rational curves of castelnuovo mumford regularity d r 1 abstract the purpose here is to show that for r 4 and d r 4 a nondegenerate rational curve of degree d in p r has a unique d r 1 secant line if and only if x fails to be d r regular this is a next case to a result of gruson lazarsfeld and peskine which asserts that for r 3 and d r 2 a nondegenerate curve x of degree d in p r has a d r 2 secant line if and only if x fails to be d r 1 regular

linear systems cut out by quadrics on projections of varieties let x be a variety embedded in p where r is an algebraically closed field let l x k denote the linear system cut out on x by the hypersurfaces of degree k then l x k is complete for large enough k the question arises given a particular embedding of x how large does k have to be before l x k is complete in the case c is a curve the problem was studied by castelnuovo 2 who showed in particular that if cc p3 then l c k is complete if k deg c 2 see szpiro 12 for a modern proof due to gruson and more recently by lazarfeld et al 5 who have shown that if c a nondegenerate curve in p then l c k is complete for f deg c n 1 in this paper we study the following special case let x be a variety in p with ideal sheaf 7 to say that l x k is complete is equivalent to saying that the sequence

castelnuovo mumford regularity bounds for singular surfaces we prove the regularity conjecture namely eisenbud goto conjecture for a normal surface with rational gorenstein elliptic and log canonical singularities along the way we bound the regularity for a dimension zero scheme by its loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree

a vanishing theorem for log canonical pairs using inversion of adjunction we deduce from nadel s theorem a vanishing property for ideals sheaves on projective varieties a special case of which recovers a result due to bertram ein lazarsfeld this enables us to generalize to a large class of projective schemes certain bounds on castelnuovo mumford regularity previously obtained by bertram ein lazarsfeld in the smooth case and by chardin ulrich for locally complete intersection varieties with rational singularities our results are tested on several examples

multigraded castelnuovo mumford regularity we develop a multigraded variant of castelnuovo mumford regularity motivated by toric geometry we work with modules over a polynomial ring graded by a finitely generated abelian group as in the standard graded case our definition of multigraded regularity involves the vanishing of graded components of local cohomology we establish the key properties of regularity its connection with the minimal generators of a module and its behavior in exact sequences for an ideal sheaf on a simplicial toric variety x we prove that its multigraded regularity bounds the equations that cut out the associated subvariety we also provide a criterion for testing if an ample line bundle on x gives a projectively normal embedding

on the derived category of the classical godeaux surface abstract we construct an exceptional sequence of length 11 on the classical godeaux surface x which is the z 5 z quotient of the fermat quintic surface in p 3 this is the maximal possible length of such a sequence on this surface which has grothendieck group z 11 z 5 z in particular the result answers kuznetsov s nonvanishing conjecture which concerns hochschild homology of an admissible subcategory in the negative the sequence carries a symmetry when interpreted in terms of the root lattice of the simple lie algebra of type e 8 we also produce explicit nonzero objects in the right orthogonal to the exceptional sequence

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

online estimation of the geometric median in hilbert spaces nonasymptotic confidence balls estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data the collected data may be contaminated by noise so that robust location indicators such as the geometric median may be preferred to the mean in this context an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by bernoulli19 2013 18 43 this work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable hilbert spaces this new result is based on the derivation of improved l 2 rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear robbins monro algorithm

noether resolutions in dimension 2 let r k x 1 ldots x n be a polynomial ring over an infinite field k and let i subset r be a homogeneous ideal with respectto a weight vector omega omega 1 ldots omega n in mathbb z n such that dim r i d in this paper we study the minimal graded free resolution of r i as a module that we call the noether resolution of r i whenever a k x n d 1 ldots x n is a noether normalization of r i when d 2 and i is saturated we give an algorithm for obtaining this resolution that involves the computation of a minimalgr obner basis of i with respect to the weighted degree reverse lexicographic order in the particular case when r i is a 2 dimensional semigroup ring we also describe the multigraded version of this resolution in terms of the underlying semigroup whenever we have the noether resolution of r i or its multigraded version we obtain formulas for the corresponding hilbert series of r i and when i is homogeneous we obtain a formula for the castelnuovo mumford regularity of r i moreover in the more general setting that r i is a simplicial semigroup ring of any dimension we provide its macaulayfication as an application of the results for 2 dimensional semigroup rings we provide a new upper bound for the castelnuovo mumford regularity of the coordinate ring of a projective monomial curve finally we describe the multigraded noether resolution and the macaulayfication of either the coordinate ring of a projective monomial curve mathcal c l subseteq mathbb p k n associated to an arithmetic sequence or the coordinate ring of any canonical projection pi r mathcal c of mathcal c to mathbb p k n 1

dispersion operators and resistant second order functional data analysis inferences related to the second order properties of functional data as expressed by covariance structure can become unreliable when the data are non gaussian or contain unusual observations in the functional setting it is often difficult to identify atypical observations as their distinguishing characteristics can be manifold but subtle in this paper we introduce the notion of a dispersion operator investigate its use in probing the second order structure of functional data and develop a test for comparing the second order characteristics of two functional samples that is resistant to atypical observations and departures from normality the proposed test is a regularized m test based on a spectrally truncated version of the hilbert schmidt norm of a score operator defined via the dispersion operator we derive the asymptotic distribution of the test statistic investigate the behaviour of the test in a simulation study and illustrate the method on a structural biology dataset

small schemes and varieties of minimal degree we prove that if x p r is any 2 regular scheme in the sense of castelnuovo mumford then x is small this means that if l is a linear space and y l x is finite then y is linearly independent in the sense that the dimension of the linear span of y is deg y 1 the converse is true and well known for finite schemes but false in general the main result of this paper is that the converse small implies 2 regular is also true for reduced schemes algebraic sets this is proven by means of a delicate geometric analysis leading to a complete classification we show that the components of a small algebraic set are varieties of minimal degree meeting in a particularly simple way from the classification one can show that if x p r is 2 regular then so is xred and so also is the projection of x from any point of x our results extend the del pezzo bertini classification of varieties of minimal degree the charac terization of these as the varieties of regularity 2 by eisenbud goto and the construction of 2 regular square free monomial ideals by froberg 0 introduction throughout this paper we will work with projective schemes x p r over an algebraically closed field k the castelnuovo mumford regularity of x p r is a basic homological measure of the complexity of x and its embedding in p r that gives a bound for the degrees of the generators of the defining ideal ix of x and for many other invariants the only schemes of regular ity 1 are the linear spaces but no classification is known for projective schemes of regularity 2 in this paper we prove a structure theorem for reduced 2 regular schemes showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet we also show that the reduced structure on any 2 regular scheme is 2 regular and thus we obtain a complete description of the reduced structures on 2 regular schemes since a high veronese re embedding of any zero dimensional scheme is 2 regular one cannot hope to characterize the isomorphism types of all 2 regular nonreduced schemes before stating our results we review some basic notions for any subscheme x p r we write span x for the smallest linear subspace of p r containing x recall that every variety reduced irreducible scheme x p r satisfies the condition deg x 1 codim x span x

projective curves of degree codimension 2 in this article we study nondegenerate projective curves x subset mathbb p d 1 of degree d which are not arithmetically cohen macaulay note that x pi p widetilde x for a rational normal curve widetilde x subset mathbb p d and a point p in mathbb p d setminus widetilde x 2 our main result is about the relation between the geometric properties of x and the position of p with respect to widetilde x we show that the graded betti numbers of x are uniquely determined by the rank hbox rk widetilde x p of p with respect to widetilde x in particular x satisfies property n 2 p if and only if p leq quad rk widetilde x p 3 therefore property n 2 p of x is controlled by quad rk widetilde x p and conversely quad rk widetilde x p can be read off from the minimal free resolution of x this result provides a non linearly normal example for which the converse to theorem 1 1 in eisenbud et al compositio math 141 1460 1478 2005 holds also our result implies that for nondegenerate projective curves x subset mathbb p d 1 of degree d which are not arithmetically cohen macaulay there are exactly lfloor frac d 2 2 rfloor distinct betti tables

a combinatorial proof of the eisenbud goto conjecture for monomial curves and some simplicial semigroup rings abstract we will give a pure combinatorial proof of the eisenbud goto conjecture for arbitrary monomial curves moreover we will show that the conjecture holds for certain simplicial affine semigroup rings

the regularity of tor and graded betti numbers let s k x1 xn let a b be finitely generated graded s modules and let m x1 xn s we give bounds for the regularity of the local cohomology of tork a b in terms of the graded betti numbers of a and b under the assumption that dim tor1 a b 1 we apply the results to syzygies grobner bases products and powers of ideals and to the relationship of the rees and symmetric algebras for example we show that any homogeneous linearly presented m primary ideal has some power equal to a power of m and if the first n 1 2 steps of the resolution of i are linear then i 2 is a power of m

a bound on the castelnuovo mumford regularity for curves recall that a projective curve in bbb p r with ideal sheaf cal i is said to be n regular if h i bbb p r cal i otimes cal o bbb p r n i 0 for every integer i 0 and that in this case it is cut out scheme theoretically by equations of degree at most n the purpose here is to show that an irreducible reduced projective curve of degree d and large arithmetic genus p a satisfies a smaller regularity bound than the optimal one d r 2 for example if p a geq r 2 then a curve is d 2r 4 regular unless it is embedded by a complete linear system of degree 2p a 2

the l 1 method for robust nonparametric regression abstract consider the problem of estimating the regression function underlying a set of data that is contaminated by a heavy tailed error distribution the two standard approaches to such a problem are each flawed robust parametric least squares is appropriate only if there is good reason to believe that the underlying function has some particular form whereas most nonparametric regression methods are asymptotically equivalent to kernel regression methods which are not resistant against outliers existing algorithms for robustifying nonparametric regression procedures use either nonlinear optimization of an influence function or iterative solution of local polynomial fitting using reweighted least squares neither of these two approaches combines computational ease with asymptotic theoretical results furthermore application of the robust procedure has been limited almost exclusively to the case of a single explanatory variable with the response variable in this article a new hybrid method is propose

formulas for the multiplicity of graded algebras let r be a standard graded noetherian algebra over an artinian local ring moti vated by the work of achilles and manaresi in intersection theory we first express the multiplicity of r by means of local j multiplicities of various hyperplane sections when applied to a homogeneous inclusion ab of standard graded noetherian algebras over an artinian local ring this formula yields the multiplicity of a in terms of that of b and of local j multiplicities of hyperplane sections along proj b our formulas can be used to find the multiplicity of special fiber rings and to obtain the degree of dual va rieties for any hypersurface in particular it gives a generalization of teissier s plucker formula to hypersurfaces with non isolated singularities our work generalizes results by simis ulrich and vasconcelos on homogeneous embeddings of graded algebras

some results on asymptotic regularity of ideal sheaves abstract let i be an ideal sheaf on p n defining a subscheme x associated to i there are two elementary invariants the invariant s which measures the positivity of i and the minimal number d such that i d is generated by its global sections it is now clear that the asymptotic behavior of reg i t is governed by s but usually not linear in this paper we first describe the linear behavior of the asymptotic regularity by showing that if s d i e s reaches its maximal value then for t large enough reg i t d t e for some positive constant e we then turn to concrete geometric settings to study the asymptotic regularity of i in the case that x is a nonsingular variety embedded by a very ample adjoint line bundle our approach also gives regularity bounds for i t once we know reg i and assume that x is a local complete intersection

automorphism groups of some ag codes in this correspondence we show that in many cases the automorphism group of a curve and the permutation automorphism group of a corresponding ag code are the same this generalizes a result of wesemeyer beyond the case of planar curves

curves of degree r 2 in pr cohomological geometric and homological aspects abstract let c p rk be a non degenerate projective curve of degree r 2 where r 3 by means of the hartshorne rao module of c we distinguish 4 different possible cases and an exceptional case which only appears if r 3 in any case c is obtained either by means of an embedding of an arbitrary smooth curve c 0 of genus 2 or by projecting an elliptic normal curve from a point or by projecting a rational normal curve from a line finally the possible minimal free resolutions of the homogeneous coordinate ring of c are studied

a castelnuovo mumford regularity bound for scrolls abstract let x p r be a scroll of codimension e and degree d over a smooth projective curve of genus g the purpose of this paper is to prove a linear castelnuovo mumford regularity bound that reg x d e 1 g e 1 this bound works over an algebraically closed field of arbitrary characteristic

robust function on function regression functional linear regression is a widely used approach to model functional responses with respect to functional inputs however classical functional linear regression models can be severely affect

approximate interval estimation of the difference in binomial parameters correction for skewness and extension to multiple tables recently beal 1987 biometrics 43 941 950 found mee s modification of anbar s approximate interval estimation for the difference in binomial parameters to be a good choice in small sample sizes as this method can be derived from the score theory of bartlett it is easily corrected for skewness exact numerical evaluation shows that this correction is not as important for this case as for the ratio of binomial parameters gart and nam 1988 biometrics 44 323 338 the score theory is also used to extend this method to the stratified or multiple table case thus good approximate interval estimates for differences ratios and odds ratios of binomial parameters can all be derived from the same general theory

alternatives to the median absolute deviation abstract in robust estimation one frequently needs an initial or auxiliary estimate of scale for this one usually takes the median absolute deviation mad n 1 4826 med xi med j x j because it has a simple explicit formula needs little computation time and is very robust as witnessed by its bounded influence function and its 50 breakdown point but there is still room for improvement in two areas the fact that mad n is aimed at symmetric distributions and its low 37 gaussian efficiency in this article we set out to construct explicit and 50 breakdown scale estimators that are more efficient we consider the estimator sn 1 1926 med med j xi xj and the estimator qn given by the 25 quantile of the distances xi x j i j note that sn and qn do not need any location estimate both sn and qn can be computed using o n log n time and o n storage the gaussian efficiency of sn is 58 whereas qn attains 82 we study sn and qn by means of their influence functions their b

properties of principal component methods for functional and longitudinal data analysis the use of principal component methods to analyze functional data is appropriate in a wide range of different settings in studies of functional data analysis it has often been assumed that a sample of random functions is observed precisely in the continuum and without noise while this has been the traditional setting for functional data analysis in the context of longitudinal data analysis a random function typically represents a patient or subject who is observed at only a small number of randomly distributed points with nonnegligible measurement error nevertheless essentially the same methods can be used in both these cases as well as in the vast number of settings that lie between them how is performance affected by the sampling plan in this paper we answer that question we show that if there is a sample of n functions or subjects then estimation of eigenvalues is a semiparametric problem with root n consistent estimators even if only a few observations are made of each function and if each observation is encumbered by noise however estimation of eigenfunctions becomes a nonparametric problem when observations are sparse the optimal convergence rates in this case are those which pertain to more familiar function estimation settings we also describe the effects of sampling at regularly spaced points as opposed to random points in particular it is shown that there are often advantages in sampling randomly however even in the case of noisy data there is a threshold sampling rate depending on the number of functions treated above which the rate of sampling either randomly or regularly has negligible impact on estimator performance no matter whether eigenfunctions or eigenvectors are being estimated

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

asymptotic confidence intervals for the difference between two binomial parameters for use with small samples some asymptotically based confidence intervals for the difference between the binomial parameters from two binomial populations are described five of these including the usual simple interval are extensively evaluated in terms of actual confidence level and interval length a new simple interval behaves much better than the usual interval as do some recently proposed iteratively computed intervals

on the complexity of the integral closure the computation of the integral closure of an affine ring has been the focus of several modern algorithms we will treat here one related problem the number of generators the integral closure of an affine ring may require this number and the degrees of the generators in the graded case are major measures of cost of the computation we prove several polynomial type bounds for various kinds of algebras and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus

bias robust estimation of scale in this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location scale family we define the class of m estimates of scale with general location and show that under certain regularity assumptions these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution and with respect to the m estimate defining score function chi we establish expressions for the maximum asymptotic bias of m estimates of scale over the contamination neighborhood as a function of the fraction of contamination using these expressions we construct asymptotically min max bias robust estimates of scale in particular we show that a scaled version of the madm median of absolute residuals about the median is approximately min max bias robust within the class of huber s proposal 2 joint estimates of location and scale we also consider the larger class of m estimates of scale with general location and show that a scaled version of the shorth the shortest half of the data is approximately min max bias robust in this class finally we present the results of a monte carlo study showing that the shorth has attractive finite sample size mean squared error properties for contaminated gaussian data

kontsevich spaces of rational curves on fano hypersurfaces we investigate the spaces of rational curves on a general hypersurface in particular we show that for a general degree d hypersurface in pn with n d 2 the space m0 0 x e of degree e kontsevich stable maps from a rational curve to x is an irreducible local complete intersection stack of dimension e n d 1 n 4 this resolves all but one case of a conjecture of coskun harris and starr

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

hypersurfaces cutting out a projective variety let x be a nondegenerate projective variety of degree d and codimension e in a projective space p n defined over an algebraically closed field we study the following two problems is the length of the intersection of x and a line l in p n at most d e 1 if l g x is the scheme theoretic intersection of all hypersurfaces of degree at most d e 1 containing x equal to x to study the second problem we look at the locus of points from which x is projected nonbirationally

penalized spline models for functional principal component analysis summary we propose an iterative estimation procedure for performing functional principal component analysis the procedure aims at functional or longitudinal data where the repeated measurements from the same subject are correlated an increasingly popular smoothing approach penalized spline regression is used to represent the mean function this allows straightforward incorporation of covariates and simple implementation of approximate inference procedures for coefficients for the handling of the within subject correlation we develop an iterative procedure which reduces the dependence between the repeated measurements that are made for the same subject the resulting data after iteration are theoretically shown to be asymptotically equivalent in probability to a set of independent data this suggests that the general theory of penalized spline regression that has been developed for independent data can also be applied to functional data the effectiveness of the proposed procedure is demonstrated via a simulation study and an application to yeast cell cycle gene expression data

regularity of prime ideals we answer several natural questions which arise from a recent paper of mccullough and peeva providing counterexamples to the eisenbud goto regularity conjecture we give counterexamples using rees algebras and also construct counterexamples that do not rely on the mayr meyer construction furthermore examples of prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal first syzygy can be made arbitrarily large are given using a result of ananyan hochster we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

comparative analysis of two rates in this paper we examine comparative analysis of rates with a view to each of the usual comparative parameters rate difference rd rate ratio rr and odds ratio or and with particular reference to first principles for rd and rr we show the prevailing statistical practices to be rather poor we stress the need for restricted estimation of variance in the chi square function underlying interval estimation and also point estimation and hypothesis testing for rr analysis we propose a chi square formulation analogous to that for rd and thus one which obviates the present practice of log transformation and its associated use of taylor series approximation of the variance as for or analysis we emphasize that the chi square function introduced by cornfield for unstratified data and extended by gart to the case of stratified analysis is based on the efficient score and thus embodies its optimality properties we provide simulation results to evince the better performance of the proposed parameter constrained procedures over the traditional ones

rational points of bounded height on projective surfaces we give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces

functional data analysis for density functions by transformation to a hilbert space functional data that are nonnegative and have a constrained integral can be considered as samples of one dimensional density functions such data are ubiquitous due to the inherent constraints densities do not live in a vector space and therefore commonly used hilbert space based methods of functional data analysis are not applicable to address this problem we introduce a transformation approach mapping probability densities to a hilbert space of functions through a continuous and invertible map basic methods of functional data analysis such as the construction of functional modes of variation functional regression or classification are then implemented by using representations of the densities in this linear space representations of the densities themselves are obtained by applying the inverse map from the linear functional space to the density space transformations of interest include log quantile density and log hazard transformations among others rates of convergence are derived for the representations that are obtained for a general class of transformations under certain structural properties if the subject specific densities need to be estimated from data these rates correspond to the optimal rates of convergence for density estimation the proposed methods are illustrated through simulations and applications in brain imaging

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

free knot spline smoothing for functional data summary the paper introduces free knot regression spline estimators for the mean and the variance components of a sample of curves the asymptotic distribution of the mean estimator is derived and asymptotic confidence bands are constructed a comparative simulation study shows that free knot splines estimate salient features of the functions such as sharp peaks more accurately than smoothing splines this adaptive behaviour is also illustrated by an analysis of weather data

s estimators for functional principal component analysis principal component analysis is a widely used technique that provides an optimal lower dimensional approximation to multivariate or functional datasets these approximations can be very useful in identifying potential outliers among high dimensional or functional observations in this article we propose a new class of estimators for principal components based on robust scale estimators for a fixed dimension q we robustly estimate the q dimensional linear space that provides the best prediction for the data in the sense of minimizing the sum of robust scale estimators of the coordinates of the residuals we also study an extension to the infinite dimensional case our method is consistent for elliptical random vectors and is fisher consistent for elliptically distributed random elements on arbitrary hilbert spaces numerical experiments show that our proposal is highly competitive when compared with other methods we illustrate our approach on a real dataset where the robust estimator discovers atypi

on estimating the difference between two probabilities with special reference to clinical trials summary the problem of estimating the difference between the probabilities of success for two binomial experiments is considered confidence intervals based on large sample theory are constructed and some of their properties are compared with those of a simple technique based on exact distributions it is shown that the former method is superior to the latter both theoretically and practically the estimation of the magnitude of the difference between the response rates to two treatments is an important aspect of the analysis of clinical trials frequently the clinical investigator is interested in the size of the difference as well as in its significance the odds ratio is the measure of the disparity between treatments that is most frequently discussed in the statistical literature although statistically the odds ratio is a natural parameter the simple difference a p2 p1 seems to be more meaningful to the clinician in most experimental situations the two proportions to be compared can be assumed to come from two independent samples under these circumstances a number of techniques for estimating a have been proposed some of the methods use large sample theory and others are based on exact usually conditional distributions the simplest large sample method is to estimate the sampling error of the difference between the two sample proportions and to use the normal approximation see for example fleiss 1981 small sample methods usually require numerical solutions to nontrivial equations for a recent discussion of small sample methods see the paper by santner and snell 1980 and the references cited therein buhrman 1977 has shown that by randomizing the patients between the two treatments in a certain way one can obtain exact confidence intervals for a his method requires a few trivial calculations and ordinary binomial tables the method of randomization is a simple nonbalanced random allocation in this communication some of the features of buhrman s method are examined and compared with the ordinary large sample method in 2 the large sample method procedure a is described briefly buhrman s method procedure b is described in 3 a comparison of the two methods is presented in 4

some effects of property n p on the higher normality and defining equations of nonlinearly normal varieties for a smooth projective variety x p embedded by the complete linear system property np has been studied for a long time 5 11 12 10 etc on the other hand castelnuovo mumford regularity conjecture and related problems have been focused for a projective variety which is not necessarily linearly normal 2 13 15 17 20 etc this paper aims to explain the influence of property np on higher normality and defining equations of a smooth variety embedded by sub linear system also we prove a claim about property np of surface scrolls which is a generalization of green s work in 11 about property np for curves

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

a bound on the geometric genus of projective varieties verifying certain flag conditions fix integers n r s1 sl and let s n r s1 sl be the set of all integral projective and nondegenerate varieties v of degree s1 and dimension n in the projective space pr such that for all i 2 l v does not lie on any variety of dimension n i 1 and degree sl we determine an upper bound gh n r s1 sl depending only on n r s1 sl for the number g n r s1 sl max pg v v s n r s1 sl where pg v denotes the geometric genus of v in case n 1 and l 2 the study of an upper bound for the geometric genus has a quite long history and for n 1 l 2 and s2 r n it has been introduced by harris we exhibit sharp results for particular ranges of our numerical data n r s1 sl for instance we extend halphen s theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension n in pn 3 as the smooth varieties of maximal geometric genus with respect to appropriate flag condition this result applies to smooth surfaces in p5 next we discuss how far gh n r s1 sl is from g n r s1 sl and show a sort of lifting theorem which states that at least in certain cases the varieties v s n r s1 sl of maximal geometric genus g n r s1 sl must in fact lie on a flag such as v v n s1 v n 1 s2 v n l 1 sl p r where v j s denotes a subvariety of pr of degree s and dimension j we also discuss further generalizations of flag conditions and finally we deduce some bounds for castelnuovo s regularity of varieties verifying flag conditions 0 introduction fix integers n r s1 sl such that 1 n r 2 2 l r n and let s n r s1 sl be the set of all integral projective and nondegenerate varieties v of degree s1 and dimension n in the projective space p such that for all i 2 l v does not lie on any variety of dimension n i 1 and degree sl we determine an upper bound g n r s1 sl depending only on n r s1 sl for the number g n r s1 sl max pg v v s n r s1 sl 0 1 received by the editors october 23 1995 1991 mathematics subject classification primary 14j99 14m07 14m10 secondary 14f17

three themes of syzygies we present three exciting themes of syzygies where major progress was made recently boij soderberg theory stillman s question and syzygies over complete intersections

robust functional principal components a projection pursuit approach in many situations data are recorded over a period of time and may be regarded as realizations of a stochastic process in this paper robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting our approach combines robust projection pursuit with different smoothing methods consistency of the estimators are shown under mild assumptions the performance of the classical and robust procedures are compared in a simulation study under different contamination schemes 1 introduction analogous to classical principal components analysis pca the projection pursuit approach to robust pca is based on finding projections of the data which have maximal dispersion instead of using the variance as a measure of dispersion a robust scale estimator sn is used for the maximization problem this approach was introduced by li and chen 1985 who proposed estimators based on maximizing or minimizing a robust scale in this way given a sample xi r d 1 i n the first robust principal component vector is defined as

testing a specified difference between proportions testing of a specified difference between binomial proportions including the test of equality may be formulated to preserve the asymptotic power intended for pairs of simple hypotheses related under a specified family of alternatives such as simple shift alternatives this formulation simply matches parameterizations for estimated and true response rates in terms of corresponding specifications for the offsets realizing the specified difference from a nuisance parameter the parameterization is then employed in estimating the variance of the difference in proportions an unbiased estimator of the appropriate variance is provided and asymptotically unbiased hypothesis testing is verified