cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

new estimates for solutions of lyapunov equations results for estimating the solution of differential and algebraic lyapunov matrix equations are obtained and some of the well known results are generalized

new upper solution bounds of the continuous algebraic riccati matrix equation by utilizing some linear algebraic techniques new upper bounds of the solution of the continuous algebraic riccati equation care are derived according to the present bounds iterative procedures are also developed for obtaining more precise estimations comparing with existing results the obtained bounds are less restrictive

solution bounds of the continuous riccati matrix equation a new approach is proposed for estimating the solution of the continuous algebraic riccati equation care upper and lower solution bounds of the care are presented comparisons show that the present bounds are more general and or tighter than existing results

gradient based iterative algorithm for solving coupled matrix equations abstract this paper is concerned with iterative methods for solving a class of coupled matrix equations including the well known coupled markovian jump lyapunov matrix equations as special cases the proposed method is developed from an optimization point of view and contains the well known jacobi iteration gauss seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle as special cases we have provided analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm simultaneously the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form the proposed approach requires less computation and is numerically reliable as only matrix manipulation is required some other existing results require either matrix inversion or special matrix products numerical examples show the effectiveness of the proposed algorithm

simultaneous eigenvalue lower bounds for the lyapunov matrix equation simultaneous eigenvalue lower bounds for the solution of the lyapunov equation are presented the n summation and the n product bounds generalize existing bounds such as those for the trace and the determinant these bounds are stronger than the majority of the relevant bounds shown in the literature

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

lower summation bounds for the discrete riccati and lyapunov equations lower eigenvalue summation including trace bounds for the solution of the discrete algebraic riccati and lyapunov matrix equations are presented these are tighter than or supplement existing results

a matrix inequality abstract a new matrix inequality associated with bounds on solutions of algebraic riccati and lyapunov equations is presented the improvement of the obtained result over the previously reported results is shown

some bounds of the solution of the algebraic riccati equation bounds on extremal eigenvalues and lower and upper bounds of the trace for the solution of the algebraic riccati equation are presented through two examples it is shown that in some special cases the presented bounds can be better than the results recently published

upper matrix bound of the solution for the discrete riccati equation a new upper matrix bound of the solution for the discrete algebraic matrix riccati equation is developed this matrix bound is then used to derive bounds on the eigenvalues trace and determinant of the same solution it is shown that these eigenvalue bounds are less restrictive than previous results

upper summation and product bounds for solution eigenvalues of the lyapunov matrix equation upper bounds for summations including the trace and for products including the determinant of the eigenvalues of the solution of the continuous algebraic lyapunov matrix equation are presented the majority of the bounds are tighter than those in the literature and some are new

gradient based iterative algorithms for generalized coupled sylvester conjugate matrix equations abstract by applying the hierarchical identification principle the gradient based iterative algorithm is suggested to solve a class of complex matrix equations with the real representation of a complex matrix as a tool the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices also we solve the problem which is proposed by wu et al 2010 finally some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper

upper bounds for the eigenvalues of the solution of the discrete algebraic lyapunov equation upper bounds for summations including the trace and for products including the determinant of the eigenvalues of the solution of the discrete algebraic lyapunov equation dale are presented all bounds are derived from the matrix series solution of the dale the majority of the bounds are tighter than those in the literature

on the lyapunov matrix differential equation a lower bound for the determinant of the solution to the lyapunov matrix differential equation is derived it is shown that this bound is obtained as a solution to a simple scalar differential equation in the limiting case where the solution to the lyapunov differential equation becomes stationary the result reduces to one of the existing bounds for the algebraic equation

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

matrix bounds of the discrete are solution this paper provides new lower and upper matrix bounds of the solution to the discrete algebraic riccati equation the lower bound always works if the solution exists the upper bounds are presented in terms of the solution of the discrete lyapunov equation and its upper matrix bound the upper bounds are always calculated if the solution of the lyapunov equation exists a numerical example shows that the new bounds are tighter than previous results in many cases

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

eigenvalue bounds for algebraic riccati and lyapunov equations the majorization result of wimmer 2 relating the eigenvalues of the matrices involved in the lyapunov equation is extended to the algebraic riccati equation this result coupled with certain results on the eigenvalue bounds for sum and product of matrices yields several lower and upper bounds for the eigenvalues of the solution matrix of the algebraic riccati and lyapunov equations

approximation for bessel functions and their application in the computation of hankel transforms we present rational approximations of the bessel functions jv x v 0 1 10 which can be used to simplify the computation of the hankel transform to the computation of two fourier transforms

a generalization of the hermitian and skew hermitian splitting iteration method for solving sylvester equations this paper is concerned with a generalization of the hermitian and skew hermitian splitting hss iteration method for solving large sparse continuous sylvester equations the analysis shows that the ghss iteration method will converge under certain assumptions numerical results show that this new method is more efficient and robust than the existing ones

new results for the bounds of the solution for the continuous riccati and lyapunov equations this paper presents upper and lower matrix bounds for the solution of the continuous algebraic matrix riccati equation furthermore a new lower matrix bound for the solution of the continuous algebraic lyapunov equation is also developed these are new results

bounds for the solution of the lyapunov matrix equation a unified approach abstract in recent years several bounds have been reported for the solution of the continuous and the discrete lyapunov equations using the unified lyapunov equation we give in this paper bounds for the solution of this equation in the limiting cases the bounds reduce to existing bounds for both the continuous and discrete lyapunov equations

diverse bounds for the eigenvalues of the continuous algebraic riccati equation summation bounds for the eigenvalues of a p pa are presented here matrices a p spl isin r sup n spl times n p is positive definite and a transpose of a these bounds are employed to derive new upper bounds for summations of eigenvalues of the solution of the continuous algebraic riccati equation

on the trace bound of a matrix product this paper generalizes the main result given by fang et al ibid vol 39 p 2489 90 1994 to provide tighter bounds on the trace of the product of two real matrices

matrix bounds of the solutions of the continuous and discrete riccati equations a unified approach in this paper a new scheme is introduced to measure the matrix bounds of the continuous and discrete riccati equations by estimating upper and lower matrix bounds of the solution of the unified algebraic riccati equation uare the same measurements for the solutions of the continuous and discrete riccati equations respectively can be obtained in limiting cases according to these obtained matrix bounds several eigenvalue bounds are also defined all the proposed results for the uare are new and more general than previous work some obtained results are compared with those of the literature via numerical examples it is shown that in some cases the presented results are tighter than the existing ones

a note on the discrete lyapunov and riccati matrix equations abstract inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic riccati and lyapunov matrix equations are presented the results give lower bounds for the product of the eigenvalues of the matrix solutions also for a discrete lyapunov equation an algorithm is presented to determine under what conditions a positive diagonal solution will exist if all the conditions are satisfied the algorithm also provides such a diagonal solution

lower bounds for the solution of the discrete algebraic lyapunov equation lower bounds for the determinant and for the trace of the matrix solution of the discrete algebraic lyapunov equation are presented the determinant is tighter than bounds in the literature and the trace supplements them

non inferiority trial design for recurrent events we describe methods for designing non inferiority trials with recurrent event responses arising from mixed poisson models sample size formulae are derived for trials in which treatment effects are expressed as relative rates and as absolute differences in cumulative mean functions at a particular time simulation studies are conducted to provide empirical validation of the frequency properties of the design and testing procedures under the null and alternative hypotheses using both mixed poisson models and robust marginal methods the robustness of the design to misspecification of the random effect distribution is also studied empirically sample size requirements based on the proposed method are contrasted with those from a design based on the time to the first event for a motivating study of patients with bone metastases at risk of skeletal complications when the between patient heterogeneity in the event rate is small there may be a considerable reduction in sample size with recurrent event outcomes copyright 2007 john wiley sons ltd

a note on eigenvalue bounds in algebraic riccati equation in this note an upper bound on the maximum eigenvalue of the solution matrix k of the algebraic riccati equation is established the approach outlined also results in several lower bounds which are more general than those derived in 1 for some of the largest eigenvalues of k

brief lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations in this paper we propose lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations we give comparisons between the parallel estimates finally we give examples showing that our bounds can be better than the previous results for some cases

iterative matrix bounds and computational solutions to the discrete algebraic riccati equation bilateral matrix bounds for the solution of the discrete algebraic riccati equation dare are presented they are new or tighter than the existing bound computational algorithms to solve the dare follow

an algorithm for computing general integer order hankel transforms a tomographic image reconstruction algorithm that we have been studying requires the nth order hankel transform for the nth function in a series where n varies over a set of integers unfortunately most previously proposed algorithms have either dealt with only zeroth order transforms or cannot be applied conveniently to our problem we propose an algorithm for computing general integer order hankel transforms the algorithm takes samples of equally spaced input functions and gives equally spaced samples of the transforms

a new approach for upper bound estimations of the solution of the continuous riccati equation by utilizing some linear algebraic techniques a new approach for measuring upper bounds of the solution of the continuous algebraic riccati equation care is developed two upper matrix bounds are developed comparing with those appeared in the literature the obtained bounds are less restrictive or more concise therefore they can be applied easily to deal with control problems

bounds in the lyapunov matrix differential equation upper and lower bounds for the trace of the solution of the lyapunov matrix differential equation are derived it is shown that they are obtained as solutions to simple scalar differential equations as a special case the bounds for the stationary solution give ones for the solution to the lyapunov algebraic equation

lower bounds for the trace of the solution of the discrete algebraic riccati equation two lower bounds for the trace of the solution of the discrete algebraic riccati equation dare are presented it is shown that in many cases these trace bounds are tighter than those in the literature and greater than the trace of the state weighting matrix even when the system matrix is singular the results are illustrated by an example

new upper solution bounds of the discrete algebraic riccati matrix equation in this note we present upper matrix bounds for the solution of the discrete algebraic riccati equation dare using the matrix bound of theorem 2 2 we then give several eigenvalue upper bounds for the solution of the dare and make comparisons with existing results the advantage of our results over existing upper bounds is that the new upper bounds of theorem 2 2 and corollary 2 1 are always calculated if the stabilizing solution of the dare exists whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions finally we give numerical examples to demonstrate the effectiveness of the derived results

unified type non stationary lyapunov matrix equation simultaneous eigenvalue bounds abstract simultaneous eigenvalue bounds for the solution of the unified non stationary lyapunov matrix equation are presented when the solution becomes stationary the results reduce to bounds of the unified type algebraic lyapunov equation in the limiting cases the results reduce to bounds for the solution of the differential and difference lyapunov equations the bounds given in this paper are a generalization of some existing bounds obtained separately for the continuous and discrete type stationary and non stationary lyapunov equations

upper and lower matrix bounds of the solution for the discrete lyapunov equation new matrix bounds of the solution for the discrete algebraic matrix lyapunov equation are established the upper and lower eigenvalue bounds such as each eigenvalue including the extreme ones the trace and the determinant are also determined by these matrix bounds for the same solution the present schemes are tighter as compared with the majority of existing results

algorithms to numerically evaluate the hankel transform abstract the efficient and accurate evaluation of the hankel transform is required in a number of applications this paper reviews a number of algorithms that have only recently been exposed in the literature results of a comparative study of some useful algorithms are presented it is found that the performance of all algorithms depends on the type of function to be transformed the projection based methods provide acceptable accuracy with better efficiency than numerical quadrature

on bounds of the riccati and lyapunov matrix equations various bounds for the traces of the solutions of the algebraic riccati and lyapunov matrix equations are established in the continuous and discrete domain respectively the presented results give lower bounds for the sum of the eigenvalue of the matrix solutions

new spectral bounds on h2 norm of linear dynamical networks abstract in this paper we obtain new lower and upper bounds for the h 2 norm of a class of linear time invariant systems subject to exogenous noise inputs we show that the h 2 norm as a performance measure can be tightly bounded from below and above by some spectral functions of state and output matrices of the system in order to show the usefulness of our results we calculate bounds for the h 2 norm of some network models with specific coupling or graph structures e g systems with normal state matrices linear consensus networks with directed graphs and cyclic linear networks as a specific example the h 2 norm of a linear consensus network over a directed cycle graph is computed and shown how its performance scales with the network size our proposed spectral bounds reveal the important role and contribution of fast and slow dynamic modes of a system in the best and worst achievable performance bounds under white noise excitation finally we use several numerical simulations to show the superiority of our bounds over the existing bounds in the literature

improved bounds for the eigenvalues of solutions of lyapunov equations bounds on the solution and on the eigenvalues of the solution of the discrete and of the continuous matrix lyapunov equations are presented these bounds are nontrivial in the case of system matrices having multiple eigenvalues especially in the latter case these bounds are better than bounds found in the literature further these bounds are sharp i e there exist systems such that the lower and upper bounds coincide yielding the solution or its eigenvalues respectively

blinded sample size reassessment in non inferiority and equivalence trials even in situations where the design and conduct of clinical trials is highly standardized there may be a considerable between study variation in the observed variability of the primary outcome variable as a consequence performing a study in a fixed sample size design implies a considerable risk of resulting in a too high or too low sample size this difficulty can be alleviated by applying a design with internal pilot study after a provisional sample size calculation in the planning stage a portion of the planned sample is recruited and the sample size is recalculated on the basis of the observed variability to comply with the requirement of some regulatory guidelines only blinded data should be used for the reassessment procedure furthermore the effect on the type i error rate should be quantified the current literature presents analytical results on the actual level in the t test situation only for superiority trials in these situations blinded sample size recalculation does not lead to an inflation of the type i error rate we extended the methodology to non inferiority and equivalence trials with normally distributed outcome variable and hypotheses formulated in terms of the ratio and difference of means surprisingly in contrast to the case of testing superiority we observed actual type i error rates above the nominal level the extent of inflation depends on the required sample size the sample size of the internal pilot study and the standardized equivalence or non inferiority margin it turned out that the elevation of the significance level is negligible for most practical situations nevertheless the consequences of sample size reassessment have to be discussed case by case and regulatory concerns with respect to the actual size of the procedure cannot generally be refuted by referring to the fact that only blinded data were used

upper bounds for the solution of the discrete riccati equation upper bounds for individual eigenvalues and for summations of eigenvalues including the trace of the solution of the discrete algebraic riccati equation are presented some are new and some supplement bounds in the literature

trace bounds on the solution of the algebraic matrix riccati and lyapunov equation lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix riccati and lyapunov equation are derived the upper trace bound obtained in this note in many cases results in a tighter bound as compared to the upper bound for the maximal eigenvalue proposed in 1 and 2

eigenvalue bounds on the solutions of coupled lyapunov and riccati equations abstract coupled lyapunov and riccati equations of the form p j 0 q a j pa j t gg t q a 0 kc 0 q a 0 kc 0 t j 1 q a j pa j t gg t k m 1 r c m pc m t dd t k t arise in the estimation problem of stochastic systems corrupted by additive and multiplicative noise the matrices p and q are the steady state and estimation error covariances in this paper trace and eigenvalue bounds for p and q are established

simultaneous eigenvalue lower bounds for the riccati matrix equation the author derives n simultaneous lower bounds on the sums of the eigenvalues of the solution of the algebraic riccati equation and of the lyapunov matrix equation these bounds extend the single inequalities shown in the literature for the trace and for the maximum eigenvalue two inequality versions are derived and evaluated for application purposes and a comparison to relevant bounds in the literature is given

eigenvalue bounds for the discrete lyapunov matrix equation upper and lower bounds for every eigenvalue of the positive definite solution to the discrete lyapunov matrix equation are provided the obtained bounds have an advantage that they can give nontrivial lower bounds even when some existing results yield trivial ones

technical communique lower bounds on the solution of coupled algebraic riccati equation lower bounds for eigenvalues and matrix lower bound of a solution to continuous time coupled algebraic riccati equation and discrete time coupled algebraic riccati equation are developed as special cases of bounds for the unified coupled algebraic riccati equation they include bounds for the minimal eigenvalues the sums of the eigenvalues the trace and the determinant

on the discrete lyapunov matrix equation some bounds for the arithmetic and the geometric means of the characteristic roots of the positive semidefinite solution to the discrete lyapunov matrix equation are derived

upper bounds for the eigenvalues of the solution of the lyapunov matrix equation eigenvalue upper bounds for the solution of the lyapunov matrix equation are presented for single eigenvalues summations of eigenvalues including the trace and products of eigenvalues including the determinant these bounds are compared to those for the trace and the extreme eigenvalues

upper matrix bounds for the discrete algebraic riccati matrix equation we propose upper matrix bounds for the discrete algebraic riccati matrix equation we show that our results are less restrictive than the previous bounds finally we give numerical examples in order to verify the effectiveness of our results

some fundamental issues with non inferiority testing in active controlled trials in an active controlled non inferiority trial without a placebo arm it is often not entirely clear what the primary objective is in many cases the considered goal is to demonstrate that the experimental treatment preserves at least some fraction of the effect of the active control the active control effect is a parameter the value of which is unknown to test the hypothesis of effect preservation the classical confidence interval approach requires specification of a non inferiority margin which is a function of the unknown active control effect when the margin is estimated it is also not clear what is the relevant type i error of making a false assertion about preservation of the active control effect the statistical uncertainty of the estimated margin arguably needs to be incorporated in evaluation of the type i error in this paper we discuss these fundamental issues we show that the classical confidence interval approach cannot attain the target type i error exactly since this error varies as the sample size or as the values of the nuisance parameters in the active controlled trial change in contrast the preservation tests as proposed in literature can attain the target type i error rate exactly regardless of the sample size and the values of the nuisance parameters but can do so only at the price of several strong assumptions holding that may not be directly verifiable one assumption is the constancy condition holding whereby the effect of the active control in the historical trial populations is assumed to carry to the population of the active control trial when this condition is violated both the confidence interval approach and the preservation test method may be problematic

a brief summary of the bounds on the solution of the algebraic matrix equations in control theory abstract in recent years several bounds have been reported for different measures of the extent or size of the solution of the algebraic matrix equations arising in control theory such as the riccati equation and the lyapunov equation this paper collects the bounds that have been presented up to now and summarizes them in an unified form this will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds

an algorithm for the computation of zeros of a special class of entire functions abstract a two stage algorithm is presented for computing the zeros of a special class of entire functions the first stage which has linear convergence produces two sided bounds for the zeros these are then used as data for the second stage which employs a cubically convergent iterative process with monotonic convergence to the zeros from below and above appropriate modifications to the algorithm in order that it be applicable to certain real polynomials are also included

some simple robust methods for the analysis of recurrent events nelson discussed a method of estimating the cumulative mean function for identically distributed processes of recurrent events we show that a similar approach can be used with more general models including regression the key idea is to use point estimates based on poisson models and to develop robust variance estimates that are valid more generally the methods are illustrated on reliability and warranty data

on solution bounds for three types of lyapunov matrix equations continuous discrete and unified equations an approach is proposed to obtain a series of solution bounds for the unified algebraic lyapunov matrix equation by making use of its feature i e a mediator between the continuous time and discrete time lyapunov equations due to the approach every existing solution bound for both the continuous and discrete equations becomes intact in the sense that it yields a corresponding bound for the unified equation the approach is endorsed by this simple fact the three types of lyapunov equations share the solution provided certain relationships are maintained among corresponding coefficient matrices of the equations some examples are presented to show that the proposed method actually produces new bounds and builds bridges over existing bounds

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

on some bounds in the algebraic riccati and lyanupov equations some bounds for the trace and determinant of the solution to the algebraic riccati and lyapunov matrix equations are presented these bounds give information on mean size of the solution to the equations in a simple way the obtained determinant bound for the lyapunov equation shows the improvement over the previously reported one

bounds on solutions to h algebraic riccati equations and h 2 properties of h central solutions abstract in this paper we give upper bounds on the stabilizing solutions of the algebraic riccati equations arising in standard h estimation and control problems for continuous time finite dimensional linear time invariant systems using these bounds we show that in h estimation or h feedback control via state feedback the so called central solution satisfies an interesting h 2 norm bound this bound may be readily used to find a central solution i e estimator or feedback controller that guarantees an equal comprise between h 2 and h objectives

on upper bounds for the unified algebraic riccati equation in recent years several eigenvalues norms and determinants bounds have been investigated separately for the solutions of continuous and discrete riccati equations in this paper an upper bound for solution of the unified riccati equation is presented in the limiting cases the result reduces to a new upper bound for the solution of continuous and discrete riccati equation

on the discrete riccati equation several bounds have been reported recently for the trace of the solution to the discrete algebraic matrix riccati equation this note adds an alternative one to them

a hybrid fast hankel transform algorithm for electromagnetic modeling a hybrid fast hankel transform algorithm has been developed that uses several complementary features of two existing algorithms anderson s digital filtering or fast hankel transform fht algorithm and chave s quadrature and continued fraction algorithm a hybrid fht subprogram called hybfht written in standard fortran 77 provides a simple user interface to call either subalgorithm the hybrid approach is an attempt to combine the best features of the two subalgorithms in order to minimize the user s coding requirements and to provide fast execution and good accuracy for a large class of electromagnetic problems involving various related hankel transform sets with multiple arguments special cases of hankel transforms of double order and double argument are discussed where use of hybfht is shown to be advantageous for oscillatory kernel functions

tight bounds for the trace of a matrix product we propose a family of new upper and lower bounds for the trace of the matrix product ab when a or b is symmetric those bounds depend on a scalar parameter and both converge monotonically to tr ab when this parameter vanishes thus providing arbitrary close approximations even large values of the parameter yield very good bounds

a note on the lyapunov matrix equation in 5 bound for the determinant of the solution to the lyapunov matrix equation was reported this note gives an another bound for this value

spectral norm and trace bounds of algebraic matrix riccati equations upper bounds on spectral norm and trace for the solution of continuous algebraic matrix riccati equations are proposed the proposed bounds are based on a form of the general analytic solution it is shown that in many cases the spectral norm and the trace bounds are better than other results

bounds for the eigenvalues of the solution of the unified algebraic riccati matrix equation abstract in recent years several eigenvalues bounds have been investigated separately for the solutions of the continuous and the discrete riccati and lyapunov matrix equations in this paper lower bounds for the eigenvalues of the solution of the unified riccati equation relatively to continuous and discrete cases are presented in the limiting cases the results reduce to some new bounds for both the continuous and discrete riccati equation

eigenvalue upper bounds of the solution of the continuous riccati equation new upper bounds for the solution eigenvalues of the continuous algebraic matrix riccati equation are developed they include bounds of the extreme eigenvalues the summation and product of eigenvalues the trace and the determinant it is shown that the majority of the present eigenvalue bounds expressed in concise forms are less restrictive and sharper than existing results

test statistics and sample size formulae for comparative binomial trials with null hypothesis of non zero risk difference or non unity relative risk when it is required to establish a materially significant difference between two treatments or alternatively to show that two treatments are equivalent standard test statistics and sample size formulae based on a null hypothesis of no difference no longer apply this paper reviews some of the test statistics and sample size formulae proposed for comparative binomial trials when the null hypothesis is of a specified non zero difference or non unity relative risk methods based on restricted maximum likelihood estimation are recommended and applied to studies of pertussis vaccine

lower bounds on the solution of lyapunov matrix and algebraic riccati equations in this note various lower bounds for all the eigenvalues of the solution matrix k of the lyapunov matrix equation are established a special case of this result is a generalization of that presented in 1 3 where lower bounds for the maximum and minimum eigenvalues of k are given moreover the approach used here enables one to establish various lower bounds for some of the largest eigenvalues of the solution matrix of the algebraic riccati equation

a stability enhancing scaling procedure for schur riccati solvers abstract scaling riccati equations so that the solution has unit norm eliminates instabilities associated with the inversion step of invariant subspace based schur methods without changing the relative condition number of the underlying problem such optimal scaling makes it easy to estimate this condition number and to bound the relative error of the computed solution moreover the determination of whether the first step of newton s method will improve this solution is also simplified

bounds for the eigenvalues of the solution of the discrete riccati and lyapunov equations and the continuous lyapunov equation we present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete riccati and lyapunov matrix equations and the continuous lyapunov matrix equation nearly all of our bounds for the discrete riccati equation are new the bounds for the discrete and continuous lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation