the parabolic map abstract in this paper we study certain aspects of bruhat order on coxeter groups letwbe a coxeter group and let p denote the set of parabolic subgroups ofw we construct a mapm w p wand examine its first properties we prove the following characterization wis the longest element of a parabolic subgroup ofwif and only if for anyv w there exists a unique maximal element with respect to bruhat order less than or equal to bothvandw

equivalence between conditional and random effects likelihoods for pair matched case control studies two approaches dominate the analysis of highly stratified data sets use of conditional likelihood and random effects models conditioning is more traditional and has attractive robustness properties contemporary random effects approaches rely more on accurate assumptions but can be applied to a larger class of problems for pair matched studies with arbitrary numbers of categorical covariates we reconcile these two approaches showing that the conditional approach has an exact interpretation as a specific type of random effects model the random effects models that provide equivalence with conditioning naturally inherit all the attractive properties of that approach and we argue they are a pragmatic model choice we also discuss desirable characteristics of the random effects model that are entirely novel and that have foundational implications for example although our model is specified entirely within a full likelihood framework its assumed random effects distribution does not need a parametric f

isogeometric kirchhoff love shell formulations for general hyperelastic materials abstract we present formulations for compressible and incompressible hyperelastic thin shells which can use general 3d constitutive models the necessary plane stress condition is enforced analytically for incompressible materials and iteratively for compressible materials the thickness stretch is statically condensed and the shell kinematics are completely described by the first and second fundamental forms of the midsurface we use c 1 continuous isogeometric discretizations to build the numerical models numerical tests including structural dynamics simulations of a bioprosthetic heart valve show the good performance and applicability of the presented methods

pattern avoidance and rational smoothness of schubert varieties abstract letwbe an element of the weyl groupsn and letxwbe the schubert variety associated towin the flag manifoldsln c b lakshmibai and sandhya showed thatxwis smooth if and only ifwavoids the patterns 4231 and 3412 using two tests for rational smoothness from carrell and peterson we show that rational smoothness ofxwis characterized by pattern avoidance for typesbandcas well a key step in the proof of this result is a sequence of rules for factoring the poincare polynomials for the cohomology ring ofxw generalizing the work of gasharov patterns can also be used to characterize actual smoothness of schubert varieties

rationally smooth schubert varieties and inversion hyperplane arrangements abstract we show that an element w of a finite weyl group w is rationally smooth if and only if the hyperplane arrangement i associated to the inversion set of w is inductively free and the product d 1 1 d l 1 of the coexponents d 1 d l is equal to the size of the bruhat interval e w where e is the identity in w as part of the proof we describe exactly when a rationally smooth element in a finite weyl group has a chain billey postnikov decomposition for finite coxeter groups we show that chain billey postnikov decompositions are connected with certain modular coatoms of i

conditional and unconditional categorical regression models with missing covariates summary we consider methods for analyzing categorical regression models when some covariates 2 are completely observed but other covariates x are missing for some subjects when data on x are missing at random i e when the probability that x is observed does not depend on the value of x itself we present a likelihood approach for the observed data that allows the same nuisance parameters to be eliminated in a conditional analysis as when data are complete an example of a matched casecontrol study is used to demonstrate our approach

closures of k orbits in the flag variety for u p q we give a pattern avoidance criterion for the conjugates of the bottom vertex of the bruhat graph attached to an on orbit mathcal o in the flag variety of gln to have degree equal to the rank of this graph as a poset this condition is known to be necessary for rational smoothness of overline mathcal o we conjecture that it is also sufficient proving that this holds for n even

billey postnikov decompositions and the fibre bundle structure of schubert varieties a theorem of ryan and wolper states that a type a schubert variety is smooth if and only if it is an iterated fibre bundle of grassmannians we extend this theorem to arbitrary finite type showing that a schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth grassmannian schubert varieties the proof depends on deep combinatorial results of billey postnikov on weyl groups we determine all smooth and rationally smooth grassmannian schubert varieties and give a new proof of peterson s theorem that all simply laced rationally smooth schubert varieties are smooth taken together our results give a fairly complete geometric description of smooth and rationally smooth schubert varieties using primarily combinatorial methods

handling missing data in matched case control studies using multiple imputation analysis of matched case control studies is often complicated by missing data on covariates analysis can be restricted to individuals with complete data but this is inefficient and may be biased multiple imputation mi is an efficient and flexible alternative we describe two mi approaches the first uses a model for the data on an individual and includes matching variables the second uses a model for the data on a whole matched set and avoids the need to model the matching variables within each approach we consider three methods full conditional specification fcs joint model mi using a normal model and joint model mi using a latent normal model we show that fcs mi is asymptotically equivalent to joint model mi using a restricted general location model that is compatible with the conditional logistic regression analysis model the normal and latent normal imputation models are not compatible with this analysis model all methods allow for multiple partially observed covariates non monotone missingness and multiple controls per case they can be easily applied in standard statistical software and valid variance estimates obtained using rubin s rules we compare the methods in a simulation study the approach of including the matching variables is most efficient within each approach the fcs mi method generally yields the least biased odds ratio estimates but normal or latent normal joint model mi is sometimes more efficient all methods have good confidence interval coverage data on colorectal cancer and fibre intake from the epic norfolk study are used to illustrate the methods in particular showing how efficiency is gained relative to just using individuals with complete data

missing exposure data in stereotype regression model application to matched case control study with disease subclassification with advances in modern medicine and clinical diagnosis case control data with characterization of finer subtypes of cases are often available in matched case control studies missingness in exposure values often leads to deletion of entire stratum and thus entails a significant loss in information when subtypes of cases are treated as categorical outcomes the data are further stratified and deletion of observations becomes even more expensive in terms of precision of the category specific odds ratio parameters especially using the multinomial logit model the stereotype regression model for categorical responses lies intermediate between the proportional odds and the multinomial or baseline category logit model the use of this class of models has been limited as the structure of the model implies certain inferential challenges with nonidentifiability and nonlinearity in the parameters we illustrate how to handle missing data in matched case control studies with finer disease subclassification within the cases under a stereotype regression model we present both monte carlo based full bayesian approach and expectation conditional maximization algorithm for the estimation of model parameters in the presence of a completely general missingness mechanism we illustrate our methods by using data from an ongoing matched case control study of colorectal cancer simulation results are presented under various missing data mechanisms and departures from modeling assumptions

inferences about exposure disease associations using probability of exposure information unconditional and conditional likelihood methods are proposed for modeling odds and odds ratios relating a binary disease variable and an exposure variable that is only known probabilistically e g via measurement of a surrogate exposure variable while adjusting appropriately for a vector of covariables assuming nondifferential errors a probabilistic structure is developed that permits analysis of the marginal relation between disease and the surrogate while maintaining important properties of the marginal relation between disease and true exposure in particular analyses conditioned on marginal totals in surrogate disease tables remove the same nuisance parameters as are removed in a conditional analysis of true exposure disease tables in addition a relation between the probability of exposure poe in the disease and disease free groups is derived permitting the use of information about exposure from populations with structure different from the study population examples are presented illustr

a criterion for smoothness of schubert varieties insp 2n b abstract in this paper we give a criterion for a schubert variety x w in sp 2 n b to be smooth the criterion is given in terms of the permutation giving w

semiparametric inference in matched case control studies with missing covariate data summary we consider the problem of matched studies with a binary outcome that are analysed using conditional logistic regression and for which data on some covariates are missing for some study participants methods for this problem involve either modelling the distribution of missing covariates or modelling the probability of data being missing for this second approach the previously proposed method did not make use of data for those persons with missing covariate data except in the model for the missingness we propose a new class of estimators that use outcome and available covariate data for all study participants and show that a particular member of this class always has better efficiency than the previously proposed estimator we illustrate the efficiency gains that are possible with our approach using simulated data

projection based stabilization of interface lagrange multipliers in immersogeometric fluid thin structure interaction analysis with application to heart valve modeling this paper discusses a method of stabilizing lagrange multiplier fields used to couple thin immersed shell structures and surrounding fluids the method retains essential conservation properties by stabilizing only the portion of the constraint orthogonal to a coarse multiplier space this stabilization can easily be applied within iterative methods or semi implicit time integrators that avoid directly solving a saddle point problem for the lagrange multiplier field heart valve simulations demonstrate applicability of the proposed method to 3d unsteady simulations an appendix sketches the relation between the proposed method and a high order accurate approach for simpler model problems

pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety abstract we give a pattern avoidance criterion to classify the orbits of sp p c sp q c resp gl n c on the flag variety of type c p q resp d n with rationally smooth closure we show that all such orbit closures fiber with smooth fiber over a smaller flag variety and hence are in fact smooth in addition we prove that the classification is insensitive to isogeny

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

likelihood methods for missing covariate data in highly stratified studies summary the paper considers canonical link generalized linear models with stratum specific nuisance intercepts and missing covariate data this family includes the conditional logistic regression model existing methods for this problem each of which uses a conditioning argu ment to eliminate the nuisance intercept model either the missing covariate data or the missingness process the paper compares these methods under a common likelihood framework the semiparametric efficient estimator is identified and a new estimator which reduces dependence on the model for the missing covariate is proposed a simulation study compares the methods with respect to efficiency and robustness to model misspecification

the nil hecke ring and singularity of schubert varieties let g be a semi simple simply connected complex algebraic group and t b a maximal torus and a borel subgroup respectively let lie t be the cartan subalgebra of the lie algebra lie g and w n t t the weyl group associated to the pair g t where n t is the normalizer of t in g we can view any element omega overline omega mod t i w as the element denoted by the corresponding german character xo of g b denned as m overline omega b for any w i w there is associated the schubert variety rm x omega overline b omega b b subset g b and the t fixed points of x w under the canonical left action are precisely rm i omega left o upsilon in wand upsilon leqslant omega right

fitting stratified proportional odds models by amalgamating conditional likelihoods classical methods for fitting a varying intercept logistic regression model to stratified data are based on the conditional likelihood principle to eliminate the stratum specific nuisance parameters when the outcome variable has multiple ordered categories a natural choice for the outcome model is a stratified proportional odds or cumulative logit model however classical conditioning techniques do not apply to the general k category cumulative logit model k 2 with varying stratum specific intercepts as there is no reduction due to sufficiency the nuisance parameters remain in the conditional likelihood we propose a methodology to fit stratified proportional odds model by amalgamating conditional likelihoods obtained from all possible binary collapsings of the ordinal scale the method allows for categorical and continuous covariates in a general regression framework we provide a robust sandwich estimate of the variance of the proposed estimator for binary exposures we show equivalence of our approach to the estimators already proposed in the literature the proposed recipe can be implemented very easily in standard software we illustrate the methods via three real data examples related to biomedical research simulation results comparing the proposed method with a random effects model on the stratification parameters are also furnished

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

smooth points of t stable varieties in g b and the peterson map let g be a connected semi simple algebraic group defined over an algebraically closed field k and let t b p be respectively a maximal torus a borel subgroup and a parabolic subgroup of g inspired by a beautiful result of dale peterson describing the singular locus of a schubert variety in g b we characterize the t fixed points in the singular locus of an arbitrary irreducible t stable subvariety of g p a t variety for short peterson s result cf the deformation theorem 1 says that if k then a schubert variety x g b is smooth at a t fixed point x if and only if it is smooth at every t fixed point y x in the bruhat chevalley order on the fixed point set x t and all the limits c x x lim z x t z x z c c t of the zariski tangent spaces t z x of x coincide as c varies over the set of all t stable curves in x with c t x y where y x using this peterson showed that if g is simply laced and defined over then every rationally smooth point of a schubert variety in g b is smooth more generally the deformation c x x is defined for any k variety x with a t action provided c is what we call good i e c is a curve of the form c overline tz where z is a smooth point of x x t and x c t our first main result theorem 1 4 says that if x x is an attractive fixed point then x is smooth at x if and only if there exist at least two good c containing x such that c x x te x x where te x x denotes the span of the tangent lines of the t stable curves in x containing x in addition if x is cohen macaulay at x and c x x te x x for even one good c then x is smooth at x our second main result theorem 1 6 says that if x is a t variety in g p where g is simply laced then c x x te x x for each good c this is not true for general g but when g has no g2 factors then c x x is always contained in the linear span of the reduced tangent cone to x at x these results lead to several descriptions of the smooth fixed points of a t variety in g p and in particular they give simple proofs of peterson s results valid for any algebraically closed field we also show cf example 7 1 that there can exist t stable subvarieties in g b where g is simply laced which have rationally smooth t fixed points in their singular loci