reconstruction from projections based on detection and estimation of objects parts i and ii performance analysis and robustness analysis the problem of reconstructing a multidimensional field from noisy limited projection measurements is approached using an object based stochastic field model objects within a cross section are characterized by a finite dimensional set of parameters which are estimated directly from limited noisy projection measurements using maximum likelihood estimation in part i the computational structure and performance of the ml estimation procedure are investigated for the problem of locating a single object in a deterministic background simulations are also presented in part ii the issue of robustness to modeling errors is addressed

canonical elements in local cohomology modules and the direct summand conjecture one of the objectives of this paper is to show that the usual homological consequences of the existence of big cohen macaulay henceforth c m modules e g the new intersection conjecture of peskine szpiro and roberts and the evans griffith syzygy conjecture follow from the direct summand conjecture when the residual characteristic of the local ring is positive this gives a new and substantially more elementary proof of the standard homological conjectures in case the characteristic of the ring itself is positive and reduces the general case of all these conjectures to one rather down to earth conjecture of course this places the direct summand conjecture in a position of central importance so that it now merits an allout attack some partial results on this problem are given in section 6 the conjecture asserts that a regular noetherian ring r is a direct summand as an r module of every module finite extension ring s 3 r the other main objective of this paper is to formulate and develop a theory of certain canonical elements in the local cohomology of special modules of syzygies neither the modules of syzygies nor the induced maps between them are canonical but the identification between the canonical elements is independent of the choices in particular a canonical element qr e hz syz k is associated see section 3 for details with each ndimensional local ring r m k rings are commutative associative with identity local ring means noetherian ring with a unique maximal ideal the conjecture that qr is nonzero for all local rings r turns out to be equivalent to the direct summand conjecture for a given r an infinite family of cases of the latter implies the former

adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth we consider nonparametric bayesian estimation inference using a rescaled smooth gaussian field as a prior for a multidimensional function the rescaling is achieved using a gamma variable and the procedure can be viewed as choosing an inverse gamma bandwidth the procedure is studied from a frequentist perspective in three statistical settings involving replicated observations density estimation regression and classification we prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax optimal up to a logarithmic factor whatever the regularity level of the data generating distribution thus the hierachical bayesian procedure with a fixed prior is shown to be fully adaptive

big cohen macaulay algebras and seeds in this article we delve into the properties possessed by algebras which we have termed seeds that map to big cohen macaulay algebras we will show that over a complete local domain of positive characteristic any two big cohen macaulay algebras map to a common big cohen macaulay algebra we will also strengthen hochster and huneke s weakly functorial existence result for big cohen macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic we also show that every seed over a positive characteristic ring maps to a balanced big cohen macaulay algebra that is an absolutely integrally closed adically separated quasilocal domain

tensor based cortical surface morphometry via weighted spherical harmonic representation we present a new tensor based morphometric framework that quantifies cortical shape variations using a local area element the local area element is computed from the riemannian metric tensors which are obtained from the smooth functional parametrization of a cortical mesh for the smooth parametrization we have developed a novel weighted spherical harmonic spharm representation which generalizes the traditional spharm as a special case for a specific choice of weights the weighted spharm is shown to be the least squares approximation to the solution of an isotropic heat diffusion on a unit sphere the main aims of this paper are to present the weighted spharm and to show how it can be used in the tensor based morphometry as an illustration the methodology has been applied in the problem of detecting abnormal cortical regions in the group of high functioning autistic subjects

the theory of d sequences and powers of ideals in this paper we study the question of finding depth r i given an ideal i in a commutative ring this problem is difficult in general even for simple examples depth r z can be difficult to compute brodmann has shown that for any commutative noetherian ring depth r z become stable for large n finding this value however is quite hard this problem originally arose through consideration of the following example suppose k is a field and x xij is an n

non noetherian cohen macaulay rings in this paper we investigate a property for commutative rings with identity which is possessed by every coherent regular ring and is equivalent to cohen macaulay for noetherian rings we study the behavior of this property in the context of ring extensions of various types and rings of invariants

dagger closure in regular rings containing a field abstract we prove that dagger closure is trivial in regular domains containing a field and that graded dagger closure is trivial in polynomial rings over a field we also prove that heitmann s full rank one closure coincides with tight closure in positive characteristic under some mild finiteness conditions furthermore we prove that dagger closure is always contained in solid closure and that the forcing algebra for an element contained in dagger closure is parasolid

galois extensions plus closure and maps on local cohomology abstract given a local domain r m of prime characteristic that is a homomorphic image of a gorenstein ring huneke and lyubeznik proved that there exists a module finite extension domain s such that the induced map on local cohomology modules h m i r h m i s is zero for each i dim r we prove that the extension s may be chosen to be generically galois and analyze the galois groups that arise

on the notion of cohen macaulayness for non noetherian rings abstract there exist many characterizations of noetherian cohen macaulay rings in the literature these characterizations do not remain equivalent if we drop the noetherian assumption the aim of this paper is to provide some comparisons between some of these characterizations in non noetherian case toward solving a conjecture posed by glaz we give a generalization of the hochster eagon result on cohen macaulayness of invariant rings in the context of non noetherian rings

an algebraic proof of zak s inequality for the dimension of the gauss image a classical theorem in complex algebraic geometry states that for any smooth projective variety the gauss map is finite in particular a smooth variety and its gauss image have the same dimension with the obvious exception of a linear space furthermore even when the variety is not smooth zak proved a lower bound on the dimension of its gauss image in terms of the dimension of its singular locus our purpose in this note is to reinterpret gauss maps within a more general algebraic framework and thus recover zak s bound on the dimension of the gauss image as a special case of an interesting new bound on the analytic spread of a module of kahler differentials this connects that classical subject in complex geometry to recent research in commutative algebra concerning integral closures of modules in particular we give a new purely algebraic proof of zak s theorem we recall a precise version of zak s theorem let x be an irreducible projective variety of dimension d defined over an algebraically closed field k considered with a fixed embedding x p v for some finite dimensional k vector space v the gauss map is the rational map from x to the grassmannian of projective d planes in p v assigning to each smooth k point of x the projective tangent plane there

on the canonical element conjecture the canonical element conjecture is proved in the following two cases i depth a dim a 1 h a is decomposable ii depth a dim a 1 hm a v is cyclic the equivalence of the c e c and the improved new intersection theorem is also established in h m hochster has assigned a canonical element qa e h syz k with each n dimensional local ring a m k i e commutative noetherian local ring a with a unique maximal ideal m and k a m and has conjectured that 71a 0 for every such a in elementary terms this conjecture means the following h for every free resolution f a8i asi i a k 0 of k and for every system of parameters x1 x of a if 4 is any map of complexes k x a f which lifts the quotient surection a xl x k then 4 kn x a asn is nonzero hochster has shown that the above conjecture henceforth c e c is equivalent to the direct summand conjecture and implies the improved new intersection conjecture which can be stated as follows improved new intersection conjecture let a m be a local ring and f be a finite free complex of finitely generated free modules o fd fo such that hi f has finite length for i 0 and ho f has a nonzero minimal generator v such that avp has finite length then dim r d it is well known that the above conjecture henceforth i n i c implies several other homological conjectures in 1 of this paper we prove that a satisfies c e c a satisfies i n i c 1 4 in proving this we come across in 1 3 another equivalent version of c e c which shows that actually c e c boils down to understanding the relations among the resolutions of homologies of k x a and itself in 2 we study the case when depth a dim a 1 we prove 2 2 that a satisfies c e c in the following cases i when h j1 a v is decomposable this implies hi x a is decomposable ii when h 71 a v is cyclic this implies hi x a v is cyclic received by the editors november 18 1985 1980 mathematics subject classification 1985 revision primary 13h10 the author was supported in part by grants from the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

a bayesian approach to nonparametric bivariate regression abstract this article outlines a general bayesian approach to estimating a bivariate regression function in a nonparametric manner it models the function using a bivariate regression spline basis with many terms binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression the regression function is estimated using an estimate of its posterior mean smoothing over the distribution of these binary indicator variables to make the computations tractable all estimates are obtained using markov chain monte carlo sampling extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data driven bivariate surface estimators prominent in the literature it is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings the first

absolute integral closures are big cohen macaulay algebras in characteristic p throughout this paper ring means commutative ring with identity and modules are unital our main interest is in local rings i e noetherian rings r m with a unique maximal ideal m in such a ring xx xn e m is a system of parameters if m has a power in the ideal x xn r and n is the krull dimension of r when r is complete and contains a field this means that r is module finite over a formal power series subring k xx xn a i is called cohen macaulay if some equivalently every system of parameters is a regular sequence in i which means that every xi x is a nonzerodivisor on r xx xt r for 0 i n 1 in the case where r is module finite over the formal power series subring a this means that r is a free 4 module for many theorems of commutative algebra and of algebraic geometry the cohen macaulay condition possibly on the local rings of a variety is just what is needed to make the theory work our main result which is given below in an algebraic form in theorem 1 and in a geometric form in theorem 2 asserts that under mild conditions on a local ring r of positive prime characteristic p one can correct the failure of the cohen macaulay condition in r itself by passing to a very large integral extension of r it is worth emphasizing that this is quite false in characteristic 0 and comes as a surprise we believe in characteristic p one can get an idea of how far these theorems are from the truth over a field of characteristic 0 from the following observation in the situation of theorem 2 when x is projectively normal the maps of cohomology are always injective by an easy trace argument despite the fact that both theorem 1 and theorem 2

uniform annihilators of local cohomology abstract in this paper we study the properties of noetherian rings containing uniform local cohomological annihilators it turns out that all such rings should be universally catenary and locally equidimensional we will prove a necessary and sufficient condition for such rings which enables us to show that if a locally equidimensional ring r is the image of a cohen macaulay ring then r has a uniform local cohomological annihilator moreover we will give a positive answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13 about excellent rings with dimension no more than 5

f singularities via alterations we give characterizations of test ideals and f rational singularities via regular alterations formally the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions lastly we establish nadel type vanishing theorems up to finite maps for test ideals and further demonstrate how these vanishing theorems may be used to extend sections

tight closure of finite length modules in graded rings in this article we look at how the equivalence of tight closure and plus closure or frobenius closure in the homogeneous m coprimary case implies the same closure equivalence in the nonhomogeneous m coprimary case in standard graded rings although our result does not depend upon dimension the primary application is based on results known in dimension 2 due to the recent work of h brenner we also demonstrate a connection between tight closure and the m adic closure of modules extended to r or r we finally show that unlike the noetherian case the injective hull of the residue field over r or r contains elements that are not killed by any power of the maximal ideal of r this fact presents an obstruction to one possible method of extending our main result to all modules

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

singularities for solutions to schrodinger equations with asymptotically constant magnetic fields this paper studies schrodinger equations with variable coefficients and magnetic potentials we suppose the perturbation is short range type in a sense similar to the work of nakamura wave front set for solutions to schrodinger equations j funct anal 256 1299 1309 2009 10 1016 j jfa 2008 06 007 we characterize the wave front set of the solutions to the equation in terms of the classical scattering data and the propagator of the unperturbed constant magnetic operator

convergence rates of posterior distributions for non i i d observations we consider the asymptotic behavior of posterior distributions and bayes estimators based on observations which are required to be neither independent nor identically distributed we give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion we then specialize our results to independent nonidentically distributed observations markov processes stationary gaussian time series and the white noise model we apply our general results to several examples of infinite dimensional statistical models including nonparametric regression with normal errors binary regression poisson regression an interval censoring model whittle estimation of the spectral density of a time series and a nonlinear autoregressive model