the estimation of the model order in exponential families estimators are sought that achieve high exponential rate of decrease in the underestimation probability while keeping the overestimation probability exponent at a certain prescribed level it is assumed that a given integer is known to upper bound the true order some examples are given of possible applications of the proposed method in specific order estimation and hypothesis testing problems

hyperbolic geometry and moduli of real curves of genus three the moduli space of smooth real plane quartic curves consists of six connected components we prove that each of these components admits a real hyperbolic structure these connected components correspond to the six real forms of a certain hyperbolic lattice over the gaussian integers we will study this gaussian lattice in detail for the connected component that corresponds to maximal real quartic curves we obtain a more explicit description we construct a coxeter diagram that encodes the geometry of this component

nonasymptotic bounds for autoregressive time series modeling the subject of this paper is autoregressive ar modeling of a stationary gaussian discrete time process based on a finite sequence of observations the process is assumed to admit an ar representation with exponentially decaying coefficients we adopt the nonparametric minimax framework and study how well the process can be approximated by a finiteorder armodel a lower bound on the accuracy of arapproximations is derived and a nonasymptotic upper bound on the accuracy of the regularized least squares estimator is established it is shown that with a proper choice of the model order this estimator is minimax optimal in order these considerations lead also to a nonasymptotic upper bound on the mean squared error of the associated one step predictor a numerical study compares the common model selection procedures to the minimax optimal order choice 1 introduction the standard methods for estimating parameters of time series are based on the assumption that the observations come from an autoregressive ar moving average ma or mixed arma model of known orders this assumption can rarely be justified in practice and the less stringent assumption is that the time series data are observations from a linear stationary process a common approach to modeling linear stationary processes is based on an arapproximation in this framework a finite order armodel is fitted to the observations the order of the armodel should provide an optimal finite arapproximation to the process and it is usually chosen by selection procedures based on the data this nonparametric arapproach to modeling linear stationary processes has been investigated by shibata 1980 bhansali 1981 1986 an chen and hannan 1982 and hannan and kavalieris 1986 shibata 1980 considered the problem of predicting a gaussian infiniteorder arprocess by fitting a finite armodel the notion of optimality for the model selection procedure proposed by shibata 1980 is based on an asymptotic lower bound on the mean squared prediction error specifically the procedure is asymptotically efficient if it attains the lower bound asymptotically shibata 1980 also established that the final prediction error fpe akaike 1970 and the aic akaike 1974 criteria are asymptotically efficient in the above sense provided that the linear process does not degenerate to a finite order autoregression a similar result has been obtained

the moduli space of cubic fourfolds via the period map we characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space it follows then that the geometric invariant theory git compactification of the moduli space of cubic fourfolds is isomorphic to the looijenga compactification associated to this arrangement this paper builds on and is a natural continuation of our previous work on the git compactification of the moduli space of cubic fourfolds

heegner divisors in the moduli space of genus three curves s kondo used periods of k3 surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6 ball in this paper we study heegner divisors in the ball quotient given by arithmetically defined hyperplane sections of the ball we show that the corresponding loci of genus three curves are given by hyperelliptic curves singular plane quartics and plane quartics admitting certain rational splitting curves

on the stability of bilinear constant velocity pressure finite elements an analysis of the babuska stability of bilinear constant finite element pairs for viscous flow calculations is given an unstable mode not of the checkerboard type is given for which the stability constant turns out to beo h thus the indicated spaces are not stable in general for numerical calculation

penalty approximation of stokes flow abstract the particular class of problems considered is stokes flow in which the constraint is the incompressibility condition u 0 on the velocity field u in part i we present first of all a simple constructive method for determining the spurious modes this is followed by an analysis of the discrete stability condition for the penalty method showing that this condition holds with a parameter that is o h we develop improved estimates of the dependence of the stability parameter on mesh size h in a more general proof than previously available in part ii we prove error estimates for the velocity and computed pressure approximations and use an aubin nitsche technique to get l 2 estimates for the velocity in the concluding numerical studies we examine the stability of the computed pressure and convergence rates with uniform mesh refinement for problems having data with varying degree of regularity

asymptotically efficient selection of the order by the criterion autoregressive transfer function on montre que la propriete d optimalite obtenue par shibata 1980 1981 est valable pour l approche cat parzen 1974 cat parzen 1977 et cat 2 bhansali 1985

the iterative penalty method for stokes equations using q1 p0 element abstract the iterative penalty method for stokes equations is an effective algorithm in this paper we discuss the bilinear constant velocity pressure element for the stokes equations and present the error estimates for the iterative penalty method using this element the key technique we use is the local checkerboard pattern numerical experiments are given which are consistent with our analysis

reflection groups in algebraic geometry after a brief exposition of the theory of discrete reflection groups in spherical euclidean and hyperbolic geometry as well as their analogs in complex spaces we present a survey of appearances of these groups in various areas of algebraic geometry

estimated wold representation and spectral density driven bootstrap for time series the second order dependence structure of purely non deterministic stationary processes is described by the coefficients of the famous wold representation these coefficients can be obtained by factorizing the spectral density of the process this relationship together with some spectral density estimator is used to obtain consistent estimators of these coefficients a spectral density driven bootstrap for time series is then developed which uses the entire sequence of estimated moving average coefficients together with appropriately generated pseudoinnovations to obtain a bootstrap pseudo time series it is shown that if the underlying process is linear and if the pseudoinnovations are generated by means of an independent and identically distributed wild bootstrap which mimics to the extent necessary the moment structure of the true innovations this bootstrap proposal asymptotically works for a wide range of statistics the relationships of the proposed bootstrap procedure to some other bootstrap procedures including the auto regressive sieve bootstrap are discussed it is shown that the latter is a special case of the spectral density driven bootstrap if a parametric auto regressive spectral density estimator is used simulations investigate the performance of the new bootstrap procedure in finite sample situations furthermore a real life data example is presented

estimation and inference by compact coding summary the systematic variation within a set of data as represented by a usual statistical model may be used to encode the data in a more compact form than would be possible if they were considered to be purely random the encoded form has two parts the first states the inferred estimates of the unknown parameters in the model the second states the data using an optimal code based on the data probability distribution implied by those parameter estimates choosing the model and the estimates that give the most compact coding leads to an interesting general inference procedure in its strict form it has great generality and several nice properties but is computationally infeasible an approximate form is developed and its relation to other methods is explored

k3 surfaces with non symplectic automorphisms of prime order in this paper we present the classification of non symplectic automorphisms of prime order on k3 surfaces i e we describe the topological structure of their fixed locus and determine their invariant lattice in cohomology we provide new results for automorphisms of order 5 and 7 and alternative proofs for higher orders moreover for any prime p we identify the irreducible components of the moduli space of k3 surfaces with a non symplectic automorphism of order p

spectral expressions of information measures of gaussian time series and their relation to aic and cat interrelations among the spectral expressions of the information measures of kullback leibler 1959 and renyi 1961 for discrimination between two stationary gaussian time series are discussed the spectral expression of fisher s information rate matrix is also treated as well as two intuitively acceptable discrimination functions it is shown that all of them are equivalent except for scalar multiplication and are expressed by fisher s information rate matrix in the sense of their second order taylor series approximation finally a relation between two criteria for order determination of models for time series data namely h akaike s 1974 information criterion aic and the criterion of autoregressive transfer functions cat is discussed in connection with these spectral expressions

persistent identification of systems with unmodeled dynamics and exogenous disturbances in this paper a novel framework of system identification is introduced to capture the hybrid features of systems subject to both deterministic unmodeled dynamics and stochastic observation disturbances using the concepts of persistent identification control oriented system modeling and stochastic analysis we investigate the central issues of irreducible identification errors and time complexity in such identification problems upper and lower bounds on errors and speed of persistent identification are obtained the error bounds are expressed as functions of observation lengths sizes of unmodeled dynamics and probability distributions of disturbances asymptotic normality and complexity lower bounds are investigated when periodic inputs and ls estimation are applied generic features of asymptotic normality are further explored to extend the asymptotic lower bounds to a wider range of signals and identification mappings

degeneration of surfaces with trivial canonical bundle the object of this note is to prove the theorem stated below the main idea of the proof and much of the detail are already to be found in v i kulikov s important paper k our contribution to the proof and the differences with kulikov s paper are discussed later in this introduction theorem let wl x d be a semistable degeneration of surfaces such that

predictive inference sufficiency entropy and an asymptotic likelihood principle summary the objective of inferring stochastic models from a set of data is to obtain the best description by using a probability model of the statistical behaviour of future samples of the process a conceptual repeated sampling experiment is considered for evaluating a predictive distribution used to describe such future observations and leads to an asymptotic likelihood principle considerations of likelihood and sufficiency lead to the use of entropy or the kullback leibler information as the natural measure of approximation to the actual distribution by a predictive distribution in repeated samples this gives a small sample justification for the use of entropy for evaluating parameter estimation as well as model order and structure determination procedures

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

on period maps that are open embeddings for certain complex projective manifolds such as k3 surfaces and their higher dimensional analogues the complex symplectic projective manifolds the period map takes values in a locally symmetric variety of type iv it is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric divisor we explain that phenomenon and get our hands on the complementary divisor in terms of geometric data

a compactification of open varieties in this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors this is done by showing that blowing up along an arrangement of subvarieties can be carried out important examples such as ulyanov s configuration spaces and complements of arrangements of linear subspaces in projective spaces etc are covered intersection ring and nonrecursive hodge polynomials are computed furthermore some general structures arising from the blowup process are also described and studied

on the moduli space of pairs consisting of a cubic threefold and a hyperplane abstract we study the moduli space of pairs x h consisting of a cubic threefold x and a hyperplane h in p 4 the interest in this moduli comes from two sources the study of certain weighted hypersurfaces whose middle cohomology admit hodge structures of k3 type and on the other hand the study of the singularity o 16 the cone over a cubic surface in this paper we give a hodge theoretic construction of the moduli space of cubic pairs by relating x h to certain lattice polarized cubic fourfolds y a period map for the pairs x h is then defined using the periods of the cubic fourfolds y the main result is that the period map induces an isomorphism between a git model for the pairs x h and the baily borel compactification of some locally symmetric domain of type iv

autoregressive aided periodogram bootstrap for time series a bootstrap methodology for the periodogram of a stationary process is proposed which is based on a combination of a time domain parametric and a frequency domain nonparametric bootstrap the parametric fit is used to generate periodogram ordinates that imitate the essential features of the data and the weak dependence structure of the periodogram while a nonparametric kernel based correction is applied in order to catch features not represented by the parametric fit the asymptotic theory developed shows validity of the proposed bootstrap procedure for a large class of periodogram statistics for important classes of stochastic processes validity of the new procedure is also established for periodogram statistics not captured by existing frequency domain bootstrap methods based on independent periodogram replicates

the moduli of curves of genus six and k3 surfaces we prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type iv by giving a period map to the moduli space of some lattice polarized k3 surfaces

the period map for cubic fourfolds the period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20 we determine the image of this period map thus confirming a conjecture of hassett and give at the same time a new proof of the theorem of voisin that asserts that this period map is an open embedding an algebraic version of our main result is an identification of the algebra of sl 6 invariant polynomials on the representation space sym 3 6 with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20 we also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a vinberg dynkin diagram

autoregressive spatial spectral estimates abstract nonparametric spectral density estimates find many uses in econometrics for stationary random fields on a regular spatial lattice we propose an autoregressive nonparametric spectral density estimate that is guaranteed positive even when suitable edge effect correction is employed and is simple to compute using least squares our estimate is based on truncating a true half plane infinite autoregressive representation while also allowing the truncation length to diverge in all dimensions to avoid the potential bias due to truncation at a fixed lag length uniform consistency of the proposed estimate is established and new criteria for order selection are also suggested and studied in practical settings the asymptotic distribution of the estimate is shown to be zero mean normal and independent at fixed distinct frequencies mirroring the behaviour for time series a small monte carlo experiment examines finite sample performance technically the key to the results is the covariance structure of stationary random fields defined on regularly spaced lattices we show the covariance matrix to satisfy a generalization of the toeplitz property familiar from time series analysis

new non arithmetic complex hyperbolic lattices we produce a family of new non arithmetic lattices in mathrm pu 2 1 all previously known examples were commensurable with lattices constructed by picard mostow and deligne mostow and fell into nine commensurability classes our groups produce five new distinct commensurability classes most of the techniques are completely general and provide efficient geometric and computational tools for constructing fundamental domains for discrete groups acting on the complex hyperbolic plane

arithmetic torelli maps for cubic surfaces and threefolds it has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety we give a construction which works for cubics over an arithmetic base and in particular identifies the moduli space of cubic surfaces with an open substack of a certain moduli space of abelian varieties this answers away from the prime 2 an old question of deligne and a recent question of kudla and rapoport

penalty finite element methods for the analysis of stokesian flows a study of a class of finite element methods for the analysis of stokes problem based on the use of exterior penalty formulations is described the effects of selective reduced integration i e the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly are investigated error estimates are derived and the numerical stability of these methods as depicted by a special babuska brezzi condition is explored in some detail the results of several numerical experiments with these methods are also given

analysis of some mixed elements for the stokes problem abstract in this paper we discuss some mixed finite element methods related to the reduced integration penalty method for solving the stokes problem we prove optimal order error estimates for bilinear constant and biquadratic bilinear velocity pressure finite element solutions the result for the biquadratic bilinear element is new while that for the bilinear constant element improves the convergence analysis of johnson and pitkaranta 1982 in the degenerate case when the penalty parameter is set to be zero our results reduce to some related known results proved in by brezzi and fortin 1991 for the bilinear constant element and bercovier and pironneau 1979 for the biquadratic bilinear element our theoretical results are consistent with the numerical results reported by carey and krishnan 1982 and oden et al 1982

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem

asymptotic properties of black box identification of transfer functions the problem of estimating the transfer function of a linear stochastic system is considered the transfer function is parametrized as a black box and no given order is chosen a priori this means that the model orders may increase to infinity when the number of observed data tends to infinity the consistency and convergence properties of the resulting transfer function estimates are investigated asymptotic expressions for the variances and distributions of these estimates are also derived for the case that the model orders increase it is shown that the variance of the transfer function estimate at a certain frequency is asymptotically given by the noise to signal ratio at that frequency mulliplied by the model order to number of data points ratio