modified numerov s method for inverse sturm liouville problems in this paper we propose a new modified numerov s method for recovering from eigenvalues a symmetric potential of a sturm liouville operator with dirichlet boundary conditions we use interpolation to refine the mesh sufficiently for numerov s method to be effective even without the asymptotic correction technique of andrew and paine accuracy and stability of the method are investigated convergence of the method is established our method is extended to deal with natural boundary conditions numerical experiments confirm its competitiveness

asymptotic correction of numerov s eigenvalue estimates with natural boundary conditions abstract asymptotic correction at negligible extra cost greatly improves the accuracy of higher eigenvalues computed by finite difference or finite element methods and generally increases the accuracy of the lower ones as well this paper gives a brief overview of the technique and describes how its previous use with numerov s method may be extended to problems with natural boundary conditions numerical results indicate that it is just as successful as with dirichlet boundary conditions

a new numerical method for solving high order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional high order differential equations with variable coefficients it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results in particular fractional problems of the fourth sixth and eighth orders are considered the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of eigenvalue problems

empirical likelihood ratio confidence regions than are those based on student s t statistic an effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable the method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain algorithms exist for finding the unique global minimum at a superlinear rate of convergence a byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points the multivariate empirical likelihood regions are justified for functions of several means such as variances correlations and regression parameters and for statistics with linear estimating equations an algorithm is given for computing profile empirical likelihoods for these statistics

correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

computing eigenelements of boundary value problems with fractional derivatives variational iteration method has been successfully implemented to handle linear and nonlinear differential equations the main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper first a general framework of the variational iteration method is presented for analytic treatment of differential equations of fractional order where the fractional derivatives are described in caputo sense second the new framework is used to compute approximate eigenvalues and the corresponding eigenfunctions for boundary value problems with fractional derivatives numerical examples are tested to show the pertinent features of this method this approach provides a new way to investigate eigenvalue problems with fractional order derivatives

matrix methods for computing eigenvalues of sturm liouville problems of order four abstract this paper examines and develops matrix methods to approximate the eigenvalues of a fourth order sturm liouville problem subjected to a kind of fixed boundary conditions furthermore it extends the matrix methods for a kind of general boundary conditions the idea of the methods comes from finite difference and numerov s methods as well as boundary value methods for second order regular sturm liouville problems moreover the determination of the correction term formulas of the matrix methods is investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q 0 are known in this case finally some numerical examples are illustrated

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

fourth order boundary value problems with finite spectrum abstract for every positive integer m we construct a class of regular self adjoint and non self adjoint fourth order boundary value problems with at most 3 m 1 eigenvalues counting multiplicity

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

the adomian decomposition method for computing eigenelements of sturm liouville two point boundary value problems we will consider the adomian decomposition method for computing the eigenelements of sturm liouville two point boundary value problem the method proved to be very successful and powerful in computing such elements numerical examples showed the competitive nature of the method comparison with the results of others will also be presented

on the kernel of a monadic transformation abstract let j b0 b1 bh a be an ideal in a noetherian ring a such that b0 is nonnilpotent and let b a x1 xh then a complete description of the asymptotic essential and quintessential prime divisors of k b0x1 b1 b0xh bh b is given and a more detailed description of these prime divisors of k and of ker b a b 1 b 0 b h b 0 is given when b0 b1 bh are an asymptotic essential or a sequence also it is shown that there exists a unique essential prime divisor of k if and only if b0 b1 bh are an essential sequence and a has exactly one prime divisor of zero

a nonparametric fiducial interval for the youden index in multi state diagnostic settings the youden index is a commonly employed metric to characterize the performance of a diagnostic test at its optimal point for tests with three or more outcome classes the youden index has been extended however there are limited methods to compute a confidence interval ci about its value often outcome classes are assumed to be normally distributed which facilitates computational formulas for the ci bounds however many scenarios exist for which these assumptions cannot be made in addition many of these existing ci methods do not work well for small sample sizes we propose a method to compute a nonparametric interval about the youden index utilizing the fiducial argument this fiducial interval ensures that ci coverage is met regardless of sample size underlying distributional assumptions or use of a complex classifier for diagnosis two alternate fiducial intervals are also considered a simulation was conducted which demonstrates the coverage and interval length for the proposed methods comparisons were made using no distributional assumptions on the outcome classes and for when outcomes were assumed to be normally distributed in general coverage probability was consistently met and interval length was reasonable the proposed fiducial method was also demonstrated in data examining biomarkers in subjects to predict diagnostic stages ranging from normal kidney function to chronic allograph nephropathy published 2015 this article is a u s government work and is in the public domain in the usa

extending the scope of empirical likelihood this article extends the scope of empirical likelihood methodology ill three directions to allow for plug in estimates of nuisance parameters in estimating equations slower than root n rates of convergence and settings in which there are a relatively large number of estimating equations compared to the sample size calibrating empirical likelihood confidence regions with plug in is sometimes intractable due to the complexity of the asymptotics so we introduce a bootstrap approximation that call be used in such situations we provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results

eigenvalues of fourth order sturm liouville problems using fliess series we shall extend our previous results chanane 1998 on the computation of eigenvalues of second order sturm liouville problems to fourth order ones the approach is based on iterated integrals and fliess series

boundary value method for inverse sturm liouville problems in this paper we present a method to recover symmetric and non symmetric potential functions of inverse sturm liouville problems from the knowledge of eigenvalues the linear multistep method coupled with suitable boundary conditions known as boundary value method bvm is the main tool to approximate the eigenvalues in each iteration step of the used newton method the bvm was extended to work for neumann neumann boundary conditions moreover a suitable approximation for the asymptotic correction of the eigenvalues is given numerical results demonstrate that the method is able to give good results for both symmetric and non symmetric potentials

computation of the eigenvalues of sturm liouville problems with parameter dependent boundary conditions using the regularized sampling method the purpose in this paper is to compute the eigenvalues of sturm liouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the regularized sampling method an improvement on the method based on shannon sampling theory which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost a few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions

maximum likelihood ratio tests for comparing the discriminatory ability of biomarkers subject to limit of detection in this article we consider comparing the areas under correlated receiver operating characteristic roc curves of diagnostic biomarkers whose measurements are subject to a limit of detection lod a source of measurement error from instruments sensitivity in epidemiological studies we propose and examine the likelihood ratio tests with operating characteristics that are easily obtained by classical maximum likelihood methodology

nonparametric bayesian covariate adjusted estimation of the youden index a novel nonparametric regression model is developed for evaluating the covariate specific accuracy of a continuous biological marker accurately screening diseased from nondiseased individuals and correctly diagnosing disease stage are critically important to health care on several fronts including guiding recommendations about combinations of treatments and their intensities the accuracy of a continuous medical test or biomarker varies by the cutoff threshold c used to infer disease status accuracy can be measured by the probability of testing positive for diseased individuals the true positive probability or sensitivity se c of the test and the true negative probability specificity sp c of the test a commonly used summary measure of test accuracy is the youden index yi max se c sp c 1 c which is popular due in part to its ease of interpretation and relevance to population health research in addition clinical practitioners benefit from having an estimate of the optimal cutoff that maximizes sensitivity plus specificity available as a byproduct of estimating yi we develop a highly flexible nonparametric model to estimate yi and its associated optimal cutoff that can respond to unanticipated skewness multimodality and other complexities because data distributions are modeled using dependent dirichlet process mixtures important theoretical results on the support properties of the model are detailed inferences are available for the covariate specific youden index and its corresponding optimal cutoff threshold the value of our nonparametric regression model is illustrated using multiple simulation studies and data on the age specific accuracy of glucose as a biomarker of diabetes

screening as an application of decision theory we develop a decision theoretical approach to setting the threshold for a screening procedure that declares each examined subject as a positive or a negative it is fundamentally different from maximising the youden index the method incorporates the consequences of the two kinds of bad decisions false positives and false negatives by means of a set of plausible loss functions elicited from a subject matter expert or committee we present details for several classes of loss functions and within group distributions of the outcomes we outline extensions related to mixture distributions and compositions of loss functions we illustrate the method on simulated examples and apply it to real datasets

different applications for the differential transformation in the differential equations the differential transformation technique which is applied to solve eigenvalue problems and to solve partial differential equations p d e is proposed in this study first using the one dimensional differential transformation to construct the eigenvalues and the normalized eigenfunctions for the differential equation of the second and the fourth order second using the two dimensional differential transformation to solve p d e of the first and second order with constant coefficients in both cases a set of difference equations is derived and the calculated results are compared closely with the results obtained by other analytical methods

eigenvalues of sturm liouville problems with discontinuity conditions inside a finite interval abstract in this work we use the regularized sampling method to compute the eigenvalues of sturm liouville problems with discontinuity conditions inside a finite interval we work out an example by computing a few eigenvalues and their corresponding eigenfunctions

note on asymptotic prime divisors and symbolic powers of primary ideals at the close of a recent survey paper on asymptotic prime divisors 12 it was commented that there were no results in the paper concerning symbolic powers of pr imary ideals since the results and methods in these two obviously related areas do not seem to blend together the purpose of the present paper is to show that at least some results in these two areas do blend together this is done by proving four theorems on asymptotic prime divisors and five related theorems on symbolic powers of primary ideals the five results on symbolic powers are rather negative in tone since they show that if all large powers of a primary ideal a noetherian ring a are primary then a must be a rather nice ring to begin with the four theorems on asymptotic prime divisors mentioned in the preceding paragraph are proved in sect 2 and the theorems on symbolic powers are proved in sect 3 to efficiently state and prove these results a fair amount of notation is needed so this section will be closed by summarizing this notation and giving a brief remark concerning it

quintessential modulated ideals abstract let r denote a commutative noetherian ring and i an ideal of r the concept of quintessential sequences over zero ideal was introduced by mcadam and ratliff 1985 8 they showed that these sequences enjoy many of the basic properties of asymptotic sequences over zero ideal it was shown that quintessential sequences over ideals i 0 r are not a good analogue of asymptotic sequences over i 0 r by making use of the new concept of quintessential grade of an ideal over another ideal we show that there exists a class of ideals i for which quintessential sequences over i are an excellent analogue of asymptotic sequences over i also we give more results on quintessential sequences over an ideal and derive generalizations of some mcadam ratliff s results 1985 8 13

construction of confidence regions in the roc space after the estimation of the optimal youden index based cut off point after establishing the utility of a continuous diagnostic marker investigators will typically address the question of determining a cut off point which will be used for diagnostic purposes in clinical decision making the most commonly used optimality criterion for cut off point selection in the context of roc curve analysis is the maximum of the youden index the pair of sensitivity and specificity proportions that correspond to the youden index based cut off point characterize the performance of the diagnostic marker confidence intervals for sensitivity and specificity are routinely estimated based on the assumption that sensitivity and specificity are independent binomial proportions as they arise from the independent populations of diseased and healthy subjects respectively the youden index based cut off point is estimated from the data and as such the resulting sensitivity and specificity proportions are in fact correlated this correlation needs to be taken into account in order to calculate confidence intervals that result in the anticipated coverage in this article we study parametric and non parametric approaches for the construction of confidence intervals for the pair of sensitivity and specificity proportions that correspond to the youden index based optimal cut off point these approaches result in the anticipated coverage under different scenarios for the distributions of the healthy and diseased subjects we find that a parametric approach based on a box cox transformation to normality often works well for biomarkers following more complex distributions a non parametric procedure using logspline density estimation can be used

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

asymptotic sequences and rees rings abstract let b1 bs be an asymptotic sequence over an ideal i in a noetherian ring r and let bi b1 bi r then it is shown that certain sequences closely related to these elements are asymptotic sequences over r r bi over ti r r i and over u r r i where r r j is the rees ring of r with respect to the ideal j these results then imply that certain other sequences are asymptotic sequences over the corresponding ideals in the associated graded rings and in the monadic transformation rings as an application of the results it is shown that if r is local then each permutation of b1 bs is an asymptotic sequence over i and that b1 bs are an asymptotic sequence in r

asymptotic prime divisors and integral extension rings let r b be rings such that r is noetherian and b is an intermediate ring between a noetherian integral extension ring c of r and the integral closure of c then it is shown that if i is a finitely generated ideal in b then the sets ass b i k a are finite and stabilize for all large k where i k a is the integral closure in b of i k moreover if i jb for some ideal j in r and if p e ass r i k a then there exists p e ass b i k b a such that p r p and the converse holds if minimal prime ideals in c lie over minimal prime ideals in r and k is large from this it is shown that most of the theorems concerning asymptotic prime divisors asymptotic sequences and asymptotic grade that are known to hold for ideals in r continue to hold for finitely generated ideals in b

computing the eigenvalues of singular sturm liouville problems using the regularized sampling method abstract this paper deals with singular sturm liouville problems we shall extend the domain of application of the regularized sampling method a method to compute the eigenvalues a few numerical examples will be presented to illustrate the merit of the method

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

essential sequences and rees rings abstract it is shown that if b 1 b s is a u essential sequence over an ideal i in at noetherian ring r then certain permutations of u tb 1 tb i b i 1 b s are a u esential sequence of ir r b i and over tir r i b i where b i b 1 h b i r 0 i s andr r j is the rees ring of r with respect to its ideal j a number of related results are also given concerning u essential sequences over i and over id or i b k d where d is the monadic transformation ring r b i b k or r i b i b k when r is local the u essential cograde of i a containment relation between the essential prime divisors of b i and the u essential prime divisors of i b i and the fact that every permutation of b 1 b s is a u essential sequence over i and is an essential sequence in r

on the asymptotic sequences over an ideal abstract let i be an ideal of a noetherian ring r ratliff has shown that very many known results concerning r sequences and classical grade have a valid analogue for asymptotic sequences and asymptotic grade as a generalization of the concepts of asymptotic sequences and asymptotic grade we introduce the concepts of asymptotic i sequences and asymptotic i grade it is shown that they play a role analogous to asymptotic sequences and asymptotic grade

spectral parameter power series for fourth order sturm liouville problems a general solution of the fourth order sturm liouville equation is presented in the form of a spectral parameter power series spps the uniform convergence of the series is proved and the coefficients of the series are calculated explicitly through a recursive intergration procedure based on the spps representation characteristic equations for spectral problems arising in mechanics and elasticity theory are obtained and it is shown that the spectral problems reduce to computation of zeros of corresponding analytic functions of the spectral parameter given by their taylor series expansions this leads to a simple and efficient numerical method for solving the spectral problems for fourth order sturm liouville equations several examples of application are discussed

essential sequences and projective equivalence it is well known that if r is a noetherian ring and b1 bh are an r sequence then there exist ideals 0 b0 b1 bh in r such that the following hold for i 1 h a bi b1 bi r b ass rbi ass r bi k k 1 c bi 1 k bi r bi 1 k for all k 1 and d if p ass rbi 1 and p bi r r then p is properly contained in some p ass rbi the main theorem in this paper shows that something quite similar to this holds for essential sequences namely if bi bh are an essential sequence in r then there exists ideals 0 j0 j1 jh in r such that the following hold for i 1 h a ji is projectively equivalent to bi bi r b ass rji ass r ji k k 1 c ji 1 k bir ji 1 k for all k 1 and d if p ass rji 1 and p bi r r then p is properly contained in some p ass rji

essential prime divisors and projectively equivalent ideals let z be an ideal in a noetherian ring r we concern ourselves with the essential prime divisors of z an interesting subset of ass r p for all large n we first take z br with b a regular element of r we show that there is a ring t with r c te r such that t is a finite r module and the essential primes of bt are exactly the prime divisors of bt we next consider an arbitrary ideal z and apply our principal arguments to the element u in the kees ring of z we thereby deduce that there is an idcal j projectively equivalent to z such that the set of essential primes of z equals the set u ass r j over ii 1 2 3 notation let z be an ideal in a noetherian ring r we will use w or

generalization of adomian polynomials to functions of several variables abstract the solution of nonlinear differential and partial differential equations by the decomposition method due to adomian 1 3 requires his an polynomials to represent nonlinearities a convenient algorithm 1 3 for generation of the an for an analytic function f u is generalized to the case of an analytic function of several variables f u v where f is not factorable into f1 u f2 v

sturm liouville problems with parameter dependent potential and boundary conditions this paper deals with the computation of the eigenvalues of sturm liouville problems with parameter dependent potential and boundary conditions we shall extend the domain of application of the method based on sampling theory to the case where the classical whittaker shannon kotel nikov theorem is not applicable a few numerical examples will be presented

sturm liouville problems with impulse effects abstract this paper deals with the computation of the eigenvalues of sturm liouville problems with several discontinuity conditions inside a finite interval and parameter dependent boundary conditions using the regularized sampling method a few examples are worked out by computing several eigenvalues and their corresponding eigenfunctions

general sampling theorem using contour integral abstract we present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions 1 y f z a 1 z n 1 exp x y 1 x f z a 1 z n 1 exp x y for some a 0 0 n 1 n 0 and any z x iy c it will be shown that many well known sampling theorems included in siam j math anal 19 1988 1198 1203 and inform control 8 1965 143 158 can be interpreted as special cases of this sampling theorem as examples we provide sampling representations for entire functions which are bounded of polynomial growth or of exponential growth on r we also provide sampling representations involving derivatives of entire functions and nonuniform sampling representations taking the set of sampling points in which a finite number of points are arbitrarily distributed we obtain a sampling representation

on the asymptotic cograde of an ideal abstract let i be an ideal in a local ring r m and let s i be the asymptotic cograde of i the maximum length of an asymptotic sequence over i rees has shown that s i altitude r l i where l i is the analytic spread of i we prove several more bounds on s i i s i min little depth p p is an asymptotic prime divisor of i ii s i grade m i the maximum length of an asymptotic sequence contained in m i iii s i grade m grade i iv grade m l i s i and v if r is quasi unmixed then grade r i n s i for all large n also it is shown that the asymptotic prime divisors of i b 1 b s r contain those of i when b 1 b s are an asymptotic sequence over i and that the residue classes modulo i of such a sequence are an asymptotic sequence in r i finally several sufficient conditions for l i to be equal to l i p for some asymptotic prime divisor p of i are given

the analysis of placement values for evaluating discriminatory measures summary the idea of using measurements such as biomarkers clinical data or molecular biology assays for classification and prediction is popular in modern medicine the scientific evaluation of such measures includes assessing the accuracy with which they predict the outcome of interest receiver operating characteristic curves are commonly used for evaluating the accuracy of diagnostic tests they can be applied more broadly indeed to any problem involving classification to two states or populations d 0 or 1 we show that the roc curve can be interpreted as a cumulative distribution function for the discriminatory measure y in the affected population d 1 after y has been standardized to the distribution in the reference population d 0 the standardized values are called placement values if the placement values have a uniform 0 1 distribution then y is not discriminatory because its distribution in the affected population is the same as that in the reference population the degree to which the distribution of the standardized measure differs from uniform 0 1 is a natural way to characterize the discriminatory capacity of y and provides a nontraditional interpretation for the roc curve statistical methods for making inference about distribution functions therefore motivate new approaches to making inference about roc curves we demonstrate this by considering the roc glm regression model and observing that it is equivalent to a regression model for the distribution of placement values the likelihood of the placement values provides a new approach to roc parameter estimation that appears to be more efficient than previously proposed methods the method is applied to evaluate a pulmonary function measure in cystic fibrosis patients as a predictor of future occurrence of severe acute pulmonary infection requiring hospitalization finally we note the relationship between regression models for the mean placement value and recently proposed models for the area under the roc curve which is the classic summary index of discrimination

inference of the youden index and associated threshold using empirical likelihood for quantiles the youden index is a widely used measure in the framework of medical diagnostics where the effectiveness of a biomarker screening marker or predictor for classifying a disease status is studied when the biomarker is continuous it is important to determine the threshold or cut off point to be used in practice for the discrimination between diseased and healthy populations we introduce two methods aimed at estimating the youden index and its associated threshold the first one is a modified version of a recent approach based on the delta method and the second one is based on the adjusted empirical likelihood for quantiles in the setting of a two sample problem we also include cis for both of them in the simulation study we compare both methods under different scenarios finally a real example of prostatic cancer well known in the literature is analysed to provide the reader with a better understanding of the new methodology copyright 2011 john wiley sons ltd

asymptotic equivalence between the cox estimator and the general ml estimators of regression and survival parameters in the cox model the usual approach to estimating regression parameters in the cox regression model uses the partial likelihood if the covariates are not time dependent the model can be stated in terms of the survival function which allows one to derive a generalized likelihood containing both regression and survival curve parameters it is shown that in the absence of ties an estimator results which is asymptotically equivalent to the partial likelihood estimator a joint information matrix leads simply to standard errors for both regression and survival curve parameters which are asymptotically correct

accurate solutions of fourth order sturm liouville problems recently we introduced a new method which we call the extended sampling method to compute the eigenvalues of second order sturm liouville problems with eigenvalue dependent potential we shall see in this paper how we use this method to compute the eigenvalues of fourth order sturm liouville problems and present its practical use on a few examples

sampling for the fourth order sturm liouville differential operator we extend the sampling method to compute the eigenvalues of a fourth order differential operator we show that the shannon sampling theorem is not applicable due to the growth of the solutions on the real line the inverse spectral theory as developed by mclaughlin and kramer s theorem allow us to express the characteristic function explicitly by a new sampling formula

joint confidence region estimation for area under roc curve and youden index in the field of diagnostic studies the area under the roc curve auc serves as an overall measure of a biomarker diagnostic test s accuracy youden index defined as the overall correct classification rate minus one at the optimal cut off point is another popular index for continuous biomarkers of binary disease status although researchers mainly evaluate the diagnostic accuracy using auc for the purpose of making diagnosis youden index provides an important and direct measure of the diagnostic accuracy at the optimal threshold and hence should be taken into consideration in addition to auc furthermore auc and youden index are generally correlated in this paper we initiate the idea of evaluating diagnostic accuracy based on auc and youden index simultaneously as the first step toward this direction this paper only focuses on the confidence region estimation of auc and youden index for a single marker we present both parametric and non parametric approaches for estimating joint confidence region of auc and youden index we carry out extensive simulation study to evaluate the performance of the proposed methods in the end we apply the proposed methods to a real data set copyright 2013 john wiley sons ltd

notes on projectively related ideals and residual division abstract let i i i i g and j j i j f be finite collections of ideals of a noetherian ring r then i and j are projectively related in case the integral closures of the ideals i i i i i i g g and j j i i j j f f are equal for some positive integers i 1 i g and j 1 j f some basic properties of this relation are proved and then these are applied when either g 1 or each i i is regular to give several necessary and sufficient conditions for the existence of an ideal k that is projectively related to i such that k n g k n for all n 1 and for all g in a given multiplicatively closed set of nonzero ideals of r it is then shown that if there exists such an ideal k in r then for certain rings a related to r there exist ideals l projectively related to i a such that l n ga l n for all n 1 and for all g

positive solutions of fourth order nonlinear singular sturm liouville eigenvalue problems abstract we consider the existence of positive solutions for the following fourth order singular sturm liouville eigenvalue problems 1 p t p t u t g t f t u u 0 0 t 1 1 u 0 1 u 0 0 1 u 1 1 u 1 0 2 u 0 2 lim t 0 p t u t 0 2 u 1 2 lim t 1 p t u t 0 where 0 g p may be singular at t 0 and or 1 moreover f t x y may also have singularity at x 0 and or y 0 by using fixed point theory in cones an explicit interval for is derived such that for any in this interval the existence of at least one positive solution to the boundary value problem is guaranteed our results extend and improve many known results including singular and nonsingular cases

spline functions the biharmonic operator and approximate eigenvalues the biharmonic operator plays a central role in a wide array of physical models such as elasticity theory and the streamfunction formulation of the navier stokes equations its spectral theory has been extensively studied in particular the one dimensional case over an interval constitutes the basic model of a high order sturm liouville problem the need for corresponding numerical simulations has led to numerous works the present paper relies on a discrete biharmonic calculus the primary object of this calculus is a high order compact discrete biharmonic operator dbo the dbo is constructed in terms of the discrete hermitian derivative however the underlying reason for its accuracy remained unclear this paper is a contribution in this direction expounding the strong connection between cubic spline functions on an interval and the dbo the first observation is that the scaled fourth order distributional derivative of the cubic spline is identical to the action of the dbo on grid functions it is shown that the kernel of the inverse of the discrete operator is up to scaling equal to the grid evaluation of the kernel of left left frac d dx right 4 right 1 and explicit expressions are presented for both kernels as an important application the relation between the infinite set of eigenvalues of the fourth order sturm liouville problem and the finite set of eigenvalues of the discrete biharmonic operator is studied the discrete eigenvalues are proved to converge at an optimal o h 4 rate to the continuous ones another consequence is the validity of a comparison principle it is well known that there is no maximum principle for the fourth order equation however a positivity result is derived both for the continuous and the discrete biharmonic equation showing that in both cases the kernels are order preserving

computing the spectrum of non self adjoint sturm liouville problems with parameter dependent boundary conditions this paper deals with the computation of the eigenvalues of non self adjoint sturm liouville problems with parameter dependent boundary conditions using the regularized sampling method a few numerical examples among which singular ones will be presented to illustrate the merit of the method and comparison made with the exact eigenvalues when they are available

exponentially convergent symbolic algorithm of the functional discrete method for the fourth order sturm liouville problems with polynomial coefficients abstract a new symbolic algorithmic implementation of the functional discrete fd method is developed and justified for the solution of fourth order sturm liouville problem on a finite interval in the hilbert space the eigenvalue problem for the fourth order ordinary differential equation with polynomial coefficients is investigated the sufficient conditions of an exponential convergence rate of the proposed approach are received the obtained estimates of the absolute errors of fd method significantly improve the accuracy of the estimates obtained earlier by i p gavrilyuk v l makarov and a m popov in 2010 our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis the number of summands in these decompositions depends on the degree of the potential coefficients and the correction number our method uses only the algebraic operations and basic operations on 2 1 column vectors and 2 2 matrices the proposed approach does not require solving any boundary value problems and computations of any integrals unlike the previous variants of fd method by i p gavrilyuk v l makarov a m popov and n m romaniuk in 2010 and 2017 the corrections to eigenpairs are computed exactly as analytical expressions the numerical examples illustrate the theoretical results the numerical results obtained with the fd method are compared with the numerical test results obtained with other existing numerical techniques

correction of finite element estimates for stum liouville eigenvalues it is shown that a simple asymptotic correction technique of paine de hoog and anderssen reduces the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by the finite element method with linear hat functions and mesh lengthh fromo k 4 h 2 too k h 2 the result still holds when the matrix elements are evaluated by simpson s rule but if the trapezoidal rule is used the error iso k 2 h 2 numerical results demonstrate the usefulness of the correction even for low values ofk

an efficient method for computing eigenelements of sturm liouville fourth order boundary value problems in this paper an efficient method based on the adomian decomposition for computing the eigenelements of fourth order sturm liouville boundary value problems is developed numerical examples show that the method proposed is easy to implement and produces accurate results

numerical methods for higher order sturm liouville problems we review some numerical methods for self adjoint and non self adjoint boundary eigenvalue problems

essential sequences over an ideal and essential cograde in 6 the concepts of essential prime divisors essential sequences and the essential grade of an ideal i in a noetherian ring were introduced and therein it was shown that these concepts are an excellent analogue of respectively associated primes r sequences and the standard grade of i in the classical theory and also of respectively asymptotic prime divisors asymptotic sequences and the asymptotic grade of i in the asymptotic theory it turns out that essential sequences and essential grade yield useful information concerning all the prime divisors of zero in the completion of a local ring thus these concepts should prove useful in many future research papers asymptotic prime divisors asymptotic sequences and asymptotic grade have recently been useful in several research papers and in 5 it was shown that asymptotic sequences over an ideal i in a noetherian ring r and the asymptotic cograde of i when r is local have some useful properties and several bounds on this cograde were established in 5 the main purpose of this paper is to show that similarly essential sequences over i behave nicely when passing to certain rings related to r and that the essential cograde of i is well defined when r is local and satisfies certain rather natural inequalities section 2 contains the definitions and a list of the basic facts concerning essential prime divisors essential sequences and essential grade that are needed in the remainder of the paper it also contains several results showing that essential sequences over an ideal i in a noetherian ring r behave nicely with respect to passing to certain rings related to r in section 3 it is shown that the essential cograde of i is well defined when r is local and likewise behaves nicely when passing to the same type of related rings in section 4 we briefly