence indicating that n is in fact 20

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

testing the rank and definiteness of estimated matrices with applications to factor state space and arma models abstract consider any consistent asymptotically normal estimate of an arbitrary rectangular or square matrix a this article derives an explicit test for the rank of a and a related test of semi definiteness of a potential applications include testing for identification of structural models testing for the number of state variables in state space models including tests for the order of autoregressive moving average arma processes consumer demand analysis applications and testing for the number of factors in factor analysis and related procedures the test is based on the gaussian elimination lower diagonal upper triangular ldu decomposition the test is illustrated with an empirical application to testing the order of arma processes

the method of finite gap integration in classical and semi classical string theory in view of proving the ads cft correspondence one day a deeper understanding of string theory on certain curved backgrounds such as ads5 s5 is required in this review we make a step in this direction by focusing on it was discovered in recent years that string theory on ads5 s5 admits a lax formulation however the complete statement of integrability requires not only the existence of a lax formulation but also that the resulting integrals of motion are in pairwise involution this idea is central to the first part of this review exploiting this integrability we apply algebro geometric methods to string theory on and obtain the general finite gap solution the construction is based on an invariant algebraic curve previously found in the ads5 s5 case however encoding the dynamics of the solution requires specification of additional marked points by restricting the symplectic structure of the string to these algebro geometric data we derive the action angle variables of the system we then perform a first principle semiclassical quantization of string theory on as a toy model for strings on ads5 s5 the result is exactly what one expects from the dual gauge theory perspective namely the underlying algebraic curve discretizes in a natural way we also derive a general formula for the fluctuation energies around the generic finite gap solution the ideas used can be generalized to ads5 s5

tau structure for the double ramification hierarchies in this paper we continue the study of the double ramification hierarchy of buryak commun math phys 336 3 1085 1107 2015 after showing that the dr hierarchy satisfies tau symmetry we define its partition function as the logarithm of the tau function of the string solution and show that it satisfies various properties string dilaton and divisor equations plus some important degree constraints we then formulate a stronger version of the conjecture from buryak 2015 for any semisimple cohomological field theory the dubrovin zhang and double ramification hierarchies are related by a normal i e preserving the tau structure dubrovin et al in adv math 293 382 435 2016 miura transformation which we completely identify in terms of the partition function of the cohft in fact using only the partition functions the conjecture can be formulated even in the non semisimple case where the dubrovin zhang hierarchy is not defined we then prove this conjecture for various cohfts trivial cohft hodge class gromov witten theory of mathbb cp 1 3 4 and 5 spin classes and in genus 1 for any semisimple cohft finally we prove that the higher genus part of the dr hierarchy is basically trivial for the gromov witten theory of smooth varieties with non positive first chern class and their analogue in fan jarvis ruan witten quantum singularity theory fan et al in ann math 178 1 1 106 2013

bihamiltonian hierarchies in 2d topological field theory at one loop approximation we compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2d topological field theory the bihamiltonian structure of the hierarchy is given by a classical w algebra we compute the central charge of this algebra we also express the generating function of elliptic gromov witten invariants via tau function of the isomonodromy deformation problem arising in the theory of wdvv equations of associativity

graphical methods for assessing logistic regression models abstract in ordinary linear regression graphical diagnostic displays can be very useful for detecting and examining anomalous features in the fit of a model to data for logistic regression models the discreteness of binary data makes it difficult to interpret such displays modifications and extensions of linear model displays lead to three methods for diagnostic checking of logistic regression models local mean deviance plots are useful for detecting overall lack of fit empirical probability plots help point out isolated departures from the fitted model partial residual plots when smoothed to show underlying structure help identify specific causes of lack of fit these methods are illustrated through the analyses of simulated and real data

algebraic curve for the so 6 sector of ads cft we construct the general algebraic curve of degree four solving the classical sigma model on s 5 up to two loops it coincides with the algebraic curve for the dual sector of scalar operators in open image in new window sym also constructed here

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

the κ ring of the moduli of curves of compact type the subalgebra of the tautological ring of the moduli of curves of compact type generated by the classes is studied in all genera relations constructed via the virtual geometry of the moduli of stable quotients are used to obtain minimal sets of generators bases and betti numbers of the rings are computed a universality property relating the higher genus rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps the g formula for hodge integrals arises as the simplest consequence