bayesian sensitivity analysis for unmeasured confounding in observational studies we consider bayesian sensitivity analysis for unmeasured confounding in observational studies where the association between a binary exposure binary response measured confounders and a single binary unmeasured confounder can be formulated using logistic regression models a model for unmeasured confounding is presented along with a family of prior distributions that model beliefs about a possible unknown unmeasured confounder simulation from the posterior distribution is accomplished using markov chain monte carlo because the model for unmeasured confounding is not identifiable standard large sample theory for bayesian analysis is not applicable consequently the impact of different choices of prior distributions on the coverage probability of credible intervals is unknown using simulations we investigate the coverage probability when averaged with respect to various distributions over the parameter space the results indicate that credible intervals will have approximately nominal coverage probability on average when the prior distribution used for sensitivity analysis approximates the sampling distribution of model parameters in a hypothetical sequence of observational studies we motivate the method in a study of the effectiveness of beta blocker therapy for treatment of heart failure

radial solutions of δu f u 0 with prescribed numbers of zeros u 0 0 u r f 0 as r 03 1 2 here n 1 is a real parameter and the function f satisfies the following hypotheses f 1 f r r is locally lipschitz continuous f2 uf u 0 and j 0 such that f u 0 on 0 pi f u 0 on uc 0 f u 0 on b 01 f u 0 on a pi where f u f r f s ds f4 f u k u jujp u g u where

testing hypotheses in order in certain circumstances one wishes to test one hypothesis only if certain other hypotheses have been rejected this ordering of hypotheses simplifies the task of controlling the probability of rejecting any true hypothesis in an example from an observational study a treated group is shown to be further from both of two control groups than the two control groups are from each other

identifiability improper priors and gibbs sampling for generalized linear models markov chain monte carlo algorithms are widely used in the fitting of generalized linear models glms such model fitting is somewhat of an art form requiring suitable trickery and tuning to obtain results in which one can have confidence a wide range of practical issues arise the focus here is on parameter identifiability and posterior propriety in particular we clarify that nonidentifiability arises for usual glms and discuss its implications for simulation based model fitting because often some part of the prior specification is vague we consider whether the resulting posterior is proper providing rather general and easily checked results for glms we also show that if a gibbs sampler is run with an improper posterior then it may be possible to use the output to obtain meaningful inference for certain model unknowns

conditional permutation tests and the propensity score in observational studies abstract in observational studies the distribution of treatment assignments is unknown and therefore randomization tests are not generally applicable however permutation tests that condition on sample information about the treatment assignment mechanism can be applicable in observational studies provided treatment assignment is strongly ignorable these tests use the conditional distribution of the treatment assignments given a sufficient statistic for the unknown parameter of the propensity score several tests that are commonly used in observational studies are particular instances of this general procedure moreover conditional permutation tests and covariance adjustment are closely related in the sense that the conditional permutation distribution of the covariance adjusted difference leads to the same inferences as the conditional permutation distribution of the unadjusted difference of sample means a backtrack algorithm is developed to permit efficient calculation of the exact conditional si

on non linear transformations for accurate numerical evaluation of weakly singular boundary integrals this paper presents a study of the performance of the non linear co ordinate transformations in the numerical integration of weakly singular boundary integrals a comparison of the smoothing property numerical convergence and accuracy of the available non linear polynomial transformations is presented for two dimensional problems effectiveness of generalized transformations valid for any type and location of singularity has been investigated it is found that weakly singular integrals are more efficiently handled with transformations valid for end point singularities by partitioning the element at the singular point further transformations which are excellent for cpv integrals are not as accurate for weakly singular integrals connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated cubic transformation is seen to be the optimum choice for single precision and quartic or quintic one for double precision computations a new approach which combines the method of singularity subtraction with non linear transformation has been proposed this composite approach is found to be more accurate efficient and robust than the singularity subtraction method and the non linear transformation methods copyright 2001 john wiley sons ltd

simplified bayesian sensitivity analysis for mismeasured and unobserved confounders we examine situations where interest lies in the conditional association between outcome and exposure variables given potential confounding variables concern arises that some potential confounders may not be measured accurately whereas others may not be measured at all some form of sensitivity analysis might be employed to assess how this limitation in available data impacts inference a bayesian approach to sensitivity analysis is straightforward in concept a prior distribution is formed to encapsulate plausible relationships between unobserved and observed variables and posterior inference about the conditional exposure disease relationship then follows in practice though it can be challenging to form such a prior distribution in both a realistic and simple manner moreover it can be difficult to develop an attendant markov chain monte carlo mcmc algorithm that will work effectively on a posterior distribution arising from a highly nonidentified model in this article a simple prior distribution for acknowledging both poorly measured and unmeasured confounding variables is developed it requires that only a small number of hyperparameters be set by the user moreover a particular computational approach for posterior inference is developed because application of mcmc in a standard manner is seen to be ineffective in this problem

a bi cubic transformation for the numerical evaluation of the cauchy principal value integrals in boundary methods the numerical strategies employed in the evaluation of singular integrals existing in the cauchy principal value cpv sense are undoubtedly one of the key aspects which remarkably affect the performance and accuracy of the boundary element method bem thus a new procedure based upon a bi cubic co ordinate transformation and oriented towards the numerical evaluation of both the cpv integrals and some others which contain different types of singularity is developed both the ideas and some details involved in the proposed formulae are presented obtaining rather simple and attractive expressions for the numerical quadrature which are also easily embodied into existing bem codes some illustrative examples which assess the stability and accuracy of the new formulae are included

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

acceleration of gauss legendre quadrature for an integrand with an endpoint singularity an asymptotic error expansion for gauss legendre quadrature is derived for an integrand with an endpoint singularity it permits convergence acceleration by extrapolation

singular basis functions and curvilinear triangles in the solution of the electric field integral equation basis functions are formulated that account for singularities in the charge density near an edge on a conducting body the formulation is general and the basis functions are valid for planar as well curvilinear geometries in principle singularities of any order can be treated but best results are obtained for so called knife edge singularities results are compared with exact solutions or measurements where available for some simple problems

singular higher order divergence conforming bases of additive kind and moments method applications to 3d sharp wedge structures we present new subsectional singular divergence conforming vector bases that incorporate the edge conditions for conducting wedges the bases are of additive kind because obtained by incrementing the regular polynomial vector bases with other subsectional basis sets that model the singular behavior of the unknown vector field in the wedge neighborhood singular bases of this kind complete to arbitrarily high order are described in a unified and consistent manner for curved quadrilateral and triangular elements the higher order basis functions are obtained as the product of lowest order functions and silvester lagrange interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties are discussed and these bases are proved to be fully compatible with the standard high order regular vector bases used in adjacent elements our singular bases guarantee normal continuity along the edges of the elements allowing for the discontinuity of tangential components adequate modelling of the divergence and removal of spurious solutions these singular high order bases provide more accurate and efficient numerical solutions of surface integral problems several test case problems are considered in the paper thereby obtaining highly accurate numerical results for the current and charge density induced on 3d sharp wedge structures the results are compared with other solutions when available and confirm the faster convergence of these bases on wedge problems

assessing the sensitivity of regression results to unmeasured confounders in observational studies this paper presents a general approach for assessing the sensitivity of the point and interval estimates of the primary exposure effect in an observational study to the residual confounding effects of unmeasured variable after adjusting for measured covariates the proposed method assumes that the true exposure effect can be represented in a regression model that includes the exposure indicator as well as the measured and unmeasured confounders one can use the corresponding reduced model that omits the unmeasured confounder to make statistical inferences about the true exposure effect by specifying the distributions of the unmeasured confounder in the exposed and unexposed groups along with the effects of the unmeasured confounder on the outcome variable under certain conditions there exists a simple algebraic relationship between the true exposure effect in the full model and the apparent exposure effect in the reduced model one can then estimate the true exposure effect by making a simple adjustment to the point and interval estimates of the apparent exposure effect obtained from standard software or published reports the proposed method handles both binary response and censored survival time data accommodates any study design and allows the unmeasured confounder to be discrete or normally distributed we describe applications on two major medical studies

on nonparametric measures of dependence for random variables in 1959 a renyi proposed a set of axioms for a measure of dependence for pairs of random variables in the same year a sklar introduced the general notion of a copula this is a function which links an n dimensional distribution function to its one dimensional margins and is itself a continuous distribution function on the unit n cube with uniform margins we show that the copula of a pair of random variables x y is invariant under a s strictly increasing transformations of x and y and that any property of the joint distribution function of x and y which is invariant under such transformations is solely a function of their copula exploiting these facts we use copulas to define several natural nonparametric measures of dependence for pairs of random variables we show that these measures satisfy reasonable modifications of renyi s conditions and compare them to various known measures of dependence e g the correlation coefficient and spearman s rho

the cross cut statistic and its sensitivity to bias in observational studies with ordered doses of treatment a common practice with ordered doses of treatment and ordered responses perhaps recorded in a contingency table with ordered rows and columns is to cut or remove a cross from the table leaving the outer corners that is the high versus low dose high versus low response corners and from these corners to compute a risk or odds ratio this little remarked but common practice seems to be motivated by the oldest and most familiar method of sensitivity analysis in observational studies proposed by cornfield et al 1959 which says that to explain a population risk ratio purely as bias from an unobserved binary covariate the prevalence ratio of the covariate must exceed the risk ratio quite often the largest risk ratio hence the one least sensitive to bias by this standard is derived from the corners of the ordered table with the central cross removed obviously the corners use only a portion of the data so a focus on the corners has consequences for the standard error as well as for bias but sampling variability was not a consideration in this early and familiar form of sensitivity analysis where point estimates replaced population parameters here this cross cut analysis is examined with the aid of design sensitivity and the power of a sensitivity analysis

error estimation of quadrature rules for evaluating singular integrals in boundary element problems the efficient numerical evaluation of integrals arising in the boundary element method is of considerable practical importance the superiority of the use of sigmoidal and semi sigmoidal transformations together with gauss legendre quadrature in this context has already been well demonstrated numerically by one of the authors in this paper the authors obtain asymptotic estimates of the truncation errors for these algorithms these estimates allow an informed choice of both the transformation and the quadrature error in the evaluation of boundary element integrals with algebraic or algebraic logarithmic singularities at an end point of the interval of integration copyright 2000 john wiley sons ltd

strong control of the familywise error rate in observational studies that discover effect modification by exploratory methods an effect modifier is a pretreatment covariate that affects the magnitude of the treatment effect or its stability when there is effect modification an overall test that ignores an effect modifier may be more sensitive to unmeasured bias than a test that combines results from subgroups defined by the effect modifier if there is effect modification one would like to identify specific subgroups for which there is evidence of effect that is insensitive to small or moderate biases in this paper we propose an exploratory method for discovering effect modification and combine it with a confirmatory method of simultaneous inference that strongly controls the familywise error rate in a sensitivity analysis despite the fact that the groups being compared are defined empirically a new form of matching strength k matching permits a search through more than k covariates for effect modifiers in such a way that no pairs are lost provided that at most k covariates are selected to group the pairs in a strength k match each set of k covariates is exactly balanced although a set of more than k covariates may exhibit imbalance we apply the proposed method to study the effects of the earthquake that struck chile in 2010

infinitely many solutions to fourth order superlinear periodic problems we present a new min max approach to the search of multiple t periodic solutions to a class of fourth order equations u iv t cu t f t u t t 0 t where f t u is continuous t periodic in t and satisfies a superlinearity assumption when u for every n n we prove the existence of a t periodic solution having exactly 2n zeroes in 0 t

transformations for evaluating singular boundary element integrals accurate numerical integration of line integrals is of fundamental importance for the reliable implementation of the boundary element method usually the regular integrals arising from a boundary element method implementation are evaluated using standard gaussian quadrature however the singular integrals which arise are often evaluated in another way sometimes using a different integration method with different nodes and weights this paper presents a straightforward transformation to improve the accuracy of evaluating singular integrals the transformation is in a sense a generalisation of the popular method of telles with the underlying idea being to utilise the same gaussian quadrature points as used for evaluating nonsingular integrals in a typical boundary element method implementation the new transformation is also shown to be equivalent to other existing transformations in certain situations comparison of the new method with existing coordinate transformation techniques shows that a more accurate evaluation of weakly singular integrals can be obtained the technique can also be extended to evaluate certain hadamard finite part integrals based on the observation of several integrals considered guidelines are suggested for the best transformation order to use i e the degree to which nodes should be clustered near the singular point

on obtaining permutation distributions in polynomial time abstract polynomial time algorithms are presented for finding the permutation distribution of any statistic that is a linear combination of some function of either the original observations or the ranks this class of statistics includes the original fisher two sample location statistic and such common nonparametric statistics as the wilcoxon ansari bradley savage and many others the algorithms are presented for the two sample problem and it is shown how to extend them to the multisample problem for example to find the distribution of the kruskal wallis and other extensions of the wilcoxon and to the single sample situation stratification ties and censored observations are also easily handled by the algorithms the algorithms require polynomial time as opposed to complete enumeration algorithms which require exponential time this savings is effected by first calculating and then inverting the characteristic function of the statistic

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

asymptotic estimation of gaussian quadrature error for a nonsingular integral in potential theory this paper considers the n point gauss jacobi approximation of nonsingular integrals of the form 11 t t log z t dt with jacobi weight and polynomial and derives an estimate for the quadrature error that is asymptotic as n the approach follows that previously described by donaldson and elliott a numerical example illustrating the accuracy of the asymptotic estimate is presented the extension of the theory to similar integrals defined on more general analytic arcs is outlined

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

adjusting for unmeasured confounding due to either of two crossed factors with a logistic regression model motivated by an investigation of the effect of surface water temperature on the presence of vibrio cholerae in water samples collected from different fixed surface water monitoring sites in haiti in different months we investigated methods to adjust for unmeasured confounding due to either of the two crossed factors site and month in the process we extended previous methods that adjust for unmeasured confounding due to one nesting factor such as site which nests the water samples from different months to the case of two crossed factors first we developed a conditional pseudolikelihood estimator that eliminates fixed effects for the levels of each of the crossed factors from the estimating equation using the theory of u statistics for independent but non identically distributed vectors we show that our estimator is consistent and asymptotically normal but that its variance depends on the nuisance parameters and thus cannot be easily estimated consequently we apply our estimator in conjunction with a permutation test and we investigate use of the pigeonhole bootstrap and the jackknife for constructing confidence intervals we also incorporate our estimator into a diagnostic test for a logistic mixed model with crossed random effects and no unmeasured confounding for comparison we investigate between within models extended to two crossed factors these generalized linear mixed models include covariate means for each level of each factor in order to adjust for the unmeasured confounding we conduct simulation studies and we apply the methods to the haitian data copyright 2016 john wiley sons ltd

integral transformations to handle corner function singularities recent approaches to model corner singularities in electromagnetic analysis require a treatment substantially different from that of edge singularities in this article new algorithms are proposed for handling the combination of corner singularities and green s function singularities on quadrilateral cells in method of moments procedures

sensitivity analysis for matched pair analysis of binary data from worst case to average case analysis in matched observational studies where treatment assignment is not randomized sensitivity analysis helps investigators determine how sensitive their estimated treatment effect is to some unmeasured confounder the standard approach calibrates the sensitivity analysis according to the worst case bias in a pair this approach will result in a conservative sensitivity analysis if the worst case bias does not hold in every pair in this paper we show that for binary data the standard approach can be calibrated in terms of the average bias in a pair rather than worst case bias when the worst case bias and average bias differ the average bias interpretation results in a less conservative sensitivity analysis and more power in many studies the average case calibration may also carry a more natural interpretation than the worst case calibration and may also allow researchers to incorporate additional data to establish an empirical basis with which to calibrate a sensitivity analysis we illustrate this with a study of the effects of cellphone use on the incidence of automobile accidents finally we extend the average case calibration to the sensitivity analysis of confidence intervals for attributable effects

multiplicity of positive and nodal solutions for scalar field equations abstract in this paper the question of finding infinitely many solutions to the problem u a x u u p 2 u in r n u h 1 r n is considered when n 2 p 2 2 n n 2 and the potential a x is a positive function which is not required to enjoy symmetry properties assuming that a x satisfies a suitable slow decay at infinity condition and moreover that its graph has some dips we prove that the problem admits either infinitely many nodal solutions or infinitely many constant sign solutions the proof method is purely variational and allows to describe the shape of the solutions

design sensitivity in observational studies summary outside the field of statistics the literature on observational studies offers advice about research designs or strategies for judging whether or not an association is causal such as multiple operationalism or a dose response relationship these useful suggestions are typically informal and qualitative a quantitative measure design sensitivity is proposed for measuring the contribution such strategies make in distinguishing causal effects from hidden biases several common strategies are then evaluated in terms of their contribution to design sensitivity a related method for computing the power of a sensitivity analysis is also developed

gaussian type quadrature rules for muntz systems a method for constructing the generalized gaussian quadrature rules for muntz polynomials on 0 1 is given such quadratures possess several properties of the classical gaussian formulae for polynomial systems such as positivity of the weights rapid convergence etc they can be applied to the wide class of functions including smooth functions as well as functions with end point singularities such as those in boundary contact value problems integral equations with singular kernels complex analysis potential theory etc the constructive method is based on an application of orthogonal muntz polynomials introduced by badalyan it akad nauk armyan ssr izv fiz mat estest tehn nauk 8 1955 pp 1 28 9 1956 pp 3 22 russian armenian summary and studied intensively by borwein erdelyi and zhang it trans amer math soc 342 1994 pp 523 542 as well as by milovanovic on a numerical procedure for evaluation of such polynomials with high precision it muntz orthogonal polynomials and their numerical evaluation in applications and computation of orthogonal polynomials internat ser numer math 131 w gautschi g h golub and g opfer eds birkhauser basel 1999 pp 179 194 the method is numerically stable some numerical examples are included

testing one hypothesis twice in observational studies in a matched observational study of treatment effects a sensitivity analysis asks about the magnitude of the departure from random assignment that would need to be present to alter the conclusions of an analysis that assumes that matching for measured covariates removes all bias the reported degree of sensitivity to unmeasured biases depends on both the process that generated the data and the chosen methods of analysis so a poor choice of method may lead to an exaggerated report of sensitivity to bias this suggests the possibility of performing more than one analysis with a correction for multiple inference say testing one null hypothesis using two or three different tests in theory and in an example it is shown that in large samples the gains from testing twice will often be large because testing twice has the larger of the two design sensitivities of the component tests and the losses due to correcting for two tests will often be small because two tests of one hypothesis will typically be highly correlated so a correction for multiple testing that takes this into account will be small an illustration uses data from the u s national health and nutrition examination survey concerning lead in the blood of cigarette smokers copyright 2012 oxford university press

strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics abstract this paper investigates singular initial problems p t u p t f u u 0 b u 0 0 on the half line 0 here b 0 is a parameter p 0 0 and p t 0 on 0 f l 0 for some l 0 and x f x 0 if l 0 x l and x 0 the existence of a strictly increasing solution to the problem for which there exists a finite c 0 such that u c l is discussed this is fundamental for the existence of a strictly increasing solution of the problem having its limit equal to l as t which has great importance in applications

multiplicity results for sign changing bound state solutions of a semilinear equation abstract in this paper we give conditions on f so that the problem u f u 0 x r n n 2 has at least two radial bound state solutions with any prescribed number of zeros and such that u 0 belongs to a specific subinterval of 0 this property will allow us to give conditions on f so that this problem has at least any given number of radial solutions having a prescribed number of zeros

numerical solution of the generalized airfoil equation for an airfoil with a flap in this paper we are concerned with the generalized airfoil equation which governs the pressure across an airfoil oscillating in a wind tunnel in particular we examine the case of an airfoil which contains a flap so that the input function has a simple jump at a point x0 1 x0 1 this generates a logarithmic singularity in the solution u at the same point x0 and considerably complicates the problem of the numerical approximation of u by introducing a nonlinear transformation of the independent variable in the integral equation we improve the behavior of the unknown function u through incorporation of the jacobian of the transformation the price is that the kernel of the new integral equation becomes more complicated nevertheless we are able to show that a galerkin method based on high order polynomials is both stable and convergent we also present some promising numerical results for the corresponding collocation method for which we have not succeeded in proving stability and convergence although those results seem to suggest that it is indeed stable and has a rate of convergence similar to that we have derived for the galerkin method

modeling junctions in sharp edge conducting structures with higher order method of moments scattering targets are often made by complex structures constituted by thin metallic plates as wings fins winglets when thin plates are connected together they define surface junctions with the possible presence of sharp edges in this paper we describe a complete procedure to handle junctions in presence of sharp edges in surface integral equation methods by defining the required basis functions and unknowns this approach is based on the use of divergence conforming higher order interpolatory vector basis functions singular vector basis functions and on kirchhoff s current law the paper presents several numerical test cases that show the instability of solutions using classical methods and the full convergence of the proposed numerical scheme

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

an adaptive mantel haenszel test for sensitivity analysis in observational studies in a sensitivity analysis in an observational study with a binary outcome is it better to use all of the data or to focus on subgroups that are expected to experience the largest treatment effects the answer depends on features of the data that may be difficult to anticipate a trade off between unknown effect sizes and known sample sizes we propose a sensitivity analysis for an adaptive test similar to the mantel haenszel test the adaptive test performs two highly correlated analyses one focused analysis using a subgroup one combined analysis using all of the data correcting for multiple testing using the joint distribution of the two test statistics because the two component tests are highly correlated this correction for multiple testing is small compared with for instance the bonferroni inequality the test has the maximum design sensitivity of two component tests a simulation evaluates the power of a sensitivity analysis using the adaptive test two examples are presented an r package sensitivity2x2xk implements the procedure

the behavior of electromagnetic fields at edges in bi isotropic and bi anisotropic materials the behavior of the electromagnetic fields is studied at the common edge of a number of wedges consisting of lossless biisotropic perfect electric conducting and or perfect magnetic conducting materials a general matrix formalism is constructed to find the transcendental equation for the singularity exponent this singularity exponent equation is examined in detail for two important cases 1 two bi isotropic wedges with common edge and 2 two bi isotropic wedges and a perfect electric conducting wedge a generalization of the theory to a special class of bi anisotropic wedges is also presented

an efficient implicit spectral element method for time dependent nonlinear diffusion equations by evaluating integrals at one quadrature point we present an implicit spectral element method to approximate the solution of time dependent nonlinear diffusion equations in complex geometries we propose a nodal expansion to approximate derivatives of an unknown function so integrals involving nonlinearities are evaluated at one quadrature point and discrete form of nonlinear diffusion equations is obtained without forming local and global stiffness matrices using the method of lines mol the nonlinear partial differential equation is reduced to a nonlinear system of ordinary differential equations such that the mol formulation is derived in terms of the differentiation matrix instead of the stiffness matrix since the arising system is stiff an implicit method is employed to solve it also with a simple algorithm we obtain a closed analytical form for the jacobian matrix of the system of nonlinear algebraic equations in addition the differentiation matrices for triangular elements and deformed quadrilateral and hexahedral elements are obtained for deformed elements an isoparametric mapping is used where the expansion coefficients are obtained with the aid of the gordon hall mapping several benchmarks including the scalar and system of these equations are carried out as physical applications the cell motion model over long time the tumour angiogenesis model on a mixed quadrangle triangle mesh and two and three dimensional p laplace equations for various choices of the parameter p are considered

nodal soliton solutions for generalized quasilinear schrodinger equations this paper is concerned with constructing nodal radial solutions for generalized quasilinear schrodinger equations in rn which arise from plasma physics fluid mechanics as well as high power ultashort laser in matter for any given integer k 0 by using a change of variables and minimization argument we obtain a sign changing minimizer with k nodes of a minimization problem

quantiles in nonrandom samples and observational studies abstract in a nonrandom sample or a nonrandomized comparison of treated and control groups methods are developed for displaying the sensitivity of confidence intervals for population quantiles to the magnitude of the departure from random selection quantiles of a single distribution and for a shift in two distributions are both discussed these confidence intervals are based on a new inequality for the distribution of the number of successes in a nonrandom sample or nonrandomized comparison the inequality has additional applications in particular it greatly accelerates the computations in a sensitivity analysis for the mantel haenszel statistic in a 2 2 s contingency table examples include a nonrandom collection of blood samples from a town with comparatively high levels of radon gas an observational study of chromosome damage from mercury and a study of the drug allopurinol as a cause of skin rash the proof of the inequality makes use of savage s lattice for ranks and holley s inequality wh

assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome this paper proposes a simple technique for assessing the range of plausible causal con clusions from observational studies with a binary outcome and an observed categorical covariate the technique assesses the sensitivity of conclusions to assumptions about an unobserved binary covariate relevant to both treatment assignment and response a medical study of coronary artery disease is used to illustrate the technique inevitably the results of clinical studies are subject to dispute in observational studies one basis for dispute is obvious since patients were not assigned to treatments at random patients at greater risk may be over represented in some treatment groups this paper proposes a method for assess ing the sensitivity of causal conclusions to an unmeasured patient characteristic relevant to both treatment assignment and response despite their limitations observational studies will continue to be a valuable source of information and therefore it is prudent to develop appropriate methods of analysis for them our sensitivity analysis consists of the estimation of the average effect of a treatment on a binary outcome variable after adjustment for observed categorical covariates and an unobserved binary covariate u under several sets of assumptions about u both cornfield et al 1959 and bross 1966 have proposed guidelines for determining whether an unmeasured binary covariate having specified properties could explain all of the apparent effect of a treatment that is whether the treatment effect after adjustment for u could be zero our method has two advantages first cornfield et al 1959 and bross 1966 adjust only for the unmeasured binary covariate u whereas we adjust for measured covariates in addition to the unmeasured covariate u second cornfield et al 1959 and bross 1966 1967 only judge whether the effect of the treatment could be zero having adjusted for u where cornfield et al 1959 employ an implicit yet extreme assumption about u in contrast we provide actual estimates of the treatment effect adjusted for both u and the observed categorical covariates under any assumption about u in principle the ith of the n patients under study has both a binary response r1i that would have resulted if he had received the new treatment and a binary response ro0 that would have resulted if he had received the control treatment in this formulation treatment effects are comparisons of r1i and roi such as r1i roi since each patient receives only one treatment either rli or ro0 is observed but not both and therefore comparisons of rli and roi imply some degree of speculation treatment effects defined as comparisons of the two potential responses r1i and roi of individual patients are implicit in fisher s 1953 randomization test of the sharp null

nodal soliton solutions for quasilinear schrodinger equations with critical exponent this paper is concerned with constructing nodal radial solutions for quasilinear schrodinger equations in rn with critical growth which have appeared as several models in mathematical physics for any given integer k 0 by using a change of variables and minimization argument we obtain a sign changing minimizer with k nodes of a minimization problem since the critical exponent appears and the lower order term may change sign we should use more delicate arguments

positive solutions of a second order singular ordinary differential equation we study the existence of positive solutions for the second order ordinary differential equation u k t u c t g u with the initial condition u 0 0 in bounded intervals 0 m and in 0 i here k g 1 c t is bounded and the growth of g at infinity is controlled by a critical exponent in our approach a shooting argument of beresticky lions and peletier is combined with variational methods c 2005 elsevier ltd all rights reserved

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

superlinear indefinite elliptic problems and pohoz aev type identities abstract we prove the existence of a nontrivial solution for a nonlinear elliptic problem u u a x g u with dirichlet boundary condition on a bounded domain where g is superlinear both at zero and at infinity a x changes sign and 0

amplification of sensitivity analysis in matched observational studies a sensitivity analysis displays the increase in uncertainty that attends an inference when a key assumption is relaxed in matched observational studies of treatment effects a key assumption in some analyses is that subjects matched for observed covariates are comparable and this assumption is relaxed by positing a relevant covariate that was not observed and not controlled by matching what properties would such an unobserved covariate need to have to materially alter the inference about treatment effects for ease of calculation and reporting it is convenient that the sensitivity analysis be of low dimension perhaps indexed by a scalar sensitivity parameter but for interpretation in specific contexts a higher dimensional analysis may be of greater relevance an amplification of a sensitivity analysis is defined as a map from each point in a low dimensional sensitivity analysis to a set of points perhaps a curve in a higher dimensional sensitivity analysis such that the possible inferences are

singular higher order complete vector bases for finite methods this paper presents new singular curl and divergence conforming vector bases that incorporate the edge conditions singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements the higher order basis functions are obtained as the product of lowest order functions and silvester lagrange interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties are discussed and these bases are proved to be fully compatible with the standard high order regular vector bases used in adjacent elements the curl divergence conforming singular bases guarantee tangential normal continuity along the edges of the elements allowing for the discontinuity of normal tangential components adequate modeling of the curl divergence and removal of spurious modes solutions these singular high order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems sample numerical results confirm the faster convergence of these bases on wedge problems

design of quadrature rules for muntz and muntz logarithmic polynomials using monomial transformation a method for constructing the exact quadratures for muntz and muntz logarithmic polynomials is presented the algorithm does permit to anticipate the precision machine precision of the numerical integration of muntz logarithmic polynomials in terms of the number of gauss legendre gl quadrature samples and monomial transformation order to investigate in depth the properties of classical gl quadrature we present new optimal asymptotic estimates for the remainder in boundary element integrals this quadrature rule can be applied to evaluate singular functions with end point singularity singular kernel as well as smooth functions the method is numerically stable efficient easy to be implemented the rule has been fully tested and several numerical examples are included the proposed quadrature method is more efficient in run time evaluation than the existing methods for muntz polynomials

radial solutions of superlinear equations on ℝn part i global variational approach we present a global variational approach to the search for multiple nodal solutions u h 1 n to a class of elliptic equations of type u x f x u x x n where n 2 f is superlinear and subcritical and f x 0

effect modification and design sensitivity in observational studies in an observational study of treatment effects subjects are not randomly assigned to treatment or control so differing outcomes in treated and control groups may reflect a bias from nonrandom assignment rather than a treatment effect after adjusting for measured pretreatment covariates perhaps by matching a sensitivity analysis determines the magnitude of bias from an unmeasured covariate that would need to be present to alter the conclusions of the naive analysis that presumes adjustments eliminated all bias other things being equal larger effects tend to be less sensitive to bias than smaller effects effect modification is an interaction between a treatment and a pretreatment covariate controlled by matching so that the treatment effect is larger at some values of the covariate than at others in the presence of effect modification it is possible that results are less sensitive to bias in subgroups experiencing larger effects two cases are considered i an a priori grouping into a few categ

radial solutions of superlinear equations on ℝn part ii the forced case we study the effect of a forcing term in the context of the search of multiple nodal solutions u h 1 n to a class of elliptic equations of type u x f x u x h x x n where f x 0 and f is superlinear but subcritical at infinity

a comparison of bayesian and monte carlo sensitivity analysis for unmeasured confounding bias from unmeasured confounding is a persistent concern in observational studies and sensitivity analysis has been proposed as a solution in the recent years probabilistic sensitivity analysis using either monte carlo sensitivity analysis mcsa or bayesian sensitivity analysis bsa has emerged as a practical analytic strategy when there are multiple bias parameters inputs bsa uses bayes theorem to formally combine evidence from the prior distribution and the data in contrast mcsa samples bias parameters directly from the prior distribution intuitively one would think that bsa and mcsa ought to give similar results both methods use similar models and the same prior probability distributions for the bias parameters in this paper we illustrate the surprising finding that bsa and mcsa can give very different results specifically we demonstrate that mcsa can give inaccurate uncertainty assessments e g 95 intervals that do not reflect the data s influence on uncertainty about unmeasured confounding using a data example from epidemiology and simulation studies we show that certain combinations of data and prior distributions can result in dramatic prior to posterior changes in uncertainty about the bias parameters this occurs because the application of bayes theorem in a non identifiable model can sometimes rule out certain patterns of unmeasured confounding that are not compatible with the data consequently the mcsa approach may give 95 intervals that are either too wide or too narrow and that do not have 95 frequentist coverage probability based on our findings we recommend that analysts use bsa for probabilistic sensitivity analysis copyright 2017 john wiley sons ltd