accurate and efficient power calculations for 2 x m tables in unmatched case control designs the aim of this article was to derive an approximation of the distribution of pearson s statistic for 2 x m contingency tables to calculate power for case control studies accurately and efficiently in terms of computer time we first prove that rather than a non central chi square distribution pearson s statistic is asymptotically equivalent with the sum of squares of m independent normal random variables whose variances are typically different under the alternative hypothesis based on this asymptotically equivalent ae we derive a cost effective approximation ce numerical results show that ce was almost as precise as ae but computationally more efficient although somewhat less costly in terms of cpu time we show that a commonly used approximation using the non central chi square distribution can be very inaccurate and overestimate power in some scenarios and underestimate it in others simulations and exact distributions on the other hand are accurate but computationally very intensive compared to our ce the ce reached its lowest accuracy under the null hypothesis where it has a chi square distribution with m 1 degree of freedom this suggests that ce can be used to calculate power for tables where researchers would feel comfortable using pearson s chi square test

real number computability and domain theory we present the different constructive definitions of real number that can be found in the literature using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them this definition of computability turns out to be equivalent to other definitions given in the literature using different methods domain theory is a useful tool to study higher order computability on real numbers an interesting connection between scott topology and the standard topologies on the real line and on the space of continuous functions on reals is stated an important result in this paper is the proof that every computable functional on real numbers is continuous w r t the compact open topology on the function space

when is a regular local ring a locality let a be a regular local ring and let r be a polynomial extension of a in a finite number of variables several attempts have been made to study the structure of projective modules over r and to find the minimal numbers of generators of ideals of r to wit z v 4 3 61 the methods employed in these references depend very much on the structure of a the cases dealt with are formal power series rings over fields and local rings at smooth points of algebraic varieties cohen s structure theorem tells us precisely what regular local rings are formal power series rings we present here the following characterization for a regular local ring to be a locality over a field

a 1 connectedness in reductive algebraic groups using sheaves of a 1 connected components we prove that the morel voevodsky singular construction on a reductive algebraic group fails to be a 1 local if the group does not satisfy suitable isotropy hypotheses as a consequence we show the failure of a 1 invariance of torsors for such groups on smooth affine schemes over infinite perfect fields we also characterize a 1 connected reductive algebraic groups over a field of characteristic 0

a computational model for metric spaces for every metric space x we define a continuous poset bx such that x is homeomorphic to the set of maximal elements of bx with the relative scott topology the poset bx is a dcpo iff x is complete and continuous iff x is separable the computational model bx is used to give domain theoretic proofs of banach s fixed point theorem and of two classical results of hutchinson on a complete metric space every hyperbolic iterated function system has a unique non empty compact attractor and every iterated function system with probabilities has a unique invariant measure with bounded support we also show that the probabilistic power domain of bx provides an continuous computational model for measure theory on a separable complete metric space x

dynamical systems measures and fractals via domain theory abstract we introduce domain theory in dynamical systems iterated function systems fractals and measure theory for a discrete dynamical system given by the action of a continuous map f x x on a metric space x we study the extended dynamical systems vx vf ux uf and lx lf where v u and l are respectively the vietoris hyperspace the upper hyperspace and the lower hyperspace functors we show that if x f is chaotic then so is ux uf when x is locally compact ux is a continuous bounded complete dcpo if x is second countable as well then ux will be continuous and can be given an effective structure we show how strange attractors attractors of iterated function systems fractals and julia sets are obtained effectively as fixed points of deterministic functions on ux or fixed points of non deterministic functions on cux where c is the convex plotkin power domain we also show that the set m x of finite borel measures on x can be embedded in pux where p is the probabilistic power domain this provides an effective framework for measure theory we then prove that the invariant measure of an hyperbolic iterated function system with probabilities can he obtained as the unique fixed point of an associated continuous function on pux

on the number of generators of ideals in polynomial rings for an ideal i in a noetherian ring r let i be the minimal number of generators of i it is well known that there is a sequence of inequalities i i i i i 1 that are strict in general however murthy conjectured in 1975 that i i i for ideals in polynomial rings whose height equals i i the purpose of this article is to prove a stronger form of the conjecture in case the base field is infinite of characteristic different from 2 namely the equality i i i holds for any ideal i irrespective of its height

effective domain representations of h x the space of compact subsets this paper gives effective domain representations of spaces h x of non empty compact subsets of effective complete metric spaces x the domain representation of h x is constructed from a domain representation of x using the plotkin power domain construction as an application of the representation an effective version of a fundamental theorem on ifs iterated function system is shown

sample size determination for logistic regression revisited there is no consensus on the approach to compute the power and sample size with logistic regression some authors use the likelihood ratio test some use the test on proportions some suggest various approximations to handle the multivariate case we advocate the use of the wald test since the z score is routinely used for statistical significance testing of regression coefficients the null variance formula became popular from early studies which contradicts modern software which utilizes the method of maximum likelihood estimation mle when the variance of the mle is estimated at the mle not at the null we derive general wald based power and sample size formulas for logistic regression and then apply them to binary exposure and confounder to obtain a closed form expression these formulas are applied to minimize the total sample size in a case control study to achieve a given power by optimizing the ratio of controls to cases approximately the optimal number of controls to cases is equal to the square root of the alternative odds ratio our sample size and power calculations can be carried out online at www dartmouth edu approximately eugened

exact inference for categorical data recent advances and continuing controversies methods for exact small sample analyses with categorical data have been increasingly well developed in recent years a variety of exact methods exist primarily using the approach that eliminates unknown parameters by conditioning on their sufficient statistics in addition a variety of algorithms now exist for implementing the methods this paper briefly summarizes the exact approaches and describes recent developments controversy continues about the appropriateness of some exact methods primarily relating to their conservative nature because of discreteness this issue is examined for two simple problems in which discreteness can be severe interval estimation of a proportion and the odds ratio in general adjusted exact methods based on the mid p value seem a reasonable way of reducing the severity of this problem

power of the likelihood ratio test in covariance structure analysis a procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived the procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified using the noncentral chi square distribution the power of the test is approximated by the asymptotic one for a sequence of local alternatives the procedure is illustrated by an example a monte carlo experiment also shows how good the approximation is for a specific case

injective stability for k 1 of classical modules in rao 1994 14 the second author and w van der kallen showed that the injective stabilization bound for k1 of the general linear group is d 1 over a regular affine algebra over a perfect c1 field where d is the krull dimension of the base ring which is finite and at least 2 in this article we prove that the injective stabilization bound for k1 of the symplectic group is d 1 over a geometrically regular ring containing a field where d is the dimension of the base ring which is finite and at least 2 using the local global principle for the transvection subgroup of the automorphism group of projective and symplectic modules we show that the injective stabilization bound is d 1 for k1 of projective and symplectic modules of global rank at least 1 and local rank at least 3 respectively in each of the two cases above

power for studies with random group sizes in any study it is essential to select the sample size carefully to ensure adequate power for many studies this is simple recruit a desired number of subjects within each group conduct measurements and perform the statistical test in some studies e g observational studies however the group membership is unknown at recruitment in this paper we examine the effect of random group sizes on power additionally we consider the situation when the group proportions are unknown and specified by a prior distribution the problem that initially motivated this research is presented power for a 2 by 2 table as are examples using continuous outcomes we find that standard estimates of power using expected group sizes can over or underestimate power sas macros are available at http www phs wfubmc edu public wambrosi randompower

on a question of quillen let r be a regular local ring and f a regular parameter of r quillen asked whether every projective r f module is free we settle this question when r is a regular local ring of an affine algebra over a field k further if r has infinite residue field we show that projective modules over laurent polynomial extensions of r f are also free

affine grassmannians of group schemes and exotic principal bundles over bbb a 1 let bf g be a simple simply connected group scheme over a regular local scheme u let mathcal e be a principal bf g bundle over bbb a 1 u trivial away from a subscheme finite over u we show that mathcal e is not necessarily trivial and give some criteria of triviality to this end we define affine grassmannians for group schemes and study their bruhat decompositions for semi simple group schemes we also give examples of principal bf g bundles over bbb a 1 u with split bf g such that the bundles are not isomorphic to pullbacks from u

a cohomological classification of vector bundles on smooth affine threefolds we give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2 as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties the proofs of these results involve three main ingredients first we give a description of the first nonstable a1 homotopy sheaf of the symplectic group second these computations can be used in concert with f morel s a1 homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques stemming from a version of the postnikov tower in a1 homotopy theory to reduce the classification results to cohomology vanishing statements third we prove the required vanishing statements

algebraic fiber bundles abstracr when x is a finite simplicial complex and g is any of a certain class of groups a classification of g principal bundles over x in terms of projective modules over a ring r g x is given this generalizes swan s classification of vector bundles and uses the results of mulvey often r can be taken to be noetherian in this case spec r is usually reducible with cohomologically trivial irreducible components information is derived concerning the nature of projective modules over such rings and some results are obtained indicating how such information reflects information about x

an application of a representation theorem for fuzzy metrics to domain theory in this paper we study the interplay between fuzzy metric spaces and domain theory using the representation theorem established in mardones perez and de prada vicente 2012 12 we will prove that suitable restrictions of completeness in fuzzy metric spaces correspond to a structure of domain in the poset of formal balls endowed with appropriate orders

affine representability results in mathbb a 1 homotopy theory i vector bundles we establish a general affine representability result in a1 homotopy theory over a general base we apply this result to obtain representability results for vector bundles in a1 homotopy theory our results simplify and significantly generalize morel s a1 representability theorem for vector bundles

the nilpotence conjecture in k theory of toric varieties it is shown that all nontrivial elements in higher k groups of toric varieties are annihilated by iterations of the natural frobenius type endomorphisms this is a higher analog of the triviality of vector bundles on affine toric varieties

on triviality of the euler class group of a deleted neighbourhood of a smooth local scheme let k be an infinite perfect field and let x specr where r is the localization of a k algebra b of finite type with respect to a regular prime ideal p specb letd be a divisor onx with a decompositiond i 1di into its irreducible components di such that each di is nonsingular and they intersect transversally consider y x d a deleted neighbourhood of the closed point x0 of x let e be a vector bundle on y in this set up e is trivial in the following cases i dim x 3 gabber ga1 ii t 1 bhatwadekar rao br iii ranke min dimr 2 t rao ra1 the above exposition is borrowed from a paper of nisnevich ni which contains some beautiful applications of these purity theorems in the same set up now let y y be a closed point and let my be the corresponding maximal ideal of oy y it is natural to ask whether my is a complete intersection observing that the conormal bundle my my is trivial we consider a more general situation let i be the defining ideal of a closed subscheme z of y such that the conormal bundle i i2 defined on z is trivial is i a complete intersection the questions we are addressing in this paper are a little stronger in nature can a given basis of i i2 be lifted to a minimal set of generators of i we remark that with the hypothesis on x each component di is defined by a single equation fi which vanishes at the closed point x0 of x let m be the maximal ideal at x0 the condition of transversality of intersection of di s is equivalent to saying that the images of f1 ft in the a m vector space m m2 are linearly independent keeping this discussion in mind we now give an excerpt of our main results in commutative algebraic terms see 4 2 4 3 4 4

stability results for projective modules over blowup rings let r be a normal affine domain of dimension n 3 over an algebraically closed field k suppose chark 0 or chark p n let g f1 fr be a r regular sequence and a r f1 g fr g let p be a stably free a module of rank n 1 then murthy proved that there exists a projective r module q such that q r a p and n 1 q r 9 theorem 2 10 as a consequence of murthy s result if f g c x1 xn with g 0 then all stably free modules over c x1 xn f g of rank n 1 are free 9 corollary 2 11 in this paper we prove the following result 3 5 which generalizes the above result of murthy

formulae and tables for the determination of sample sizes and power in clinical trials for testing differences in proportions for the two sample design a review this paper is a compendium of exact and asymptotic formulae and tables for estimating the sample size in a clinical trial with two treatment groups and a dichotomous outcome the paper provides separate formulae for equal and unequal treatment group sizes formulae for the calculation of power given the sample size and complete references for all formulae and tables cited

embedding metric spaces into cpo s abstract we show that metric spaces with polish topology mp can be represented very well by bounded complete algebraic cpo s with continuous weight and distance cwd an operator cwd mp is defined such that 1 and 2 hold 1 1 d can be embedded topologically into d such that a partial function on d is metrically continuous if it can be extended to a total continuous function on d 2 2 every m mp is isometric to some d therefore the continuous cpo functions are exactly adequate for describing the continuous functions on the corresponding metric spaces the approach could be refined in order to study computability on metric spaces with polish topology examples of application are the euclidean space the space of total functions on the natural numbers baire space and spaces of infinite trees

domain representations of topological spaces a domain representation of a topological space x is a function usually a quotient map from a subset of a domain onto x several dierent classes of domain representations are introduced and studied it is investigated when it is possible to build domain representations from existing ones it is for example discussed whether there exists a natural way to build a domain representation of a product of topological spaces from given domain representations of the factors it is shown that any t0 topological space has a domain representation these domain representations are very large however smaller domain representations are also constructed for large classes of spaces for example each second countable regular hausdor space has a domain representation with a countable base domain representations of functions and function spaces are also studied c 2000 elsevier science b v all rights reserved

on the complete intersection conjecture of murthy abstract suppose a k x 1 x 2 x n is a polynomial ring over a field k and i is an ideal in a m p murthy conjectured that i i i 2 where denotes the minimal number of generators recently fasel 3 settled this conjecture affirmatively when k is an infinite perfect field with 1 2 k always we are able to do the same when k is an infinite field in fact we prove similar results for ideals i in a polynomial ring a r x that contains a monic polynomial and r is essentially smooth algebra over an infinite field k or r is a regular ring over a perfect field k

domain representations of partial functions with applications to spatial objects and constructive volume geometry a partial spatial object is a partial map from space to data data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry cvg where space and data are subspaces of n dimensional euclidean space we investigate the computability of partial spatial object data types in general and in volume graphics using the theory of effective domain representations for topological algebras the basic mathematical problem considered is to classify which partial functions between topological spaces can be represented by total continuous functions between given domain representations of the spaces we prove theorems about partial functions on regular hausdorff spaces and their domain representations and apply the results to partial spatial objects and cvg algebras

efficient generation of zero dimensional ideals in polynomial rings let a be a commuta t ive noether ian ring and i an ideal in a it is a wel l known fact that it i 2 2 see 5 when n 1 and r is a regular local ring such as a power series r ing over a field or a regular spot over an infinite perfect field bha twadekar has shown that it ill 2 i for all maximal i d e a l s i n r t see 2 in this paper we generalise these results for the case when i is a zero d imens iona l ideal of the fol lowing kind i fo r any zero d imensional ideal i in a r t 1 t we have it i i2 it i if n 2 2 let i b e a zero d imensional ideal in a r t then it i i2 i in each o f the following cases a r is a power series ring over a field b r is a regular local r ing of an affine a lgebra over an infinite perfect field in sect 2 we state some theorems without proof which we require later in sect 3 we prove the analogues of dav i s ge rami t a theorems in sect 4 we prove the result for a power series r ing or for a regular local r ing of an affine algebra the results of this pape r go th rough for laurent po lynomia l rings

the serre problem for discrete hodge algebras all rings in this paper will be commutative and noetherian let a b be a split injection of rings a finitely generated projective b module is called extended from a if there exists a finitely generated projective a module q such that p q b furthermore we call an c eglr b resp slr b extended from a if can be written as f i with fi glr a resp slr a and 7 e b the group of elementary matrices now let a be a ring in this paper we will consider a algebras of the following type a x1 xn i where i is an ideal generated by monomials such an a algebra is called a discrete hodge algebra over a and vice versa every discrete hodge algebra over a is of this type 5 for the proofs of most of our results we can restrict ourselves to the special class of square free discrete hodge algebras over a by which is meant that i can be generated by square free monomials the varieties over a corresponding to these square flee discrete hodge algebras are unions of intersections of coordinate hyperplanes h i x 1 xn anixi 0 i 1 n typical examples are a x 1 xn and a x 1 x 2 x i x 2 the main result of this paper is the following

power comparison of two sided exact tests for association in 2 x 2 contingency tables using standard mid p and randomized test versions exact pearson s chi square likelihood ratio lr and fisher s tests are obtained from the conditional distribution of its test statistic given the row and column sums of the contingency table the power and obtained significance level of the standard mid p and randomized versions of these tests are compared for two sided tests in 2 x 2 tables using binomial and multinomial sampling the mid p type i error probabilities seldom exceed the nominal significance level the mid p and randomized test versions have approximately the same power and higher power than the standard test version the power of the pearson s chi square lr and fisher s test differ and they differ in approximately the same way for standard mid p and randomized test versions for any given set of parameters there is no general ranking between the three tests in many cases pearson s chi square and fisher s tests have almost equal power and higher power than lr in a few cases perhaps characterized by poorly balanced designs lr performs best fisher s test seems to be slightly more robust even if the design is poor

the euler class group of a polynomial algebra with coefficients in a line bundle let a be a commutative noetherian ring and p be a projective a module of rank text dim a 1 an intriguing open question is to find the precise obstruction for p to split as p simeq q oplus a for some a module q in this paper we settle this question when a r t for some ring r containing the field of rationals and p is a projective a module of rank text dim r

euler class groups and the homology of elementary and special linear groups abstract we improve homology stability ranges for elementary and special linear groups over rings with many units our result implies stability for unstable quillen k groups and proves a conjecture of bass for commutative local rings with infinite residue fields we show that the obstruction to further stability is given by milnor witt k theory as an application we construct euler classes of projective modules with values in the cohomology of the milnor witt k theory sheaf for d dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the euler class is necessary and sufficient for an oriented projective module p of rank d to split off a rank 1 free direct summand along the way we obtain a new presentation of milnor witt k theory and of symplectic k 2 simplifying the classical matsumoto moore presentation

a simple analysis of a simple crossover trial with a dichotomous outcome measure for the two treatment two period crossover trial with a dichotomous outcome measure a simple and straightforward analysis is proposed compute the treatment difference for each of the treatment sequences ab and ba and then average over the sequence groups the mean difference is used in the statistical test and in the computation of the 95 per cent confidence interval the method is generalized to the case of a crossover trial including more than two treatments

planning and analysis of three arm non inferiority trials with binary endpoints three arm trials including an experimental treatment an active control and a placebo group are frequently preferred for the assessment of non inferiority in contrast to two arm non inferiority studies these designs allow a direct proof of efficacy of a new treatment by comparison with placebo as a further advantage the test problem for establishing non inferiority can be formulated in such a way that rejection of the null hypothesis assures that a pre defined portion of the unknown effect the reference shows versus placebo is preserved by the treatment under investigation we present statistical methods for this study design and the situation of a binary outcome variable asymptotic test procedures are given and their actual type i error rates are calculated approximate sample size formulae are derived and their accuracy is discussed furthermore the question of optimal allocation of the total sample size is considered power properties of the testing strategy including a pre test for assay sensitivity are presented the derived methods are illustrated by application to a clinical trial in depression

the milnor patching property abstract this paper gives a complete answer to the following question if r is a commutative ring in which every height 1 ideal is a local set theoretic complete intersection e g r could be regular and r 1 and r 2 are r algebras which are flat or of finite type when is it the case that projective r modules are precisely those r modules constructed by patching a projective r 1 module to a projective r 2 module over r 1 r r 2 in other words we give a classification theorem for situtations in which the conclusion to the milnor patching theorem holds our answer takes the form of a necessary and sufficient condition whose truth or falsity is very easy to check in most particular situations

the monoid structure on homotopy obstructions abstract let a be a commutative noetherian ring containing a field k with 1 2 k dim a d and let p be a projective a module with r a n k p n let l o p denote the set of all pairs i where i is an ideal of a and p i i 2 is a surjective map the homotopy relations on l o p induced by l o p t leads to a set 0 l o p of equivalence classes in l o p there are two distinguished elements e 0 e 1 0 l o p respectively the images of 0 0 and a 0 define the obstruction class e p e 0 0 l o p to be called the nori homotopy class of p the following results are under suitable smoothness or regularity hypotheses we prove if 2 n d 2 then 0 l o p has a natural structure of a monoid which is a group if p q a when 2 n d 3 we prove p q a e p e 1 the additive zero further we give a definition of a euler class group e p under suitable smoothness hypotheses we prove if p q a and 2 n d 3 then there is natural isomorphism e p 0 l o p of groups

sample size determination in clinical trials with multiple co primary binary endpoints clinical trials often employ two or more primary efficacy endpoints one of the major problems in such trials is how to determine a sample size suitable for multiple co primary correlated endpoints we provide fundamental formulae for the calculation of power and sample size in order to achieve statistical significance for all the multiple primary endpoints given as binary variables on the basis of three association measures among primary endpoints we discuss five methods of power and sample size calculation the asymptotic normal method with and without continuity correction the arcsine method with and without continuity correction and fisher s exact method for all five methods the achieved sample size decreases as the value of association measure increases when the effect sizes among endpoints are approximately equal in particular a high positive association has a greater effect on the decrease in the sample size on the other hand such a relationship is not very strong when the effect sizes are different

on maximal ideals in polynomial and laurent polynomial rings abstract when r is a complete local ring containing a field of characteristic zero we show that any maximal ideal in r t t 1 is projectively generated if r is a regular ring containing a field of characteristic zero we show that any maximal ideal m of r t such that ht m dim r is efficiently generated

on the number of generators of modules over polynomial affine rings preprint enviat per a la seva publicacio en una revista cientifica mathematische zeitschrift 1991 vol 208 p 11 21 https doi org 10 1007 bf02571506

power domains and iterated function systems we introduce the notion of weakly hyperbolic iterated function system ifs on a compact metric space which generalises that of hyperbolic ifs based on a domain theoretic model which uses the plotkin power domain and the probabilistic power domain respectively we prove the existence and uniqueness of the attractor of a weakly hyperbolic ifs and the invariant measure of a weakly hyperbolic ifs with probabilities extending the classic results of hutchinson for hyperbolic ifss in this more general setting we also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure extending the corresponding algorithms for hyperbolic ifss we then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent ifs and obtain an algorithm to generate the invariant distribution on the digitised screen the generalised riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution for hyperbolic recurrent ifss and lipschitz maps one can estimate the integral up to any threshold of accuracy

vector bundles on contractible smooth schemes we discuss algebraic vector bundles on smooth k schemes x contractible from the standpoint of a homotopy theory when k c the smooth manifolds x c are contractible as topological spaces the integral algebraic k theory and integral motivic cohomology of such schemes are that of spec k one might hope that furthermore and in analogy with the classification of topological vector bundles on manifolds algebraic vector bundles on such schemes are all isomorphic to trivial bundles this is almost certainly true when the scheme is affine however in the non affine case this is false we show that essentially every smooth a contractible strictly quasi affine scheme that admits a u torsor whose total space is affine for u a unipotent group possesses a non trivial vector bundle indeed we produce explicit arbitrary dimensional families of non isomorphic such schemes with each scheme in the family equipped with as many i e arbitrary dimensional moduli of non isomorphic vector bundles of every sufficiently large rank n as one desires neither the schemes nor the vector bundles on them are distinguishable by algebraic k theory we also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible but that are not necessarily a contractible