inferences on multivariate measures of interclass and intraclass correlations in familial data summary multivariate measures of interclass and intraclass correlations are introduced to assess the degree of resemblance between family members with respect to more than one characteristic unified estimators for the multivariate measures are proposed as the eigenvalues of certain random matrices constructed by the matrices of the weighted sums of squares and products of observations asymptotic distributions of the proposed estimators are derived under the assumption of multivariate normality the asymptotic results produce approximate confidence intervals and test procedures in a simple way

estimation of interclass correlations in familial data summary the maximum likelihood estimates of the parent child interclass correlation and other parameters from familial data when the families have unequal numbers of offspring is given also given are two alternative sets of estimates which are easier to compute than the maximum likelihood estimates using these estimates a test for the interclass correlation to be zero requires only the ordinary student s t distribution similarly other tests use only known distributions in the analysis of familial data the degree of mother sib resemblance is measured by the interclass correlation the correlation between two different types of family member several estimates of this correlation have been proposed in the literature see rosner donner hennekens 1977 for details and their monte carlo comparisons of mean squared errors in particular they gave maximum likelihood estimates when the sib sizes are equal however when the sib sizes are not equal rosner 1979 proposed an algorithm for finding the maximum likelihood estimates which involves iterative maximization of an implicit function of two parameters this algorithm is difficult to implement and may not even converge for some sets of data recently mak ng 1981 used the linear model approach of kempthorne tandan 1953 to obtain the maximum likelihood estimates this reduces the problem to iterative maximization of a function of one parameter however nothing is known about the convergence of the procedure another iterative method of finding the maximum likelihood estimates has been given by smith 1980 in this paper an alternative approach is given which requires solving only one equation also two alternative sets of estimates are given these estimates are easier to compute than the maximum likelihood estimates using these estimates a test for the interclass correlation to be zero is given it requires only the ordinary student s t distribution similarly other tests use only known results

on the arithmetic and homology of algebras of linear type three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras e g cohen macaulayness factoriality the main tools are certain fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing up rings

torsion freeness of symmetric powers of ideals let i be an ideal in a noetherian commutative ring r with unit let k 2 be an integer and let k i s k i i k be the canonical surjective r module homomorphism from the kth symmetric power of i to the kth power of i when pd r i 1 or when i is a perfect gorenstein ideal of grade 3 we provide a necessary and sufficient condition for k to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of i when i m is a maximal ideal of r we show that is an isomorphism if and only if r m is a regular local ring in all three cases for i our results yield that if is an isomorphism then at is also an isomorphism for each 1 t k

evolutions symbolic squares and fitting ideals given a reduced local algebra t over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections r t which induce isomorphisms r k t t k such maps called evolutions arise naturally in the study of hecke algebras as they implicitly do in the recent work of wiles taylor wiles and flach we show that the existence of non trivial evolutions of an algebra t can be characterized in terms of the symbolic square of an ideal defining t we give a characterization of the symbolic square in terms of fitting ideals using this and other techniques we show that certain classes of reduced algebras codimension 2 cohen macaulay codimension 3 gorenstein licci algebras in general and some others admit no nontrivial evolutions on the other hand we give examples showing that non trivial evolutions of reduced cohenmacaulay algebras of codimension 3 do exist in every positive characteristic

cohen macaulay rees algebras and degrees of polynomial relations e mail aron sunrnp vfba br 2 department of mathematics michigan state university east lansing mi 48824 usa e mail 21144bfu msu edu 3 department of mathematics rutgers university new brunswick nj 08903 usa e mail vasconce rings rutgers edu received 5 october 1993

semiparametric bayesian modeling of random genetic effects in family based association studies we consider the inference problem of estimating covariate and genetic effects in a family based case control study where families are ascertained on the basis of the number of cases within the family however our interest lies not only in estimating the fixed covariate effects but also in estimating the random effects parameters that account for varying correlations among family members these random effects parameters though weakly identifiable in a strict theoretical sense are often hard to estimate due to the small number of observations per family a hierarchical bayesian paradigm is a very natural route in this context with multiple advantages compared with a classical mixed effects estimation strategy based on the integrated likelihood we propose a fully flexible bayesian approach allowing nonparametric modeling of the random effects distribution using a dirichlet process prior and provide estimation of both fixed effect and random effects parameters using a markov chain monte carlo numerical integration scheme the nonparametric bayesian approach not only provides inference that is less sensitive to parametric specification of the random effects distribution but also allows possible uncertainty around a specific genetic correlation structure the bayesian approach has certain computational advantages over its mixed model counterparts data from the prostate cancer genetics project a family based study at the university of michigan comprehensive cancer center including families having one or more members with prostate cancer are used to illustrate the proposed methods a small scale simulation study is carried out to compare the proposed nonparametric bayes methodology with a parametric bayesian alternative

rees algebras of modules we study rees algebras of modules within a fairly general framework we introduce an approach through the notion of bourbaki ideals that allows the use of deformation theory one can talk about the essentially unique generic bourbaki ideal i e of a module e which in many situations allows one to reduce the nature of the rees algebra of e to that of its bourbaki ideal i e properties such as cohen macaulayness normality and being of linear type are viewed from this perspective the known numerical invariants such as the analytic spread the reduction number and the analytic deviation of an ideal and its associated algebras are considered in the case of modules corresponding notions of complete intersection almost complete intersection and equimultiple modules are examined in some detail special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way for these modules one is able to estimate the reduction number and other invariants in terms of the buchsbaum rim multiplicity

on linear complete intersections or some of its localizations becomes a complete intersection is a major factor in section 1 we describe a jacobian formulation of serre s normality criterion for symmetric algebras it is used to provide several classes of examples of normal and non normal rees algebras of prime ideals in regular domains adding to the catalog of cohen macaulay domains that are symmetric algebras we examine almost complete intersections these bear a close relationship to certain height three ideals of

robust asymptotic sampling theory for correlations in pedigrees methods to unravel the genetic determinants of non mendelian diseases lie at the next frontier of statistical approaches for human genetics it is generally agreed that before proceeding with segregation or linkage analysis the trait under study ought to be shown to exhibit familial correlation by coding dichotomous traits as binary variables a single robust approach in the estimation of pedigree correlations rather than two distinct approaches can be used to assess the potential heritability of a trait and latterly to examine the mode of inheritance the asymptotic theory to conduct hypothesis tests and confidence intervals for correlations among different members of nuclear families is well established but is applicable only if the nuclear families are independent as a further contribution to the literature we derive the asymptotic sampling distribution of correlations between random variables among arbitrary pairs of members in extended families for the pearson product moment estimator with generalized weights this derivation is done without assuming normality of the traits the sampling distribution is shown to be asymptotically normal to first order and hence large sample hypothesis tests and confidence intervals with estimates of the variances and correlation coefficients are proposed discussion concludes with an example and a suggestion for future research

cremona transformations and some related algebras abstract one proves a general characteristic free criterion for a rational map between projective varieties to be birational in terms of ideal theoretic and modulo theoretic conditions this criterion is more inclusive than that of f russo a simis compositio math 126 2001 335 358 and moreover differs from previous criteria in its nature in that the syzygies of the base ideal of the map are not directly involved in its formulation however a great deal of the consequences are phrased by means of those very syzygies avoided in the formulation of the criterion in any case the criterion is stated in effective terms so it yields an efficient computable test of birationality one also introduces a so called linear obstruction principle for base ideals of linear type thus raising a basic question concerning the structure of a certain related bilinear algebra

rees algebras of truncations of complete intersections abstract in this paper we describe the defining equations of the rees algebra and the special fiber ring of a truncation i of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m to describe explicitly the rees algebra r i in terms of generators and relations we map another rees ring r m onto it where m is the direct sum of powers of m we compute a grobner basis of the ideal defining r m it turns out that the normal domain r m is a koszul algebra and from this we deduce that in many instances r i is a koszul algebra as well

a fully implicit model of the three dimensional thermohaline ocean circulation 685 in this paper a fully implicit numerical model of the three dimensional thermohaline ocean circulation is presented with this numerical model it is possible to follow branches of steady states in parameter space and monitor their linear stability also transient flows can be computed allowing much larger time steps than those possible with explicit schemes by using recently developed solvers for linear systems of equations and for generalized eigenvalue problems results for reasonable spatial resolution can be obtained bifurcation diagrams and transient flows are computed for typical flows in a single hemispheric basin situation with focus on i the performance of the methodology and ii the new type of information which can be obtained on these flows

linearly presented perfect ideals of codimension 2 in three variables abstract the goal of this paper is the fine structure of the ideals in the title with emphasis on the properties of the associated rees algebra and the special fiber the watershed between the present approach and some of the previous work in the literature is that here one does not assume that the ideals in question satisfy the usual generic properties one exception is a recent work of n p h lan which inspired the present work here we recover and extend his work we strongly focus on the behavior of the ideals of minors of the corresponding so called hilbert burch matrix and on conjugation features of the latter we apply the results to three important models linearly presented ideals of plane fat points reciprocal ideals of hyperplane arrangements and linearly presented monomial ideals

estimation of interclass and intraclass correlations in multivariate familial data asymptotically normal estimators of interclass and intraclass correlations are derived for more than two quantitative characteristics of parent and siblings in a simple random sampling of families that have different numbers of offspring these estimators are proposed as an alternative to the maximum likelihood estimators which can be found only by iterative methods requiring prohibitively large amounts of computation the asymptotic variances of the proposed estimators are also given in an illustrative example these easily computable estimators are seen to be comparable to the corresponding maximum likelihood estimators

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

an incomplete factorization preconditioning using repeated red black ordering the convergence of the conjugate gradient method for the iterative solution of large systems of linear equations depends on proper preconditioning matrices we present an efficient incomplete factorization preconditioning based on a specific repeated red black ordering scheme and cyclic reduction for the dirichlet model problem we prove that the condition number increases asymptotically slower with the number of equations than for usual incomplete factorization methods numerical results for symmetric and non symmetric test problems and on locally refined grids demonstrate the performance of this method especially for large linear systems

on the divisor class group of rees algebras let r be a normal noetherian domain and let i be an ideal whose associated graded ring gr r is reduced barshay 2 showed that under these conditions the rees algebra s r it r 0 it 0 izt2 a e r t is normal a much simpler proof was given later by huneke 111 let p r it t be the extended rees algebra then pt e r t t 1 is normal and p tp e gr r is reduced it follows that p itself is normal this implies easily the normality of s as well we are interested in the class group cl s of s to obtain a neat result we strengthen our hypothesis and assume that gr r is a domain hochster 6 showed in this case that p is factorial if r is so the only tool needed to prove this is the following elementary but powerful theorem of nagata

principal component analysis for multivariate familial data summary the use of a principal component analysis is considered for multivariate data on families with different numbers of siblings the coefficients in principal components are given as the eigenvectors of the weighted sums of squares and products matrix from the sibling data asymptotic distributions of the eigenvalues and eigenvectors of the estimated covariance matrix are obtained for an elliptical population asymptotic distributions of statistics associated with reduction of dimensionality are also derived the results can be used to construct approximate confidence intervals for the coefficients and variances of principal components

strongly cohen macaulay schemes and residual intersections this paper studies the local properties of closed subschemes y in cohen macaulay schemes x such that locally the defining ideal of y in x has the property that its koszul homology is cohen macaulay whenever this occurs y is said to be strongly cohen macaulay in x this paper proves several facts about such embeddings chiefly with reference to the residual intersections of y in x the main result states that any residual intersection of y in x is again cohen macaulay introduction our purpose in this paper is to investigate a property of a closed subscheme y in a noetherian cohen macaulay scheme x which we call strong cohen macaulayness this is a local property which we first describe for a local ring let x spec r where r is a cohen macaulay local ring and let y spec r i

cohen macaulay rees algebras and their specialization the rees algebra of an ideal j in a commutative ring r is by definition the graded algebra where t is an indeterminate in this paper we are concerned with proving that certain rees algebras are cohen macaulay and with answering under strong hypotheses the question if r s n is a factor ring of s and i jr when is i r a specialization of j s that is when is the natural epimorphism r a j s r i r an isomorphism these questions have interest partly because if s and s j s are cohen macaulay then so is gr s s j j j 16 1 and under these hypotheses if n is perfect and r s j s i r then r and 7 1 r are cohen macaulay too thus gr r is cohen macaulay and its torsion freeness and normality for exmple can be characterized in terms of analytic spreads as in 161 see section 3 for details we deal with the specialization question in section 1 it is answered for the case where as above s and 2 j s are cohen macaulay and n is 202

symbolic powers of perfect ideals of codimension 2 and birational maps abstract this work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field assuming that the entries of the corresponding presentation matrix are general linear forms the main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal a full description of the corresponding symbolic rees algebra is given in some cases one application is an affirmative solution of a conjecture of eisenbud mazur in 11 section 2

ideals having the expected reduction number 1 introduction let r be a noetherian local ring with infinite residue field k and let be an r ideal when studying algebraic properties of the blow up of spec along v l it is often important to have good upper bounds for the reduction number of the ideal recall that an ideal c is a reduction of if the extension of rees algebras r jt c r it is module finite or equivalently if ir l j r for some r 0 32 the least such r is denoted by rj i a reduction is minimal if it is minimal with respect to inclusion and the reduction number r l of is defined as min o j a minimal reduction of finally the analytic spread i of is the krull dimension of the fiber ring r lt r k or equivalently the minimal number of generators jbi j of any minimal reduction j of 1 32 philosophically speaking j is a simplification of with the reduction number r l measuring how closely the two ideals are related of the numerous recent results about rees algebras and associated graded rings of ideals having small reduction number 18 19 38 35 12 13 14 4 3

effective normality criteria for algebras of linear type abstract the algebras studied here are subalgebras of rings of polynomials generated by 1 forms so called rees algebras with coefficients in a noetherian ring given a normal domain r and a torsionfree module e with a free resolution f 2 f 1 f 0 e 0 we study the role of the matrices of syzygies in the normality of the rees algebra of e when the rees algebra r e and the symmetric algebra s e coincide the main results characterize normality in terms of the ideal ic s e and of the completeness of the first s symmetric powers of e where c rank and s rankf0 ranke it requires that r be a regular domain special results under broader conditions on r are still more effective

rees algebras and almost linearly presented ideals abstract consider a grade 2 perfect ideal i in r k x 1 x d which is generated by forms of the same degree assume that the presentation matrix is almost linear that is all but the last column of consist of entries which are linear for such ideals we find explicit forms of the defining ideal of the rees algebra r i we also introduce the notion of iterated jacobian duals

the implicitization problem for ϕ pn p1 n 1 abstract we develop in this paper methods for studying the implicitization problem for a rational map p n p 1 n 1 defining a hypersurface in p 1 n 1 based on computing the determinant of a graded strand of a koszul complex we show that the classical study of macaulay resultants and koszul complexes coincides in this case with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions under suitable hypotheses these techniques enable us to obtain the implicit equation up to a power and up to some extra factor we give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of and what are the extra varieties that appear we also give some applications to the problem of computing sparse discriminants

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

quantifying singularities with differential operators abstract the f signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties for example this invariant detects non singularity and strong f regularity however it is very difficult to compute motivated by different aspects of the f signature we define a numerical invariant for rings of characteristic zero or p 0 that exhibits many of the useful properties of the f signature we also compute many examples of this invariant including cases where the f signature is not known we also obtain a number of results on symbolic powers and bernstein sato polynomials

computing eigenvalues occurring in continuation methods with the jacobi davidson qz method continuation methods are well known techniques for computing several stationary solutions of problems involving one or more physical parameters in order to determine whether a stationary solution is stable and to detect the bifurcation points of the problem one has to compute the rightmost eigenvalues of a related generalized eigenvalue problem the recently developed jacobi davidson qz method can be very effective for computing several eigenvalues of a given generalized eigenvalue problem in this paper we will explain how the jacobi davidson qz method can be used to compute the eigenvalues needed in the application of continuation methods as an illustration the two dimensional rayleigh benard problem has been studied with the rayleigh number as a physical parameter we investigated the stability of stationary solutions and several bifurcation points have been detected the jacobi davidson qz method turns out to be very efficient for this problem

a characteristic free criterion of birationality abstract one develops ab initio the theory of rational birational maps over reduced but not necessarily irreducible projective varieties in arbitrary characteristic a virtual numerical invariant of a rational map is introduced called the jacobian dual rank it is proved that a rational map in this general setup is birational if and only if the jacobian dual rank is well defined and attains its maximal possible value even in the classical case where the source variety is irreducible there is some gain for this invariant over the degree of the map because on one hand it relates naturally to constructs in commutative algebra and on the other hand is effectively computable applications are given to results only known so far in characteristic zero one curious byproduct is an alternative approach to deal with the result of dolgachev concerning the degree of a plane polar cremona map

combined maximum likelihood estimates for the equicorrelation coefficient combined maximum likelihood estimates for equicorrelation covariance matrices are considered the case of a common equicorrelation rho and possibly different standard deviations sigma 1 sigma k among k experimental groups is examined first and the estimation of rho sigma 1 sigma k is discussed second under the assumption of a common standard deviation and possibly different equicorrelations the estimation of rho 1 rho k sigma is considered in each case maximum likelihood solutions and corresponding large sample variances are presented