the underlying structure of nonnested hypothesis tests summary a comparison between fitted and observed log likelihoods is used to study likelihood ratio statistics for both nested and nonnested hypothesis tests a study of the geometry involved in the nonnested case clarifies the relationship between cox s likelihood ratio test and the complete parametric wald encompassing test of mizon and richard further it is shown why the asymptotic distribution of these test statistics is sometimes degenerate also a new test statistic is proposed which combines the cox and mizon richard tests

generating a residual intersection let i be an ideal of height g 0 in a commutative noetherian ring r and let 2i be a proper sub ideal of i generated by 3 g elements if the ideal j 21 i has height at least s then j is called an s residual intersection of i see artin and nagata 1 huneke 19 or huneke and ulrich 22 the case s g corresponds to the theory of linkage and is well understood at least from the homolo cal point of view the cornerstone of this theory is that if i is perfect then a free resolution of r j can be produced from a free resolution of r z and the koszul complex on a set of generators of cu very little is known about resolutions of s residual intersections for s g a few special cases have been resolved s 10 301 and the end of the resolution or in other words a set of generators for the canonical module of r j has been described 22 however in general even the generators of j remain unknown here we show that j can be approximated by a sum of links of i if s g 1 and we give conditions under which this approximation in fact gives a full set of generators of j since these generators arise from comparison maps from koszul complexes to known resolutions at least assuming that the resolution of r i is in some sense known it is feasible to count them compute their degrees in the graded case or even to give explicit formulas for them after stating our main results theorem 2 15 and 2 16 and corollary 2 18 precisely in section 2 we lay out the argument in sections 3 4 and 5 in section 6 we are concerned with the question of when a g t 1 residual intersection j is

an l bound for solutions of the cahn hilliard equation in this paper we consider the equation partial t u varepsilon delta u varepsilon sum limits i j 1 n partial ij f ij u varepsilon x varepsilon 2 delta 2 u varepsilon with fij constant when u e is large and we obtain l bounds independent of 0

classification of the tor algebras of codimension four almost complete intersections let r m k be a local ring in which 2 is a unit assume that every element of k has a square root in k we classify the algebras tor r r j k as j varies over all grade four almost complete intersection ideals in r the analogous classification has already been found when j varies over all grade four gorenstein ideals 21 and when j varies over all ideals of grade at most three 5 30 the present paper makes use of the classification in 21 of the tor algebras of codimension four gorenstein rings as well as the usually nonminimal dg algebra resolution of a codimension four almost complete intersection which is produced in 25 and 26

the singular limit of a vector valued reaction diffusion process we study the asymptotic behaviour of the solution to the vector valued reaction diffusion equation e t e4 1 e w 0 in t where e t 0 t r2 we assume that the the potential w depends only on the modulus of and vanishes along two concentric circles we present a priori estimates for the solution and in the spatially radially symmetric case we show rigorously that in the singular limit as e 0 two phases are created the interface separating the bulk phases evolves by its mean curvature while evolves according to a harmonic map flow on the respective circles coupled across the interfaces by a jump condition in the gradient

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

phase segregation dynamics in particle systems with long range interactions ii interface motion we study properties of the solutions of a family of second order integrodifferential equations which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics we first establish existence and uniqueness as well as some properties of the instantonic solutions then we concentrate on formal asymptotic sharp interface limits we argue that the obtained interface evolution laws a stefan like problem and the mullins sekerka solidification model coincide with the ones which can be obtained in the analogous limits from the cahn hilliard equation the fourth order pde which is the standard macroscopic model for phase segregation with one conservation law

a return to an old paper tests of separate families of hypotheses summary two papers of 50 years ago concerning the comparison of non nested models are reviewed and some subsequent developments and problems are sketched

regularity conditions for cox s test of non nested hypotheses abstract in this article we provide for the first time general regularity conditions and a rigorous proof of the asymptotic normality of cox s statistic for testing separate families of hypotheses the cox test for choosing between competing linear regression models is discussed as an example

robust properties of likelihood ratio tests summary the usual asymptotic chi squared distribution for the likelihood ratio test statistic is based on the assumptions that the data come from the parametric model under consideration and that the parameter satisfies the null hypothesis in this paper we examine the distribution of the likelihood ratio statistic when the data do not come from the parametric model but when the nearest member of the parametric family still satisfies the null hypothesis in general the likelihood ratio statistic no longer follows an asymptotic chi squared distribution and an alternative statistic based on the unionintersection approach is proposed

convergence of the cahn hilliard equation to the mullins sekerka problem in spherical symmetry abstract we show that as e 0 the solution of the cahn hilliard equation formula converges to a solution of the mullins sekerka problem formula where denotes a normal v the normal velocity and k the sum of principal curvatures of the interface provided the solutions are radially symmetric we use energy type estimates to show that the solution of the cahn hilliard equation can be approximated by the well known stationary wave solution that corresponds to the potential w

approximate solutions of the cahn hilliard equation via corrections to the mullins sekerka motion we develop an alternative method to matched asymptotic expansions for the construction of approximate solutions of the cahn hilliard equation suitable for the study of its sharp interface limit the method is based on the hilbert expansion used in kinetic theory besides its relative simplicity it leads to calculable higher order corrections to the interface motion

huneke ulrich almost complete intersections of cohen macaulay type two abstract let r m k be a commutative noetherian local ring n 2 an integer x a 2n 1 2n 1 alternating matrix with entries from m y a 1 2n 1 matrix with entries from m i the ideal i i1 yx and a the quotient ring r i assume that the grade of i is at least 2n in this case i is a perfect ideal of grade equal to 2n and i is minimally generated by 2n 1 elements we prove that the minimal resolution of a by free r modules is a dg algebra furthermore we identify the algebra torr a k and prove that if r is regular and char k 0 or n 2 2 char k then the poincare series pma z i 0 dimk torai m k zi is a rational function for every finitely generated a module m as a consequence we deduce that if the projective dimension of m is infinite then eventually the betti numbers of m form an increasing sequence with strong exponential growth

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

a multi phase mullins sekerka system matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem we propose a generalisation of the mullins sekerka problem to model phase separation in multi component systems the model includes equilibrium equations in bulk the gibbs thomson relation on the interfaces young s law at triple junctions together with a dynamic law of stefan type using formal asymptotic expansions we establish the relationship to a transition layer model known as the cahn hilliard system we introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds together with measure theoretical tools through an implicit time discretisation we construct approximate solutions by stepwise minimisation under the assumption that there is no loss of area as the time step tends to zero we show the existence of a weak solution

algorithms for computing motion by mean curvature we propose a finite element algorithm for computing the motion of a surface moving by mean curvature the algorithm uses the level set formulation so that changes in topology of the surface can be accommodated stability is deduced by showing that the discrete solutions satisfy both l infty and w 1 1 bounds existence of discrete solutions and connections with brakke flows are established some numerical examples and application to related problems such as the phase field equations are also presented

connectivity of phase boundaries in strictly convex domains we consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problem inf int omega u dx m int omega frac 1 varepsilon w u frac varepsilon 2 big nabla u big 2 dx here w is a double well potential and omega subset r n is a strictly convex domain for e small this is closely related to the problem of partitioning into two subdomains of fixed volume where the subdomain boundaries correspond to the transitional boundary between phases motivated by this geometry problem we show that in a strictly convex domain stable critical points of the original variational problem have a connected thin transition layer separating the two phases this relates to work in gm where special geometries such as cylindrical domains were treated and is analogous to the results in cho which show that in a convex domain stable critical points of the corresponding unconstrained problem are constant the proof of connectivity employs tools from geometric measure theory including the co area formula and the isoperimetric inequality on manifolds the thinness of the transition layer follows from a separate calculation establishing spatial decay of solutions to the pure phases

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

on stable parametric finite element methods for the stefan problem and the mullins sekerka problem with applications to dendritic growth we introduce a parametric finite element approximation for the stefan problem with the gibbs thomson law and kinetic undercooling which mimics the underlying energy structure of the problem the proposed method is also applicable to certain quasi stationary variants such as the mullins sekerka problem in addition fully anisotropic energies are easily handled the approximation has good mesh properties leading to a well conditioned discretization even in three space dimensions several numerical computations including for dendritic growth and for snow crystal growth are presented

the encompassing principle and its application to testing nonnested hypotheses the encompassing principle is used to develop a testing framework which unifies the literature on non nested testing allowing analysis of the relationship between alternative tests and in particular enabling asymptotic and finite sample equivalences and identities to be established easily when they exist as well as embracing nested hypothesis testing the concept of the implicit null hypothesis of a test is introduced to show that the effective acceptance region for some tests extends beyond the acceptance region corresponding to the null of interest and so such tests can be inconsistent against fixed alternatives closely related to the nominal null and alternative the analysis is illustrated by an application to two non nested linear regression models and it is shown that the conventional f test as well as all the one degree of freedom non nested tests has an encompassing interpretation and that the f test is a complete parametric encompassing test

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

complexes which arise from a matrix and a vector resolutions of divisors on certain varieties of complexes abstract consider the polynomial ring r r0 x y where r0 is a normal domain and x1 g and yg f are matrices of indeterminates the r ideal j i1 xy imin f g y defines a variety of complexes over r0 the divisor class group of r j is isomorphic to cl r0 z i where i is an ideal of r j generated by appropriately chosen lower order minors of y we produce the minimal r free resolution of i i for all integers i 1 if f is greater than or equal to g then j is a generic residual intersection of the generic grade g complete intersection i1 x the resolutions that we produce in this case are in many ways analogous to resolutions of divisors on generic residual intersections of grade two perfect ideals or grade three gorenstein ideals

on two families of transformations to additivity for binary response data summary two families of power transformations for probabilities are introduced to model symmetric and asymmetric departures from the logistic model the aim is to provide in a comprehensive way a representation of alternative scales in terms of which to carry out the analysis of binary response data maximum likelihood methods are employed to estimate an appropriate transformation using glim tests for detection of symmetric and asymmetric departures are proposed

on the slow dynamics for the cahn hilliard equation in one space dimension we study the limiting behaviour of the solution of the cahn hilliard equation using energy type methods we assume that the initial data has a transition layer structure i e u epsilon approx pm 1 except near finitely many transition points we show that in the limit as epsilon rightarrow 0 the solution maintains its transition layer structure and the transition layers move slower than any power of epsilon

information gain and a general measure of correlation summary given a parametric model of dependence between two random quantities x and y the notion of information gain can be used to define a measure of correlation this definition of correlation generalizes both the usual product moment correlation coeffi cient for the bivariate normal model and the multiple correlation coefficient in the standard linear regression model the use of this information based correlation in a descriptive statistical analysis is examined and several examples are given if the dependence between two random quantities x and y is modelled parametri cally then the concept of information gain can be used to define a measure of correlation this correlation coefficient can appear in two possible contexts depending on whether one models the joint distribution of x and y or just the conditional distribution of y given x an important motivating feature for this information based correlation coefficient is the fact that it generalizes both the usual product moment correlation coefficient for the bivariate normal model and the usual multiple correlation coefficient for the standard multiple regression model with normal errors our intuition is well developed for these usual correlation coefficients and hopefully our intuition will still be applicable for the information based correlation in more general modelling situations of parametric dependence further since our correlation coefficient is based on information gain we might hope to extend our intuition to interpret the information gain in any statistical modelling situation where we want to assess how much better a more complicated model is than a simpler model the concept of information gain for general statistical models is described in 2 this concept is then used to define an information based measure of correlation the joint case is covered in 3 and the conditional case in 4 estimation of the correlation coefficient is carried out by estimating the corresponding information gain see 5 7 the use of information gain for the purpose of model choice in a descriptive statistical analysis is discussed in 8 and a comparison between our approach and akaike s information criterion is given in 9 some examples of the use of this information based correlation are given in 10 and 11

the minimal free resolutions of the huneke ulrich deviation two gorenstein ideals structure theorems exist for gorenstein ideals of small grade a different parameter one can use for organizing ideals is deviation or complete intersection defect the deviation of the ideal i is the minimal number of generators of i minus the grade of i a deviation zero ideal is a complete intersection kunz s proved that there are no deviation one gorenstein ideals presently people are trying to find all deviation two gorenstein ideals see for example 3 a hypersurface section of a grade g 1 deviation two gorenstein ideal is a grade g deviation two gorenstein ideal we consider this to be a trivial example very few non trivial examples of deviation two gorenstein ideals are known the buchsbaum eisenbud structure theorem 2 describes all grade three gorenstein ideals in particular every deviation two grade three gorenstein ideal is generated by the maximal order pfaflians of a 5 x 5 alternating matrix no non trivial deviation two gorenstein ideals of even grade are known herzog and miller 3 have taken a step toward proving that all grade four deviation two gorenstein ideals are trivial let i be a grade four deviation two gorenstein ideal if i is a generic complete intersection and i i is cohenmacaulay then i is a hypersurface section the only known non trivial deviation 2 gorenstein ideals of grade at least four are the ideals of huneke and ulrich 4 which are the ideals studied in this paper for each integer s let y be a 1 x s matrix of indeterminates x be an s x s alternating matrix of indeterminates g be the product y x and ct be the sequence gp g let a be the ideal of

measures of dependence for censored survival data in the linear regression model with normal errors the squared product moment correlation provides the standard measure of dependence between the explanatory variable and the response variable using the concept of information gain a measure of dependence can also be defined for more general regression models used in survival analysis such as the weibull regression model or cox s proportional hazards model further this measure of dependence can be conveniently estimated even when the response variable is subject to censoring and a simple approximation is available

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

analysis of mixed interior penalty discontinuous galerkin methods for the cahn hilliard equation and the hele shaw flow this paper proposes and analyzes two fully discrete mixed interior penalty discontinuous galerkin dg methods for the fourth order nonlinear cahn hilliard equation both methods use the backward euler method for time discretization and interior penalty dg methods for spatial discretization they differ from each other on how the nonlinear term is treated one of them is based on fully implicit time stepping and the other uses energy splitting time stepping the primary goal of the paper is to prove the convergence of the numerical interfaces of the dg methods to the interface of the hele shaw flow this is achieved by establishing error estimates that depend on epsilon 1 only in some low polynomial orders instead of exponential orders similar to x feng and a prohl numer math 74 2004 pp 47 84 the crux is to prove a discrete spectrum estimate in the discontinuous galerkin finite element space however the validity of such a result is not obvious because the dg space is not a subspa